Chain: A Dynamic Double Auction Framework for Matching Patient Agents
aa r X i v : . [ c s . G T ] O c t Journal of Artificial Intelligence Research 30 (2007) 133–179 Submitted 3/07; published 9/07
Chain: A Dynamic Double Auction Framework for MatchingPatient Agents
Jonathan Bredin [email protected]
Dept. of Mathematics and Computer Science, Colorado CollegeColorado Springs, CO 80903, USA
David C. Parkes [email protected]
Quang Duong [email protected]
School of Engineering and Applied Sciences, Harvard UniversityCambridge, MA 02138, USA
Abstract
In this paper we present and evaluate a general framework for the design of truthfulauctions for matching agents in a dynamic, two-sided market. A single commodity, suchas a resource or a task, is bought and sold by multiple buyers and sellers that arriveand depart over time. Our algorithm,
Chain , provides the first framework that allows atruthful dynamic double auction (DA) to be constructed from a truthful, single-period (i.e.static) double-auction rule. The pricing and matching method of the
Chain constructionis unique amongst dynamic-auction rules that adopt the same building block. We examineexperimentally the allocative efficiency of
Chain when instantiated on various single-periodrules, including the canonical McAfee double-auction rule. For a baseline we also considernon-truthful double auctions populated with “zero-intelligence plus”-style learning agents.
Chain -based auctions perform well in comparison with other schemes, especially as arrivalintensity falls and agent valuations become more volatile.
1. Introduction
Electronic markets are increasingly popular as a method to facilitate increased efficiencyin the supply chain, with firms using markets to procure goods and services. Two-sidedmarkets facilitate trade between many buyers and many sellers and find application totrading diverse resources, including bandwidth, securities and pollution rights. Recentyears have also brought increased attention to resource allocation in the context of on-demand computing and grid computing. Even within settings of cooperative coordination,such as those of multiple robots, researchers have turned to auctions as methods for taskallocation and joint exploration (Gerkey & Mataric, 2002; Lagoudakis et al., 2005; Lin &Zheng, 2005).In this paper we consider a dynamic two-sided market for a single commodity, for in-stance a unit of a resource (e.g. time on a computer, some quantity of memory chips) ora task to perform (e.g. a standard database query to execute, a location to visit). Eachagent, whether buyer or seller, arrives dynamically and needs to be matched within a timeinterval. Cast as a task-allocation problem, a seller can perform the task when allocatedwithin some time interval and incurs a cost when assigned. A buyer has positive value forthe task being assigned (to any seller) within some time interval. The arrival time, accept-able time interval, and value (negative for a seller) for a trade are all private information c (cid:13) redin, Parkes and Duong to an agent. Agents are self-interested and can choose to misrepresent all and any of thisinformation to the market in order to obtain a more desirable price.The matching problem combines elements of online algorithms and sequential decisionmaking with considerations from mechanism design. Unlike traditional sequential decisionmaking, a protocol for this problem must provide incentives for agents to report truthfulinformation to a match-maker. Unlike traditional mechanism design, this is a dynamicproblem with agents that arrive and leave over time. We model this problem as a dynamicdouble auction (DA) for identical items. The match-maker becomes the auctioneer. Eachseller brings a task to be performed during a time window and each buyer brings thecapability to perform a single task. The double-auction setting also is of interest in its ownright as a protocol for matching in a dynamic business-to-business exchange.Uncertainty about the future coupled with the two-sided nature of the market leads toan interesting mechanism design problem. For example, consider the scenario where theauctioneer must decide how (and whether) to match a seller with reported cost of $6 at theend of its time interval with a present and unmatched buyer, one of which has a reportedvalue of $8 and one a reported value of $9. Should the auctioneer pair the higher bidderwith the seller? What happens if a seller, willing to sell for $4, arrives after the auctioneeracts upon the matching decision? How should the matching algorithm be designed so thatno agent can benefit from misstating its earliest arrival, latest departure, or value for atrade? Chain provides a general framework that allows a truthful dynamic double auction tobe constructed from a truthful, single-period (i.e. static) double-auction rule. The auctionsconstructed by
Chain are truthful, in the sense that the dominant strategy for an agent,whatever the future auction dynamics and bids from other agents, is to report its truevalue for a trade (negative if selling) and true patience (maximal tolerance for trade delay)immediately upon arrival into the market. We also allow for randomized mechanisms and,in this case, require strong truthfulness: the DA should be truthful for all possible randomcoin flips of the mechanism. One of the DAs in the class of auctions implied by
Chain is a dynamic generalization of McAfee’s (1992) canonical truthful, no-deficit auction for asingle period. Thus, we provide the first examples of truthful, dynamic DAs that allow fordynamic price competition between buyers and sellers. The main technical challenge presented by dynamic DAs is to provide truthfulness with-out incurring a budget deficit, while handling uncertainty about future trade opportunities.Of particular concern is to ensure that an agent does not indirectly affect its price throughthe effect of its bid on the prices faced by other agents and thus other supply and demandin the market. We need to preclude this because the availability of trades depends on theprice faced by other agents. For example, a buyer that is required to pay $4 in the DA totrade might like to decrease the price that a potentially matching seller will receive from $6to $3 to allow for trade.
Chain is a modular approach to auction design, which takes as a building block a single-period matching rule and provides a method to invoke the rule in each of multiple periodswhile also providing for truthfulness. We characterize properties that a well-defined single-
1. The closest work in the literature is due to Blum et al. (2006), who present a truthful, dynamic DA forour model that matches bids and asks based on a price sampled from some bid-independent distribution.We compare the performance of our schemes with this scheme in Section 6. hain: An Online Double Auction period matching rule must satisfy in order for
Chain to be truthful. We further identify thetechnical property of strong no-trade , with which we can isolate agents that fail to trade inthe current period but can nevertheless survive and be eligible to trade in a future period.An auction designer defines the strong no-trade predicate, in addition to providing a well-defined single-period matching rule. Instances within this class include those constructedin terms of both “price-based” matching rules and “competition-based” matching rules.Both can depend on history and be adaptive, but only the competition-based rules use theactive bids and asks to determine the prices in the current period, facilitating a more directcompetitive processes.In proving that
Chain , when combined with a well-defined matching rule and a validstrong no-trade predicate, is truthful we leverage a recent price-based characterization fortruthful online mechanisms (Hajiaghayi et al., 2005). We also show that the pricing andmatching rules defined by
Chain are unique amongst the family of mechanisms that areconstructed with a single-period matching rule as a building block. Throughout our work weassume that a constant limits every buyer and seller’s patience. To motivate this assumptionwe provide a simple environment in which no truthful, no-deficit DA can implement someconstant fraction of the number of the efficient trades, for any constant.We adopt allocative efficiency as our design objective, which is to say auction protocolsthat maximize the expected total value from the sequence of trades. We also consider netefficiency , wherein any net outflow of payments to the marketmaker is also accounted for inconsidering the quality of a design. Experimental results explore the allocative efficiency of Chain when instantiated to various single-period matching rules and for a range of differentassumptions about market volatility and maximal patience. For a baseline we consider theefficiency of a standard (non-truthful) open outcry DA populated with simple adaptivetrading agents modeled after “zero-intelligence plus” (ZIP) agents (Cliff & Bruten, 1998;Preist & van Tol, 1998). We also compare the efficiency of
Chain with that of a truthfulonline DA due to Blum et al. (2006), which selects a fixed trading price to guaranteecompetitiveness in an adversarial model.From within the truthful mechanisms we find that adaptive, price-based instantiationsof
Chain are the most effective for high arrival intensity and low volatility. Even defininga single, well-chosen price that is optimized for the market conditions can be reasonablyeffective in promoting efficient trades in low volatility environments. On the other hand,for medium to low arrival intensity and medium to high volatility we find that the
Chain -based DAs that allow for dynamic price competition, such as the McAfee-based rule, aremost efficient. The same qualitative observations hold whether one is interested in alloca-tive efficiency or net efficiency, although the adaptive, price-based methods have betterperformance in terms of net efficiency. The Blum et al. (2006) rule fairs poorly in ourtests, which is perhaps unsurprising given that it is optimized for worst-case performancein an adversarial setting. When populated with ZIP agents, we find that non-truthful DAscan provide very good efficiency in low volatility environments but poor performance inhigh volatility environments. The good performance of the ZIP-based market occurs whenagents learn to bid approximately truthfully; i.e., when the market operates as if truthful,but without incurring the stringent cost (e.g., through trading constraints) of imposingtruthfulness explicitly. An equilibrium analysis is available only for the truthful DAs; we redin, Parkes and Duong have no way of knowing how close the ZIP agents are to playing an equilibrium, and notethat the ZIP agents do not even consider performing time-based manipulations.
Section 2 introduces the dynamic DA model, including our assumptions, and presentsdesiderata for online DAs and a price-based characterization for the design of truthfuldynamic auctions. Section 3 defines the
Chain algorithm together with the building blockof a well-defined, single-period matching rule and the strong no-trade predicate. Section 4gives a number of instantiations to both price-based and competition-based matching rules,including a general method to define the strong no-trade predicate given a price-based in-stantiation. Section 5 proves truthfulness, no-deficit and feasibility of the
Chain auctionsand also establishes their uniqueness amongst auctions constructed from the same single-period matching-rule building block. The importance of the assumption about maximalagent patience is established. Section 6 presents our empirical analysis, including a descrip-tion of the simple adaptive agents that we use to populate a non-truthful open-outcry DAand provide a benchmark. Section 7 gives related work. In Section 8 we conclude with adiscussion about the merits of truthfulness in markets and present possible extensions.
2. Preliminaries: Basic Definitions
Consider a dynamic auction model with discrete, possibly infinite, time periods T = { , , . . . } , indexed by t . The double auction (DA) provides a market for a single commodity.Agents are either buyers or sellers interested in trading a single unit of the commodity. Anagent’s type, θ i = ( a i , d i , w i ) ∈ Θ i , where Θ i is the set of possible types for agent i , definesan arrival a i , departure d i , and value w i ∈ R for trade. If the agent is a buyer, then w i > w i ≤
0. We assume a maximal patience K , so that d i ≤ a i + K for all agents.The arrival time models the first time at which an agent learns about the market orlearns about its value for a trade. Thus, information about its type is not available beforeperiod a i (not even to agent i ) and the agent cannot engage in trade before period a i . Thedeparture time, d i , models the final period in which a buyer has positive value for a trade,or the final period in which a seller is willing to engage in trade. We model risk-neutralagents with quasi-linear utility, w i − p when a trade occurs in t ∈ [ a i , d i ] and payment p is collected (with p < bid to refer, generically, to a claim that an agent –either a buyer or a seller – makes to a DA about its type. In addition, when we need to bespecific about the distinction between claims made by buyers and claims made by sellerswe refer to the bid from a buyer and the ask from a seller. Consider the following naive generalization of the (static) trade-reduction DA (Lavi & Nisan,2005; McAfee, 1992) to this dynamic environment. A bid from an agent is a claim about hain: An Online Double Auction its type ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ), necessarily made in period t = ˆ a i . Bids are active while t ∈ [ˆ a i , ˆ d i ]and no trade has occurred.Then in each period t , use the trade-reduction DA to determine which (if any) of theactive bids trade and at what price. These trades occur immediately. The trade-reductionDA (tr-DA) works as follows: Let B denote the set of bids and S denote the set of asks.Insert a dummy bid with value + ∞ into B and a dummy ask with value 0 into S . When | B | ≥ | S | ≥ B and S in order of decreasing value. Let ˆ w b ≥ ˆ w b ≥ . . . andˆ w s ≥ ˆ w s ≥ . . . denote the bid and ask values with ( b , s ) denoting the dummy bid-askpair. Let m ≥ w b m + ˆ w s m ≥ w b m +1 + ˆ w s m +1 <
0. When m ≥ { b , . . . , b m − } and asks { s , . . . , s m − } trade and payment ˆ w b m is collected from each winning buyer and payment − ˆ w s m is made to each winning seller.First consider a static tr-DA with the following bids and asks: B S i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b ∗ s ∗ -2 b s -2 s -5The line indicates that bids (1–4) and asks (1–4) could be matched for efficient trade.By the rules of the tr-DA, bids (1–3) and asks (1–3) trade, with payments $3 collected fromwinning buyers and payment $2 made to winning sellers. The auctioneer earns a profit of$3. The asterisk notation indicates the bids and asks that trade. The tr-DA is truthful , inthe sense that it is a dominant-strategy for every agent to report its true value whateverthe reports of other agents. For intuition, consider the buy-side. The payment made bywinners is independent of their bid price while the losing bidder could only win by biddingmore than $4, at which point his payment would be $4 and more than his true value.Now consider a dynamic variation with buyer types { (1 , , , (1 , , , (1 , , , (2 , , } and seller types { (1 , , − , (2 , , − , (1 , , − , (2 , , − , (1 , , − } . When agents are truth-ful, the dynamic tr-DA plays out as follows:period 1 period 2 B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b s -2 b s -2 b s -5 b s -5In period 1, buyer 1 and seller 1 trade at payments of $10 and $2 respectively. Inperiod 2, buyer 2 and seller 2 trade at payments of $4 and $2 respectively. But now we canconstruct two kinds of manipulation to show that this dynamic DA is not truthful. First,buyer 1 can do better by delaying his reported arrival until period 2: redin, Parkes and Duong period 1 period 2 B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b s -2 b s -2 s -5 b s -5Now, buyer 2 trades in period 1 and does not set the price to buyer 1 in period 2. Instead,buyer 1 now trades in period 2 and makes payment $4.Second, buyer 3 can do better by increasing his reported value:period 1 period 2 B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b ∗ s ∗ -2 b s -2 b s -5 s -5Now, buyers 1 and 2 both trade in period 1 and this allows buyer 3 to win (at a pricebelow his true value) in period 2. This is a particularly interesting manipulation becausethe agent’s manipulation is by increasing its bid above its true value. By doing so, it allowsmore trades to occur and makes the auction less competitive in the next period. We consider only direct-revelation, dynamic DAs that restrict the message that an agentcan send to the auctioneer to a single, direct claim about its type. We also consider “closed”auctions so that an agent receives no feedback before reporting its type and cannot conditionits strategy on the report of another agent. Given this, let θ t denote the set of agent types reported in period t , θ = ( θ , θ , . . . , θ t , . . . , )denote a complete type profile (perhaps unbounded), and θ ≤ t denote the type profile re-stricted to agents with (reported) arrival no later than period t . A report ˆ θ i = (ˆ a i , ˆ d i , ˆ w i )represents a commitment to buy (sell) one unit of the commodity in any period t ∈ [ˆ a i , ˆ d i ]for a payment of at most ˆ w i . Thus, if a seller reports a departure time ˆ d i > d i , it mustcommit to complete a trade that occurs after her true departure and even though a selleris modeled as having no utility for payments received after her true departure.A dynamic DA, M = ( π, x ), defines an allocation policy π = { π t } t ∈ T and paymentpolicy x = { x t } t ∈ T , where π ti ( θ ≤ t ) ∈ { , } indicates whether or not agent i trades in period t given reports θ ≤ t , and x ti ( θ ≤ t ) ∈ R indicates a payment made by agent i , negative if this isa payment received by the agent. The auction rules can also be stochastic , so that π ti ( θ ≤ t )and x ti ( θ ≤ t ) are random variables. For a dynamic DA to be well defined, it must hold that π ti ( θ ≤ t ) = 1 in at most one period t ∈ [ a i , d i ] and zero otherwise, and the payment collectedfrom agent i is zero except in periods t ∈ [ a i , d i ].In formalizing the desiderata for dynamic DAs, it will be convenient to adopt ( π ( θ ) , x ( θ ))to denote the complete sequence of allocation decisions given reports θ , with shorthand
2. The restriction to direct-revelation, online mechanisms is without loss of generality when combined witha simple heart-beat message from an agent to indicate its presence in any period t during its reportedarrival-departure interval. See the work of Pai and Vohra (2006) and Parkes (2007). hain: An Online Double Auction π i ( θ ) ∈ { , } and x i ( θ ) ∈ R to indicate whether agent i trades during its reported arrival-departure interval, and the total payment made by agent i , respectively. By a slight abuseof notation, we write i ∈ θ ≤ t to denote that agent i reported a type no later than period t .Let B denote the set of buyers and S denote the set of sellers.We shall require that the dynamic DA satisfies no-deficit , feasibility , individual-rationality and truthfulness . No-deficit ensures that the auctioneer has a cash surplus in every period: Definition 1 (no-deficit)
A dynamic DA, M = ( π, x ) is no-deficit if: X i ∈ θ ≤ t X t ′ ∈ [ a i , min( t,d i )] x t ′ i ( θ ≤ t ′ ) ≥ , ∀ t, ∀ θ (1)Feasibility ensures that the auctioneer does not need to take a short position in thecommodity traded in the market in any period: Definition 2 (feasible trade)
A dynamic DA, M = ( π, x ) is feasible if: X i ∈ θ ≤ t ,i ∈ S X t ′ ∈ [ a i , min( t,d i )] π t ′ i ( θ ≤ t ′ ) − X i ∈ θ ≤ t ,i ∈ B X t ′ ∈ [ a i , min( t,d i )] π t ′ i ( θ ≤ t ′ ) ≥ , ∀ t, ∀ θ (2)This definition of feasible trade assumes that the auctioneer can “hold” an item thatis matched between a seller-buyer pair, for instance only releasing it to the buyer uponhis reported departure. See the remark concluding this section for a discussion of thisassumption.Let v i ( θ i , π ( θ ′ i , θ − i )) ∈ R denote the value of an agent with type θ i for the allocationdecision made by policy π given report ( θ ′ i , θ − i ), i.e. v i ( θ i , π ( θ ′ i , θ − i )) = w i if the agenttrades in period t ∈ [ a i , d i ] and 0 if it trades outside of this interval and is a buyer, or −∞ ifit trades outside of this interval and is a seller. Individual-rationality requires that agent i ’sutility is non-negative when it reports its true type, whatever the reports of other agents: Definition 3 (individual-rational)
A dynamic DA, M = ( π, x ) is individual-rational(IR) if v i ( θ i , π ( θ )) − x i ( θ ) ≥ for all i , all θ . In order to define truthfulness, we introduce notation C ( θ i ) ⊆ Θ i for θ i ∈ Θ i to denotethe set of available misreports to an agent with true type θ i . In the standard model adoptedin offline mechanism design, it is typical to assume C ( θ i ) = Θ i with all misreports available.Here, we shall assume no early-arrival misreports, with C ( θ i ) = { ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ) : a i ≤ ˆ a i ≤ ˆ d i } . This assumption of limited misreports is adopted in earlier work on online mechanismdesign (Hajiaghayi et al., 2004), and is well-motivated when the arrival time is the firstperiod in which a buyer first decides to acquire an item or the period in which a seller firstdecides to sell an item. Definition 4 (truthful)
Dynamic DA, M = ( π, x ) , is dominant-strategy incentive-compatible, or truthful, given limited misreports C if: v i ( θ i , π ( θ i , θ ′− i )) − x i ( θ i , θ ′− i ) ≥ v i ( θ i , π (ˆ θ i , θ ′− i )) − x i (ˆ θ i , θ ′− i ) . for all ˆ θ i ∈ C ( θ i ) , all θ i , all θ ′− i ∈ C ( θ − i ) , all θ − i ∈ Θ − i . redin, Parkes and Duong This is a robust equilibrium concept: an agent maximizes its utility by reporting itstrue type whatever the reports of other agents. Truthfulness is useful because it simplifiesthe decision problem facing bidders: an agent can determine its optimal bidding strategywithout a model of either the auction dynamics or the other agents. In the case thatthe allocation and payment policy is stochastic , then we adopt the requirement of strongtruthfulness so that an agent maximizes its utility whatever the random sequence of coinflips within the auction.
Remark.
The flexible definition of feasibility, in which the auctioneer is able to take along position in the commodity, allows the auctioneer to time trades by receiving the unitsold by a seller in one period but only releasing it to a buyer in a later period. This allowsfor truthfulness in environments in which bidders can overstate their departure period. Insome settings this is an unreasonable requirement, however, for instance when the com-modity represents a task that is performed, or because a physical good is being tradedin an electronic market. In these cases, the definition of feasibility strengthened to re-quire exact trade-balance in every period. The tradeoff is that available misreports mustbe further restricted, with agents limited to reporting no late-departures in addition to noearly-arrivals (Lavi & Nisan, 2005; Hajiaghayi et al., 2005). For the rest of the paper wework in the “relaxed feasibility, no early-arrival” model. The
Chain framework can be im-mediately extended to the “strong-feasibility, no early-arrival and no late-departure” modelby executing trades immediately rather than delaying the trade until a buyer’s departure.
3. Chain: A Framework for Truthful Dynamic DAs
Chain provides a general algorithmic framework with which to construct truthful dynamicDAs from well-defined single-period matching rules, such as the tr-DA rules described inthe earlier section.Before introducing
Chain we need a few more definitions: Bids reported to
Chain are active while t ≤ ˆ d i (for reported departure period ˆ d i ), and while the bid is unmatchedand still eligible to be matched. In each period, a single-period matching rule is used todetermine whether any of the active bids will trade and also which (if any) of the bids thatdo not match will remain active in the next period.Now we define the building blocks, well-defined single-period matching rules, and intro-duce the important concept of a strong no-trade predicate, which is defined for a single-period matching rule. In defining a matching rule, it is helpful to adopt b t ∈ R m> and s t ∈ R n ≤ to denote theactive bids and active asks in period t , where there are m ≥ n ≥ history in period t , denoted H t ∈ R h where h ≥ matching rule ), M mr = ( π mr , x mr ) definesan allocation rule π mr ( H t , b t , s t , ω ) ∈ { , } ( m + n ) and a payment rule x mr ( H t , b t , s t , ω ) ∈
3. Note that if the task is a computational task, then tasks can be handled within this model by requiringthat the seller performs the task when it is matched but with a commitment to hold onto the result untilthe matched buyer is ready to depart. hain: An Online Double Auction function
SimpleMatch ( H t , b t , s t )matched := ∅ p t := mean( | H t | ) while ( b t = ∅ )&( s t = ∅ ) do i := 0, b i := − ǫ , j := 0, s j := −∞ while ( b i < p t )&( b t = ∅ ) do i := random( b t ), b t := b t \ { i } end whilewhile ( s j < − p t )&( s t = ∅ ) do j := random( s t ), s t := s t \ { j } end whileif ( i = 0)&( j = 0) then matched := matched ∪ { ( i, j ) } end ifend whileend function Figure 1:
A well-defined matching rule defined in terms of the mean bid price in the history. R ( m + n ) . Here, we include random event ω ∈ Ω to allow explicitly for stochastic matchingand allocation rules.
Definition 5 (well-defined matching rule)
A matching rule M mr = ( π mr , x mr ) is well-defined when it is strongly truthful, no-deficit, individual-rational, and strong-feasible. Here, the properties of truthfulness, no-deficit, and individual-rationality are exactlythe single-period specializations of those defined in the previous section. For instance,a matching rule is truthful in this sense when the dominant strategy for an agent in aDA defined with this rule, and in a static environment, is to bid truthfully and for allpossible random events ω . Similarly for individual-rationality. No-deficit requires that thetotal payments are always non-negative. Strong-feasibility requires that exactly the samenumber of asks are accepted as bids, again for all random events.One example of a well-defined matching rule is the tr-DA, which is invariant to thehistory of bids and asks. For an example of a well-defined, adaptive (history-dependent)and price-based matching rule, consider procedure SimpleMatch in Figure 1. The
Sim-pleMatch matching rule computes the mean of the absolute value of the bids and asks inthe history H t and adopts this as the clearing price in the current period. It is a stochasticmatching rule because bids and asks are picked from the sets b t and s t at random andoffered the price. We can reason about the properties of SimpleMatch as follows:(a) truthful: the price p t is independent of the bids and the probability that a bid (orask) is matched is independent of its bid (or ask) price(b) no-deficit: payment p t is collected from each matched buyer and made to eachmatched seller(c) individual-rational: only bids b i ≥ p t and asks s j ≥ − p t are accepted.(d) feasible: bids and asks are introduced to the “matched” set in balanced pairs redin, Parkes and Duong In addition to defining a matching rule M mr , we allow a designer to (optionally) designatea subset of losing bids that satisfy a property of strong no-trade. Bids that satisfy strongno-trade are losing bids for which trade was not possible at any bid price (c.f. ask pricefor asks), and moreover for which additional independence conditions hold between bidsprovided with this designation.We first define the weaker concept of no-trade. In the following, notation π mr,i ( H t , b t , s t , ω | ˆ w i ) indicates the allocation decision made for bid (or ask) i when its bid(ask) price is replaced with ˆ w i : Definition 6 (no-trade)
Given matching rule M mr = ( π mr , x mr ) then the set of agents, NT t , for which no trade is possible in period t and given random events ω are those forwhich π mr,i ( H t , b t , s t , ω | ˆ w i ) = 0 , for every ˆ w i ∈ R > when i ∈ b t and for every ˆ w i ∈ R ≤ when i ∈ s t . It can easily happen that no trade is possible, for instance when the agent is a buyerand there are no sellers on the other side of the market. Let SNT t ⊆ NT t denote the setof agents designated with the property of strong no-trade . Unlike the no-trade property,strong no-trade need not be uniquely defined for a matching rule. To be valid, however, theconstruction offered by a designer for strong no-trade must satisfy the following: Definition 7 (strong no-trade)
A construction for strong no-trade,
SNT t ⊆ NT t , is valid for a matching rule when:(a) ∀ i ∈ NT t with ˆ d i > t , whether or not i ∈ SNT t is unchanged for all alternate reports θ ′ i = ( a ′ i , d ′ i , w ′ i ) = ˆ θ i while d ′ i > t ,(b) ∀ i ∈ SNT t with ˆ d i > t , the set { j : j ∈ SNT t , j = i, ˆ d j > t } is unchanged for allreports θ ′ i = ( a ′ i , d ′ i , w ′ i ) = ˆ θ i while d ′ i > t , and independent even of whether or not agent i ispresent in the market. The strong no-trade conditions must be checked only for agents with a reported depar-ture later than the current period. Condition (a) requires that such an agent in NT t cannotaffect whether or not it satisfies the strong no-trade predicate as long as it continues toreport a departure later than the current period. Condition (b) is defined recursively, andrequires that if such an agent is identified as satisfying strong no-trade, then its own reportmust not affect the designation of strong no-trade to other agents, with reported departurelater than the current period, while it continues to report a departure later than the currentperiod – even if it delays its reported arrival until a later period.Strong no-trade allows for flexibility in determining whether or not a bid is eligible formatching. Specifically, only those bids that satisfy strong no-trade amongst those that losein the current period can remain as a candidate for trade in a future period. The propertyis defined to ensure that such a “surviving” agent does not, and could not, affect the set ofother agents against which it competes in future periods. hain: An Online Double Auction Example 1
Consider the tr-DA matching rule defined earlier with bids and asks
B S i ˆ w i i ˆ w i b ∗ s ∗ − b s − b s − Bid 1 and ask 1 trade at price and − respectively. NT t = ∅ because bids 2 and 3 couldeach trade if they had (unilaterally) submitted a bid price of greater than 10. Similarly forasks 2 and 3. Now consider the order book B S i ˆ w i i ˆ w i b s − b s − b s − No trade occurs. In this case, NT t = { b , b , b , s } . No trade is possible for any bids, evenbids 2 and 3, because ˆ w b + ˆ w s = 8 − < . But, trade is possible for asks 2 and 3, because ˆ w b + ˆ w s = 7 − ≥ and either ask could trade by submitting a low enough ask price. Example 2
Consider the tr-DA matching rule and explore possible alternative construc-tions for strong no-trade.(i) Dictatorial: in each period t , identify an agent that could be present in the period ina way that is oblivious to all agent reports. Let i denote the index of this agent. If i ∈ NT t ,then include SNT t = { i } . Strong no-trade condition (a) is satisfied because whether or not i is selected as the “dictator” is agent-independent, and given that it is selected, then whetheror not trade is possible is agent-independent. Condition (b) is trivially satisfied because | SNT t | = 1 and there is no cross-agent coupling to consider.(ii) SNT t := NT t . Consider the order book B S i ˆ w i i ˆ w i b s − b s − b s − Suppose all bids and asks remain in the market for at least one more period. Clearly, NT t = { b , b , b , s , s , s } . Consider the candidate construction SNT t = NT t . Strong no-trade condition (a) is satisfied because whether or not i is in set NT t is agent-independent.Condition (b) is not satisfied, however. Consider bid 2. If bid 2’s report had been insteadof 2 then trade would be possible for bids 1 and 3, and SNT t = NT t = { b , s , s , s } . Thus,whether or not bids 1 and 3 satisfy the strong no-trade predicate depends on the value of bid2. This is not a valid construction for strong no-trade for the tr-DA matching rule.(iii) SNT t = NT t if | b t | < or | s t | < , and SNT t = ∅ otherwise. As above, strongno-trade condition (a) is immediately satisfied. Moreover, condition (b) is now satisfiedbecause trade is not possible for any bid or ask irrespective of bid values because there aresimply not enough bids or asks to allow for trade with tr-DA (which needs at least 2 bidsand at least 2 asks). redin, Parkes and Duong Figure 2:
The decision process in
Chain upon arrival of a new bid. If admitted, then the bidparticipates in a sequence of matching events while it remains unmatched and in thestrong no-trade set. The bid matches at the first available opportunity and is pricedimmediately.
Example 3
Consider a variant of the
SimpleMatch matching rule, defined with fixedprice 9. We can again ask whether
SNT t := NT t is a valid construction for strong no-trade.Throughout this example suppose all bids and asks remain in the market for at least onemore period. First consider a bid with ˆ w b = 8 and two asks with values ˆ w s = − and ˆ w s = − . Here, NT t = { s , s } because the asks cannot trade whatever their price sincethe bid is not high enough to meet the fixed trading price of 9. Moreover, SNT t = { s , s } is a valid construction; strong no-trade condition (a) is satisfied as above and condition (b)is satisfied because whether or not ask 2 is in NT t (and thus SNT t ) is independent of theprice on ask 1, and vice versa. But consider instead a bid with ˆ w b = 8 and an ask with ˆ w s = − . Now, NT t = { b , s } and SNT t = { b , s } is our candidate strong no-trade set.However if bid 1 had declared value 10 instead of 8 then NT t = { b } and ask 1 drops out of SNT t . Thus, strong no-trade condition (b) is not satisfied. We see from the above examples that it can be quite delicate to provide a valid, non-trivial construction of strong no-trade. Note, however, that SNT t = ∅ is a (trivial) validconstruction for any matching rule. Note also that the strong no-trade conditions (a) and(b) require information about the reported departure period of a bid. Thus, while thematching rules do not use temporal information about bids, this information is used in theconstruction for strong no-trade. The control flow in
Chain is illustrated in Figure 2. Upon arrival of a new bid, an admissiondecision is made and bid i is admitted if its value ˆ w i is at least its admission price q i . Anadmitted bid competes in a sequence of matching events , where a matching event simplyapplies the matching rule to the set of active bids and asks. If a bid fails to match in someperiod and is not in the strong no-trade set ( i / ∈ SNT t ), then it is priced out and leaves themarket without trading. Otherwise, if it is still before its departure time ( t ≤ ˆ d i ) , then it isavailable for matching in the next period.Each bid is always in one of three states: active , matched or priced-out . Bids are activeif they are admitted to the market until t ≤ ˆ d i , or until they are matched or priced-out. An hain: An Online Double Auction active bid becomes matched in the first period (if any) when it trades in the single-periodmatching rule. An active bid is marked as priced-out in the first period in which it losesbut is not in the strong no-trade set. As soon as a bid is no longer active, it enters thehistory, H t , and the information about its bid price can be used in defining matching rulesfor future periods.Let E t denote the set of bids that will expire in the current period. A well-definedmatching rule, when coupled with a valid strong no-trade construction, must provide Chain with the following information, given history H t , active bids b t and active asks s t , andexpiration set E t in period t :(a) for each bid or ask, whether it wins or loses(b) for each winning bid or ask, the payment collected (negative for an ask)(c) for each losing bid or ask, whether or not it satisfies the strong no-trade conditionNote that the expiration set E t is only used for the strong no-trade construction. Thisinformation is not made available to the matching rule. The following table summarizes theuse of this information within Chain . Note that a winning bid cannot be in set SNT t : ¬ SNT t SNT t Lose priced-out surviveWin matched n/aWe describe
Chain by defining the events that occur for a bid upon its arrival into themarket, and then in each period in which it remains active:
Upon arrival : Consider all possible earlier arrival periods t ′ ∈ [ ˆ d i − K, ˆ a i −
1] consistentwith the reported type. There are no such periods to consider if the bid is maximallypatient. If the bid would lose and not be in SNT t ′ for any one of these arrival periods t ′ , then it is not admitted. Otherwise, the bid would win in all periods t ′ for which i / ∈ SNT t ′ , and define the admission price as: q (ˆ a i , ˆ d i , θ − i , ω ) := max t ′ ∈ [ ˆ d i − K, ˆ a i − ,i/ ∈ SNT t ′ [ p t ′ i , −∞ ] , (3)where p t ′ i is the payment the agent would have made (negative for a seller) in arrivalperiod t ′ (as determined by running the myopic matching rule in that period). Whenthe agent would lose in all earlier arrival periods t ′ (and so i ∈ SNT t ′ for all t ′ ), orthe bid is maximally patient, then the admission price defaults to −∞ and the bid isadmitted. While active : Consider period t ∈ [ˆ a i , ˆ d i ]. If the bid is selected to trade by the myopicmatching rule, then mark it as matched and define final payment: x ti ( θ ≤ t ) = max( q (ˆ a i , ˆ d i , θ − i , ω ) , p ti ) , (4)where p ti is the price (negative for a seller) determined by the myopic matching rule inthe current period. If this is a buyer, then collect the payment but delay transferringthe item until period ˆ d i . If this is a seller, then collect the item but delay makingthe payment until the reported departure period. If the bid loses and is not in SNT t , then mark the bid as priced-out . redin, Parkes and Duong We illustrate
Chain by instantiating it to various matching rules in the next section.In Section 5 we prove that
Chain is strongly truthful and no-deficit when coupled with awell-defined matching rule and a valid strong no-trade construction. We will see that thedelay in buyer delivery and seller payment ensures truthful revelation of a trader’s departureinformation. For instance, in the absence of this delay, a buyer might be able to do better byover-reporting departure information, still trading early enough but now for a lower price.
We choose not to allow the single-period matching rules to use the reported arrival anddeparture associated with active bids and asks. This maintains a clean separation betweennon-temporal considerations (in the matching rules) and temporal considerations (in thewider framework of
Chain ). This is also for simplicity. The single-period matching rulescan be allowed to depend on the reported arrival-departure interval, as long as the (single-period) rules are monotonic in tighter arrival-departure intervals, in the sense that an agentthat wins for some ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ) continues to win and for an improved price if it insteadreports ( a ′ i , d ′ i , ˆ w i ) with [ a ′ i , d ′ i ] ⊂ [ˆ a i , ˆ d i ]. However, whether or not trade is possible mustbe independent of the reported arrival-departure interval and similarly for strong no-trade.Determinations such as these would need to be made with respect to the most patient type( ˆ d i − K, ˆ d i , ˆ w i ) given report ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ).
4. Practical Instantiations: Price-Based and Competition-Based Rules
In this section we offer a number of instantiations of the
Chain online DA framework. Wepresent two different classes of well-defined matching rules: those that are price-based andcompute simple price statistics based on the history which are then used for matching, andthose that we refer to as competition-based and leverage the history but also consider directcompetition between the active bids and asks in any period. In each case, we establish thatthe matching rules are well-defined and provide a valid strong no-trade construction.
Each one of these rules constructs a single price, p t , in period t based on the history H t ofearlier bids and asks that are traded or expired. For this purpose we define variations on areal valued statistic, ξ ( H t ), that is used to define this price given the history. Generalizingthe SimpleMatch procedure, as introduced in Section 3.1, the price p t is used to determinethe trades in period t . We also provide a construction for strong no-trade in this context.The main concern in setting prices is that they may be too volatile, with price updatesdriving the admission price higher (via the max operator in the admission rule of Chain )and having the effect of pricing bids and asks out of the market. We describe various formsof smoothing and windowing, all designed to provide adaptivity while dampening short-term variations. In each case, the parameters (e.g. the smoothing factor , or the windowsize ) can be determined empirically through off-line tuning.We experiment with five price variants:
History-EWMA:
Exponentially-weighted moving average. The bid history, H t , is usedto define price p t in period t , computed as p t := λ ξ ( H t ) + (1 − λ ) p t − , where λ ∈ (0 ,
1] is a hain: An Online Double Auction smoothing constant and ξ ( H t ) is a statistic defined for bids and asks that enter the historyin period t . Experimentally we find that the mean statistic, ξ mean ( H t ) , of the absolutevalues of bids and asks that enter the history performs well with λ of 0.05 or lower for mostscenarios that we test. For cases in which ξ ( H t ) is not well-defined because of too few (orzero) new bids or asks, then we set p t := p t − . History-median:
Compute price p t from a statistic over a fixed-size window of the mostrecent history, p t := ξ ( H t , ∆) where ∆ is the window-size, i.e. defining bids introducedto history H t in periods [ t − ∆ , . . . , t ]. Experimentally, we find that the median statistic, ξ median ( H t , ∆) , of the absolute bid and ask values performs well for the scenarios we test,with the window size depending inversely with the volatility of agents’ valuations. Typically,we observe optimal window sizes of 20 and 150, depending on volatility. For cases in which ξ ( H t , ∆) is not well-defined because of too few (or zero) new bids or asks, then we set p t := p t − . History-clearing:
Identical to the history-median rule except the statistic ξ ( H t , ∆) isdefined as ( b m − s m ) / b m and s m are the lowest value pair of trades that would beexecuted in the efficient (value-maximizing) trade given all bids and asks to enter history H t in periods [ t − ∆ , . . . , t ]. Empirically, we find similar optimal window sizes for history-clearing as for history-median. History-McAfee:
Define the statistic ξ ( H t , ∆) to represent the McAfee price, defined inSection 4.2, for the bids in H t had they all simultaneously arrived. Fixed price:
This simple rule computes a single fixed price p t := p ∗ for all trading periods,with the price optimized offline to maximize the average-case efficiency of the dynamic DAgiven Chain and the associated single-period matching rule that leverages price p ∗ as thecandidate trading price.For each pricing variant, procedure Match (see Figures 3–4) is used to determine whichbids win (at price p t ), which lose, and, of those that lose, which satisfy the strong no-trade predicate. The subroutine used to determine the current price is referred to as determineprice in Match . We provide as input to
Match the set E t in addition to( H t , b t , s t ) because Match also constructs the strong no-trade set, and E t is used exclu-sively for this purpose.The proof of the following lemma is technical and is postponed until the Appendix. Lemma 1
Procedure
Match defines a valid strong no-trade construction.
Theorem 1
Procedure
Match defines a well-defined matching rule and a valid strong no-trade construction.
Proof:
No-deficit, feasibility, and individual-rationality are immediate by the constructionof
Match since bids and asks are added to matched in pairs, with the same payment, andonly if the payment is less than or equal to their value. Truthfulness is also easy to see: theorder with which a bid (or ask) is selected is independent of its bid price, and the price itfaces, when selected, is independent of its bid. If the price is less than or equal to its bid,then whether or not it trades depends only on its order. The rest of the claim follows fromLemma 1. (cid:3) redin, Parkes and Duong function
Match ( H t , b t , s t , E t )matched := ∅ , lose := ∅ , NT t := ∅ , SNT t := ∅ stop := false p t := determineprice( H t ) while ¬ stop do i := 0, j := 0, checked B := ∅ , checked S := ∅ while ((checked B ⊂ b t )&( i =0)) ∨ ((checked S ⊂ s t )&( j =0)) doif ( i = 0)&( j = 0) then k := random( b t \ checked B S s t \ checked S ) else if ( i = 0) then k := random( b t \ checked B ) else if ( j = 0) then k := random( s t \ checked S ) end ifif ( k ∈ b t ) then checked B := checked B ∪ { k } if ( b k ≥ p t ) then i := k end ifelse checked S := checked S ∪ { k } if ( s k ≥ − p t ) then j := k end ifend ifend whileif ( i = 0)&( j = 0) then matched := matched S { ( i, j ) } lose := lose S (checked B \ { i } ) S (checked S \ { j } ) b t := b t \ checked B , s t := s t \ checked S else stop := true end ifend whileend function Figure 3: The procedure used for single-period matching in applying
Chain to the price-based rules. The algorithm continues in Figure 4. hain: An Online Double Auction function
Match (continued)( H t , b t , s t , E t ) if ( i = 0)&( j = 0) then ⊲ Ilose := lose S s t , NT t := b t if ( ∃ k ∈ b t · (( b k ≥ p t )&( ˆ d k = t ))) ∨ ( ∀ k ∈ s t · ( ˆ d k = t )) then SNT t := b t else ⊲ I-aSNT t := b t \ checked B end ifelse if ( j = 0)&( i = 0) then ⊲ IIlose := lose S b t , NT t := s t if ( ∃ k ∈ s t · (( s k ≥ − p t )&( ˆ d k = t ))) ∨ ( ∀ k ∈ b t · ( ˆ d k = t )) then SNT t := s t else SNT t := s t \ checked S end ifelse if ( i = 0)&( j = 0) then ⊲ IIINT t := b t S s t if ( ∀ k ∈ b t · ( ˆ d k = t )) ∨ ( ∀ k ∈ s t · ( ˆ d k = t )) then SNT t := b t S s t end ifend ifend function Figure 4: Continuing procedure from Figure 3 for single-period matching in applying
Chain to the price-based rules.
Example 4 (i) Bid b t = { } , ask s t = {− } , indexed { , } and price p t = 9 . The outer while loop in Figure 3 terminates with j = 2 and i = 0 in Case II. The bid is marked asa loser while NT t = { } . If the bid will depart immediately, then SNT t = { } , otherwise SNT t = ∅ .(ii) Bid b t = { } , asks s t = {− , − } , indexed { , , } , and price p t = 9 . Suppose thatask 2 is selected before ask 3 in the outer while loop. Then the loop terminates with j = 2 and i = 0 in Case II and NT t = { , } . Suppose the bid and asks leave the market laterthan this period. Then SNT t = { } because checked S = { } .(iii) Bid b t = { } and ask s t = {− } , indexed { , } , price p t = 9 and both the bid andthe ask is patient. The outer while loop terminates with i = 0 and j = 0 in Case III sothat NT t = { , } . However, SNT t = ∅ . Each one of these rules determines which bids match in the current period through pricecompetition between the active bids. We present three variations:
McAfee, Windowed-McAfee and Active-McAfee . The latter two rules are hybrid rules in that they leverage redin, Parkes and Duong history of past offers, in smoothing prices generated by the competition-based matchingrules.
McAfee:
Use the static DA protocol due to McAfee as the matching rule. Let B denotethe set of bids and S denote the set of asks. If min( | B | , | S | ) < , then there is no trade.Otherwise, first insert two dummy bids with value {∞ , } and two dummy asks with value { , −∞} into the set of bids and asks. Let b ≥ b ≥ . . . ≥ b m and s ≥ s ≥ . . . ≥ s n . . . denote the bid and ask values with ( b , s ) denoting dummy pair ( ∞ ,
0) and ( b m , s n )denoting dummy pair (0 , −∞ ) and ties otherwise broken at random. Let m ≥ b m + s m ≥ b m +1 + s m +1 <
0. When m ≥ , consider the following two cases: • (Case I) If price p m +1 = b m +1 − s m +1 ≤ b m and − p m +1 ≤ s m then the first m bids andasks trade and payment p m +1 is collected from each winning buyer and made to eachwinning seller. • (Case II) Otherwise, the first m − b m is collectedfrom each winning buyer and payment − s m is made to each winning seller.To define NT t , replace a bid that does not trade with a bid reporting a very large valueand see whether this bid trades. To determine whether trade is possible for an ask thatdoes not trade: replace the ask with an ask reporting value ǫ >
0, some small ǫ . Saythat there is a quorum if and only if there are at least two bids and at least two asks, i.e.min( | b t | , | s t | ) ≥
2. Define strong no-trade as follows: set SNT t := NT t = b t ∪ s t whenthere is no quorum and SNT t := ∅ otherwise. Lemma 2
For any bid b i in the McAfee matching rule, then for any other bid (or ask) j there is some bid ˆ b i that will make trade possible for bid (or ask) j when there is a quorum. Proof:
Without loss of generality, suppose there are three bids and three asks. Label thebids ( a, c, e ) and the asks ( b, d, f ), both ordered from highest to lowest so that ( a, b ) is themost competitive bid-ask pair. Proceed by case analysis on bids. The analysis is symmetricfor asks and omitted. Let tp ( i ) ∈ { , } denote whether or not trade is possible for bid i , sothat i ∈ NT t ⇔ tp ( i ) = 0. For bid a : when b ≥ − ( a − d ) / tp ( c ) = tp ( e ) = 1 and thisinequality can always be satisfied for a large enough a; when a ≥ ( c − d ) / tp ( b ) = 1and when a ≥ ( c − b ) / tp ( d ) = tp ( f ) = 1, and both of these inequalities are satisfiedfor a large enough a . For bid c : when b ≥ − ( c − d ) / tp ( a ) = 1 and when, in addition, c > a , then tp ( e ) = 1 and each one of these inequalities are satisfied for a large enough c ;similarly when c ≥ ( a − d ) / tp ( b ) = 1 and when c ≥ ( a − b ) / tp ( d ) = tp ( f ) = 1.Analysis for bid e follows from that for bid c . (cid:3) Lemma 3
The construction for strong no-trade is valid and there is no valid strong no-trade construction that includes more than one losing bid or ask that will not depart in thecurrent period for any period in which there is a quorum.
Proof:
To see that this is a valid construction, notice that strong no-trade condition (a)holds since any bid (or ask) is always in both NT t and SNT t . Similarly, condition (b)trivially holds (with the other bids and asks remaining in SNT t even if any bid is not hain: An Online Double Auction present in the market). To see that this definition is essentially maximal, consider now thatmin( | b t | , | s t | ) ≥
2. For contradiction, suppose that two losing bids { i, j } with departureafter the current period are designated as strong no-trade. But, strong no-trade condition(b) fails because of Lemma 2 because either bid could have submitted an alternate bid pricethat would remove the other bid from NT t and thus necessarily also from SNT t . (cid:3) The construction offered for SNT t cannot be extended even to include one agent selectedat random from the set i ∈ NT t that will not depart immediately, in the case of a quorum.Such a construction would fail strong no-trade condition (b) when the set NT t containsmore than one bid (or ask) that does not depart in the current period, because bid i ’sabsence from the market would cause some other agent to be (randomly) selected as SNT t . Windowed-McAfee:
This myopic matching rule is parameterized on window size ∆.Augment the active bids and asks with the bids and asks introduced to the history H t inperiods t ′ ∈ { t − ∆ + 1 , . . . , t } . Run McAfee with this augmented set of bids and asks anddetermine which of these bids and asks would trade. Denote this candidate set C . Someactive agents identified as matching in C may not be able to trade in this period because C can also contain non-active agents.Let B ′ and S ′ denote, respectively, the active bids and active asks in set C . Windowed-McAfee then proceeds by picking a random subset of min( | B ′ | , | S ′ | ) bids and asks to trade.When | B ′ | 6 = | S ′ | , then some bids and asks will not trade.Define strong no-trade for this matching rule as:(i) if there are no active asks but active bids, then SNT t := b t (ii) if there are no active bids but active asks, then SNT t := s t (iii) if there are fewer than 2 asks or fewer than 2 bids in the augmented bid set, thenSNT t := b t ∪ s t ,and otherwise set SNT t := ∅ . In all cases it should be clear that SNT t ⊆ NT t . Lemma 4
The strong no-trade construction for windowed-McAfee is valid.
Proof:
That this is a valid SNT criteria in case (iii) follows immediately from the validityof the SNT criteria for the standard McAfee matching rule. Consider case (i). Case (ii)is symmetric and omitted. For strong no-trade condition (a), we see that all bids i ∈ NT t and also i ∈ SNT t , and whether or not they are designated strong no-trade is independentof their own bid price but simply because there are no active asks. Similarly, for strongno-trade condition (b), we see that all bids (and never any asks) are in SNT t whatever thebid price of any particular bid (and even whether or not it is present). (cid:3) Empirically, we find that the efficiency of Windowed-McAfee is sensitive to the size of H t , but that frequently the best choice is a small window size that includes only the activebids. Active-McAfee:
Active-McAfee augments the active bids and asks to include all un-expired but traded or priced-out offers. It proceeds as in Windowed-McAfee given thisaugmented bid set. redin, Parkes and Duong
We next provide two stylized examples to demonstrate the matching performed by
Chain using both a price-based and a competition-based matching rule. For both examples, weassume a maximal patience of K = 2. Moreover, while we describe when Chain determinesthat a bid or an ask trades, remember that a winning buyer is not allocated the good untilits reported departure and a winning seller does not receive payment until its reporteddeparture.
Example 5
Consider
Chain using an adaptive, price-based matching rule. The particulardetails of how prices are determined are not relevant. Assume that the prices in periods 1and 2 are { p , p } = { , } and the maximal patience is three periods. Now consider period3 and suppose that the order book is empty at the end of period 2 and that the bids and asksin Table 1 arrive in period 3. B S i ˆ w i ˆ d i ˆ d i − K q i p i SNT? i ˆ w i ˆ d i ˆ d i − K q i p i SNT? b * 15 4 2 7 7 N s -1 4 2 -7 n/a Y b * 10 3 1 8 8 N s * -3 5 3 −∞ -6.5 N b s -4 3 1 -7 n/a Y b −∞ n/a N s * -5 4 2 -7 -6.5 N s -10 5 3 −∞ n/a YTable 1: Bids and asks that arrive in period 3. Bids { b , b } match with asks { s , s } (as indicatedwith a *). Bid b is priced-out upon admission because q b > ˆ w b (indicated with a strike-through). The admission price is q i and the payment made by an agent that trades is p i . Column ‘SNT?’ indicates whether or not the bid or ask satisfies the strong no-tradepredicate. Asks { s , s } survive into the next period because they are in SNT and have d i > Bids { b , b , b } and asks { s , .., s } are admitted. Bid b is priced out because q b =max( p , p , −∞ ) = max(8 , , −∞ ) = 8 > ˆ w b = 7 by Eq. (3). Note that b and s areadmitted despite low bids (asks) because they have maximal patience and their admissionprices are −∞ . Now, suppose that p := 6 . is defined by the matching rule and considerapplying Match to the admitted bids and asks.Suppose that the bids are randomly ordered as ( b , b , b ) and the asks as ( s , s , s , s , s ) . Bid b is picked first but priced-out because ˆ w b = 6 < p = 6 . . Bid b is tentatively accepted ( ˆ w b = 10 ≥ p = 6 . ) and then ask s is accepted ( w s = − ≥ p = − . ). Bid b is matched with ask s , with payment max( q b , p ) = max(8 , .
5) = 8 for b by Eq. (4) and payment max( q s , p ) = max( −∞ , − .
5) = − . for s . Bid b is thententatively accepted ( ≥ . ) and then matched with ask s , which is accepted because − ≥ − . . The payments are max(7 , .
5) = 7 for b and max( −∞ , − .
5) = − . for s .Ask s expires but asks s and s survive and are marked i ∈ SNT in this period becausethey were never offered the chance to match with any bid. These asks will be active in period4. Note the role of the admission price in truthfulness. Had bid b delayed arrival untilperiod 4, its admission price would be max( p , p , −∞ ) = max(7 , .
5) = 7 and its payment hain: An Online Double Auction in period 4 (if it matches) at least 7. Similarly, had ask s delayed arrival, then its admissionprice would be max( − , − . , −∞ ) = − . and the maximal payment it can receive in period4 is 6.5. Example 6
Consider
Chain using the McAfee-based matching rule with K = 3 and withthe same bids and asks arriving in period 3. Suppose that the prices in periods 1 and 2 thatwould have been faced by a buyer are { p b , p b } = { , } and { p s , p s } = {− , − } for a seller.These prices are determined by inserting an additional bid (with value ∞ ) or an additionalask (with value 0) into the order books in each of periods 1 and 2. We will illustrate thisfor period 3. Consider now the bids and asks in period 3 in Table 2. B S i w i d i d i − K q i p i SNT? i w i d i d i − K q i p i SNT? b * 15 4 2 7 7 N s * -1 4 2 -6 -4 N b * 10 3 1 8 8 N s * -3 5 3 −∞ -4 N b s -4 3 1 -6 n/a N b −∞ n/a N s -5 4 2 -6 n/a N s -10 5 3 −∞ n/a NTable 2: Bids and asks that arrive in period 3. Bids { b , b } match with asks { s , s } (as indicatedwith a *). Bid b is priced-out upon admission because q b > ˆ w b . The admission price is q i and the payment made by an agent that trades is p i . Column ‘SNT?’ indicates whetheror not the bid or ask satisfies the strong no-trade predicate. No asks or bids survive intothe next period. As before bid b is not admitted. The myopic matching rule now runs the (static) McAfeeauction rule on bids { b , b , b } and asks { s , .., s } . Consider bids and asks in decreasingorder of value, the last efficient trade is indexed m = 3 with ˆ w b + ˆ w s = 6 − ≥ . But p m +1 = (0 − ( − / . (inserting a dummy bid with value 0 as described in Section 4.2).Price − p m +1 = − . > s = − and this trade cannot be executed by McAfee. Instead,buyers { b , b } trade and face price p bm = ˆ w b = 6 and sellers { s , s } trade and face price p sm = ˆ w s = − . Bids b and asks { s , s , s } are priced-out and do not survive into thenext round. Ultimately, payment max( q b , p bm ) = max(7 ,
6) = 7 is collected from buyer b and payment max( q b , p bm ) = max(8 ,
6) = 8 is collected from buyer b . For sellers, payment max( − , −
4) = − and max( −∞ , −
4) = − for s and s respectively.The prices p b and p s that are used in Eq. (3) to define the admission price for bids andasks with arrivals in periods 4 and 5 are determined as follows. For the buy-side price, weintroduce an additional bid with bid-price ∞ . With this the bid values considered by McAfeewould be ( ∞ , , , , and the ask values would be ( − , − , − , − , − , where a dummybid with value 0 is included on the buy-side. The last efficient pair to trade is m = 4 with − ≥ and p m +1 = (0 − ( − / , which satisfies this bid-ask pair. Therefore thebuy-side price, p b := 5 . On the sell-side, we introduce an additional ask with ask-price so that the bid values considered by McAfee are (15 , , , (again, with a dummy bidincluded) and the ask values are (0 , − , − , − , − , − . This time m = 3 and the lastefficient pair to trade is − ≥ . Now p m +1 = (0 − ( − / and this price does not redin, Parkes and Duong satisfy s , with − p m +1 > s and price p sm +1 = s = − is adopted. Therefore the sell-sideprice, p s := − .Again, we can see that bidder 1 cannot improve its price by delaying its entry until period4. The admission price for the bidder would be max( p b , p b ) = max(7 , p b ) ≥ and thus itspayment in period 4, if it matches, will be at least 7. Similarly for ask s , which would faceadmission price max { p s , p s } = max {− , − } = − and can receive a payment of at most 4in period 4. We leave it as an exercise for the reader to verify that p s = − if ask s delaysits arrival until period 4 (in comparison, p s = − when ask s is truthful). Because the McAfee-based pricing scheme computes a price and clears the order bookfollowing every period in which there are at least two bids and two asks, the bid activityperiods tend to be short in comparison to the adaptive, price-based rules where orders canbe kept active longer when there is an asymmetry in the number of bids and asks in themarket. In fact, one interesting artifact that occurs with adaptive, price-based matchingrules is that the admission-price and SNT can perpetuate this kind of bid-ask asymmetry.Once the market has more asks than bids, SNT becomes likely for future asks, but not bids.Therefore, bids are much more likely than asks to be immediately priced out of the marketby failing to meet the admission price constraint.
5. Theoretical Analysis: Truthfulness, Uniqueness, and JustifyingBounded-Patience
In this section we prove that
Chain combined with a well-defined matching rule and a validstrong no-trade construction generates a truthful, no-deficit, feasible and individual-rationaldynamic DA. In Section 5.2, we establish that uniqueness of
Chain amongst dynamic DAsthat are constructed from single-period matching rules as building blocks. In Section 5.3,we establish the importance of the existence of a maximal bound on bidder patience bypresenting a simple environment in which no truthful, no-deficit DA can implement even asingle trade despite the number of efficient trades can be increased without bound.
It will be helpful to adopt a price-based interpretation of a valid single-period matching rule.Given rule M mr , define an agent-independent price , z i ( H t , A t \ i, ω ) ∈ R where A t = b t ∪ s t ,such that for all i , all bids b t , all asks s t , all history H t , and all random events ω ∈ Ω. Wehave: (A1) ˆ w i − z i ( H t , A t \ i, ω ) > ⇒ π mr,i ( H t , b t , s t , ω ) = 1, and ˆ w i − z i ( H t , A t \ i, ω ) < ⇒ π mr,i ( H t , b t , s t , ω ) = 0(A2) payment x mr,i ( H t , b t , s t , ω ) = z i ( H t , A t \ i, ω ) if π mr,i ( H t , b t , s t , ω ) = 1 and x mr,i ( H t , b t , s t , ω ) = 0 otherwise
4. We can check that p b := 6 in this case. Suppose that bidder 1 were not present in period 3. Now considerintroducing an additional bid with value ∞ so that the bids values are {∞ , , , } (with a dummy bid)with ask values {− , − , − , − , − } . Then m = 3 and p m +1 = (0 − ( − / .
5, which does notsupport the trade between bid b and ask s . Instead, p bm = ˆ w b = 6 is adopted, and we would have p b := 6. Of course, this is exactly the price determined by McAfee for bid b in period 3 when the bidderis truthful. hain: An Online Double Auction The interpretation is that there is an agent-independent price, z i ( H t , A t \ i, ω ), that isat least ˆ w i when the agent loses and no greater than ˆ w i otherwise. In particular, z i ( H t , A t \ i, ω ) = ∞ when i ∈ NT t . Although an agent’s price is only explicit in a matching rule whenthe agent trades, it is well known that such a price exists for any truthful, single-parametermechanism; e.g., see works by Archer and Tardos (2001) and Goldberg and Hartline (2003). Moving forward we adopt price z i to characterize the matching rule used as a building blockfor Chain , and assume without loss of generality properties (A1) and (A2).Given this, we will now establish the truthfulness of
Chain by appeal to a price-basedcharacterization due to Hajiaghayi et al. (2005) for truthful, dynamic mechanisms. We state(without proof) a variant on the characterization result that holds for stochastic policies( π, x ) and strong -truthfulness. The theorem that we state is also specialized to our DAenvironment. We continue to adopt ω ∈ Ω to capture the realization of stochastic eventsinternal to the mechanism:
Theorem 2 (Hajiaghayi et al., 2005) A dynamic DA M = ( π, x ) , perhaps stochastic, isstrongly truthful for misreports limited to no early-arrivals if and only if, for every agent i ,all ˆ θ i , all θ − i , and all random events ω ∈ Ω , there exists a price p i (ˆ a i , ˆ d i , θ − i , ω ) such that:(B1) the price is independent of agent i ’s reported value(B2) the price is monotonic-increasing in tighter [ a ′ i , d ′ i ] ⊂ [ˆ a i , ˆ d i ] (B3) trade π i (ˆ θ i , θ − i ) = 1 whenever p i (ˆ a i , ˆ d i , θ − i , ω ) < ˆ w i and π i (ˆ θ i , θ − i ) = 0 whenever p i (ˆ a i , ˆ d i , θ − i , ω ) > ˆ w i , and the trade is performed for a buyer upon its departure period ˆ d i .(B4) the agent’s payment is x i (ˆ θ i , θ − i ) = p i (ˆ a i , ˆ d i , θ − i , ω ) when π i (ˆ θ i , θ − i ) = 1 , with x i (ˆ θ i , θ − i ) = 0 otherwise, and the payment is made to a seller upon its departure, ˆ d i .where random event ω is independent of the report of agent i in as much as it affects theprice to agent i . Just as for the single-period, price-based characterization, the price p i ( a i , d i , θ − i , ω ) neednot always be explicit in Chain . Rather, the theorem states that given any truthful dynamicDA, such as
Chain , there exists a well-defined price function with these properties of value-independence (B1) and arrival-departure monotonicity (B2), and such that they define thetrade (B3) and the payment (B4).To establish the truthfulness of
Chain , we prove that it is well-defined with respect tothe following price function: p i (ˆ a i , ˆ d i , θ − i , ω ) = max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p i (ˆ a i , ˆ d i , θ − i , ω )) , (5)where ˇ q (ˆ a i , ˆ d i , θ − i , ω ) = max t ∈ [ ˆ d i − K, ˆ a i − ,i/ ∈ SNT t ( z i ( H t , A t \ i, ω ) , −∞ ) (6)
5. A single-parameter mechanism is one in which the private information of an agent is limited to onenumber. This fits the single-period matching problem because the arrival and departure informationis discarded. Moreover, although there are both buyers and sellers, the problem is effectively single-parameter because no buyer can usefully pretend to be a seller and vice versa . redin, Parkes and Duong and ˇ p (ˆ a i , ˆ d i , θ − i , ω ) = (cid:26) z i ( H t ∗ , A t ∗ \ i, ω ) , if decision ( i ) = 1+ ∞ , otherwise (7)where decision ( i ) = 0 indicates that i ∈ SNT t for all t ∈ [ˆ a i , ˆ d i ] and decision ( i ) = 1otherwise, and where t ∗ ∈ [ˆ a i , ˆ d i ] is the first period in which i / ∈ SNT t . We refer to thisas the decision period . Term ˇ q (ˆ a i , ˆ d i , θ − i , ω ) denotes the admission price, and is defined onperiods t before the agent arrives for which i / ∈ SNT t had it arrived in that period. Notecarefully that the rules of Chain are implicit in defining this price function. For instance,whether or not i ∈ SNT t in some period t depends, for example, on the other bids thatremain active in that period.We now establish conditions (B1)–(B4). The proofs of the technical lemmas are deferreduntil the Appendix. The following lemma is helpful and gets to the heart of the strong no-trade concept. Lemma 5
The set of active agents (other than i ) in period t in Chain is independent of i ’s report while agent i remains active, and would be unchanged if i ’s arrival is later thanperiod t . The following result establishes properties (B1) and (B2).
Lemma 6
The price constructed from admission price ˇ q and post-arrival price ˇ p is value-independent and monotonic-increasing when the matching rule in Chain is well-defined,the strong no-trade construction is valid, and agent patience is bounded by K . Having established properties (B1) and (B2) for price function p i (ˆ a i , ˆ d i , θ − i , ω ), we justneed to establish (B3) and (B4) to show truthfulness. The timing aspect of (B3) and (B4),which requires that the buyer receives an item and the seller receives its payment uponreported departure, is already clear from the definition of Chain . Theorem 3
The online DA
Chain is strongly truthful, no-deficit, feasible and individual-rational when the matching rule is well-defined, the strong no-trade construction is valid,and agent patience is bounded by K . Proof:
Properties (B1) and (B2) follow from Lemma 6. The timing aspects of (B3) and(B4) are immediate. To complete the proof, we first consider (B3). If ˇ q > ˆ w i , then agent i is priced out at admission by Chain because this reflects that z i ( H t , A t \ i, ω ) > ˆ w i insome t ∈ [ ˆ d i − K, ˆ a i −
1] with i / ∈ SNT t , and thus the bid would lose if it arrived in thatperiod (either because it could trade, but for a payment greater than its reported value, orbecause i ∈ NT t ). Also, if there is no decision period, then ˇ p = ∞ , which is consistent with Chain , because there is no bid price at which a bid will trade when i ∈ SNT t for all periods t ∈ [ˆ a i , ˆ d i ]. Suppose now that there is a decision period t ∗ and ˇ q < ˆ w i . If ˇ p > ˆ w i , then thereshould be no trade. This is the case in Chain , because the price z i ( H t ∗ , A t ∗ \ i, ω ) in t ∗ isgreater than ˆ w i and thus the agent is priced-out. If ˇ p < ˆ w i then the bid should trade andindeed it does, again because the price z i in that period satisfies (A1) and (A2) with respectto the matching rule. Turning to (B4), it is immediate that the payments collected in Chain hain: An Online Double Auction are equal to price p i (ˆ a i , ˆ d i , θ − i , ω ), because if bid i trades then p i (ˆ a i , ˆ d i , θ − i , ω ) ≤ ˆ w i andthus ˇ q ≤ ˆ w i and ˇ p ≤ ˆ w i . The admission price q (ˆ a i , ˆ d i , θ − i , ω ) = ˇ q (ˆ a i , ˆ d i , θ − i , ω ) when ˇ q ≤ ˆ w i because price z i is well-defined by properties (A1) and (A2). Similarly, the payment p t ∗ defined by the matching rule in Chain in the decision period is equal to ˇ p .That Chain is individual-rational and feasible follows from inspection.
Chain is no-deficit because the payment collected from every agent (whether a buyer or a seller) is atleast that defined by a valid matching rule in the decision period t ∗ (it can be higher whenthe admission price is higher than this matching price), the matching rules are themselvesno-deficit, and because the auctioneer delays making a payment to a seller until its reporteddeparture but collects payment from a buyer immediately upon a match. (cid:3) We remark that information can be reported to bidders that are not currently partici-pating in the market, for instance to assist in their valuation process. If this informationis delayed by at least the maximal patience of a bidder, so that the bid of a current biddercannot influence the other bids and asks that it faces, then this is without any strategicconsequences. Of course, without this constraint, or with bidders that participate in themarket multiple times, the effect of such feedback would require careful analysis and bringus outside of the private values framework.
In what follows, we establish that
Chain is unique amongst all truthful, dynamic DAs thatadopt well-defined, myopic matching rules as simple building blocks. For this, we definethe class of canonical, dynamic DAs , which take a well-defined single period matching rulecoupled with a valid strong no-trade construction, and satisfy the following requirements:(i) agents are active until they are matched or priced-out,(ii) agents participate in the single-period matching rule while active(iii) agents are matched if and only if they trade in the single-period matching rule.We think that these restrictions capture the essence of what it means to construct adynamic DA from single-period matching rules. Notice that a number of design elements areleft undefined, including the payment collected from matched bids, when to mark an activebid as priced-out, what rule to use upon admission, and how to use the strong no-tradeinformation within the dynamic DA. In establishing a uniqueness result, we leverage thenecessary and sufficient price-based characterization in Theorem 2, and exactly determinethe price function p i (ˆ a i , ˆ d i , θ − i , ω ) to that defined in Eq. (4) and associated with Chain .The proofs for the two technical lemmas are deferred until the Appendix.
Lemma 7
A strongly truthful, canonical dynamic DA must define price p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ z i ( H t ∗ , A t ∗ \ i, ω ) where t ∗ is the decision period for bid i (if it exists). Moreover, the bidmust be priced-out in period t ∗ if it is not matched. Lemma 8
A strongly truthful, canonical and individual-rational dynamic DA must defineprice p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , and a bid with ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) must be priced-out upon admission. redin, Parkes and Duong Theorem 4
The dynamic DA algorithm
Chain uniquely defines a strongly truthful,individual-rational auction among canonical dynamic DAs that only designate bids as priced-out when necessary.
Proof:
If there is no decision period, then we must have p i (ˆ a i , ˆ d i , θ − i , ω ) = ∞ , by canonical(iii) coupled with (B3). Combining this with Lemmas 7 and 8, we have p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )). We have also established that a bid must be priced-out if its bid value is less than the admission price, or it fails to match in its decisionperiod. Left to show is that the price is exactly as in Chain , and that a bid is admittedwhen its value ˆ w i ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) and retained as active when it is in the strong no-trade set. The last two control aspects are determined once we choose a rule that “onlydesignates bids as priced-out when necessary.” We prefer to allow a bid to remain activewhen this does not compromise truthfulness or individual-rationality. Finally, supposefor contradiction that p ′ = p i (ˆ a i , ˆ d i , θ − i , ω ) > max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )). Thenan agent with max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )) < w i < p ′ would prefer to bid ˆ w i =ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )) − ǫ and avoid winning – otherwise its payment would begreater than its value. (cid:3) Chain depends on a maximal bound on patience used to calculate the admission price facedby a bidder on entering the market with Eq. (3). To motivate this assumption about theexistence of a maximal patience, we construct a simple environment in which the number oftrades implemented by a truthful, no-deficit DA can be made an arbitrarily small fractionof the number of efficient trades with even a small number of bidders having potentially un-bounded patience. This illustrates that a bound on bidder patience is required for dynamicDAs with reasonable performance.In achieving this negative result, we impose the additional requirement of anonymity ,This anonymity property is already satisfied by
Chain , when coupled with matching rulesthat satisfy anonymity, as is the case with all the rules presented in Section 4. In defininganonymity, extend the earlier definition of a dynamic DA, M = ( π, x ), so that allocationpolicy π = { π t } t ∈ T defines the probability π ti ( θ ≤ t ) ∈ [0 ,
1] that agent i trades in period t given reports θ ≤ t . Payment, x = { x t } t ∈ T , continues to define the payment x ti ( θ ≤ t ) by agent i in period t , and is a random variable when the mechanism is stochastic. Definition 8 (anonymity)
A dynamic DA, M = ( π, x ) is anonymous if allocation policy π = { π t } t ∈ T defines probability of trade π ti ( θ ≤ t ) in each period t that is independent ofidentity i and invariant to a permutation of ( θ ≤ t \ i ) and if the payment x ti ( θ ≤ t ) , contingenton trade by agent i , is independent of identity i and invariant to a permutation of ( θ ≤ t \ i ) . We now consider the following simple environment. Informally, there will be a randomnumber of high-valued phases in which bids and asks have high value and there might be asingle bidder with patience that exceeds that of the other bids and asks in the phase. Thesehigh-valued phases are then followed by some number, perhaps zero, of low-valued phaseswith bounded-patience bids and asks. Formally, there are T h ≥ high-valued phases(a random variable, unknown to the auction), each of duration L ≥ k ∈ { , , . . . , T h − } and each with: hain: An Online Double Auction • N or N − kL, ( k + 1) L, v H ), • kL, d, αv H ) for some mark-up parameter, α > d ∈ T , • N asks with type (1 + kL, ( k + 1) L, − ( v H − ǫ )),followed by some number (perhaps zero) of low-valued phases, also of duration L , andindexed k ∈ { T h , . . . , ∞} , with: • N or N − kL, ( k + 1) L, v L ) • N asks with type (1 + kL, ( k + 1) L, − ( v L − ǫ )),where N ≥
1, 0 < v L < v H , and bid-spread parameter ǫ >
0. Note that any phase canbe the last phase, with no additional bids or asks arriving in the future.
Definition 9 (reasonable DA)
A dynamic DA is reasonable in this simple environmentif there is some parameterization of new bids, N ≥ , and periods-per-phase, L ≥ , forwhich it will execute at least one trade between new bids and new asks in each phase ,for any choice of high value v H , low value v L < v H , bid-spread ǫ > , mark-up α > , highpatience d . All of the dynamic DAs presented in Section 4 can be parameterized to make themreasonable for a suitably large N ≥ L ≥
1, and without the possibility of a bid withan unbounded patience.
Theorem 5
No strongly truthful, individual-rational, no-deficit, feasible, anonymous dy-namic DA can be reasonable when a bidder’s patience can be unbounded.
Proof:
Fix any N ≥ L ≥
1, and for the number of high-valued phases, T h ≥
1, set thedeparture of a high-patience agent to d = ( T h + 1) L . Keep v H > v L > ǫ >
0, and α > k = 0 and with N − , L, v H ), N of type (1 , L, − ( v H − ǫ )) and 1 agent of patient type,(1 , ( T h + 1) L, αv H ). If the patient bid deviates to (1 , L, v H ) , then the bids are all identical,and with probability at least 1 /N the bid would win by anonymity and reasonableness.Also, by anonymity, individual-rationality and no-deficit we have that the payment madeby any winning bid is the same, and must be p ′ ∈ [ v H − ǫ, v H ]. (If the payment had been lessthan this, the DA would run at a deficit since the sellers require at least this much paymentfor individual-rationality.) Condition now on the case that the patient bid would win if itdeviates and reports (1 , L, v H ). Suppose the bidder is truthful, reports (1 , ( T h + 1) L, αv H )but does not trade in this phase. But, if phase k = 0 is the last phase with new bids andasks, then the bid will not be able to trade in the future and for strong-truthfulness theDA would need to make a payment of at least αv H − v H = ( α − v H in a later phase toprevent the bid having a useful deviation to (1 , L, v H ) and winning in phase k = 0. But, if: N ǫ < ( α − v H , (8) redin, Parkes and Duong then the DA cannot make this payment without failing no-deficit (because N ǫ is an upper-bound on the surplus the auctioneer could extract from bidders in this phase withoutviolating individual-rationality). We will later pick values of α, ǫ and v H , to satisfy Eq. (8).So, the bid must trade when it reports (1 , ( T h + 1) L, αv H ), in the event that it would winwith report (1 , L, v H ), as “insurance” against this being the last phase with new bids andasks. Moreover, it should trade for payment, p ′ ∈ [ v H − ǫ, v H ], to ensure an agent with truetype (1 , L, v H ) cannot benefit by reporting (1 , ( T h + 1) L, αv H ).Now suppose that this was not the last phase with new bids and asks, and T h > k = 0 deviated and reported(1 + T h L, ( T h + 1) L, v L ). As before, this bid would win with probability at least 1 /N byanonymity and reasonableness, but now with some payment p ′′ ∈ [ v L − ǫ, v L ]. Conditionnow on the case that the patient bid would win, both with a report of (1 , L, v H ) and with areport of (1 + T h L, ( T h + 1) L, v L ). When truthful, it trades in phase k = 0 with payment atleast v H − ǫ . If it had reported (1 + T h L, ( T h + 1) L, v L ) , it would trade in phase k = T h forpayment at most v L . For strong truthfulness, the DA must make an additional payment tothe patient agent of at least ( v H − v L ) − ( v H − ( v H − ǫ )) = v H − v L − ǫ . But, suppose thatthe high and low values are such that,( T h + 1) N ǫ < v H − v L − ǫ. (9)Making this payment in this case would violate no-deficit, because ( T h +1) N ǫ is an upper-bound on the surplus the auctioneer can extract from bidders across all phases, includingthe current phase, without violating individual-rationality. But now we can fix any v L > ǫ < v L and choose v H > ( T h + 1) N ǫ + v L + ǫ to satisfy Eq. (9) and α > ( N ǫ/v H ) + 1to satisfy Eq. (8). Thus, we have proved that no truthful dynamic DA can choose a bid-ask pair to trade in period k = 0. The proof can be readily extended to show a similarproblem with choosing a bid-ask pair in any period k < T h , by considering truthful type of(1 + kL, ( T h + 1) L, αv H ). (cid:3) To drive home the negative result: notice that the number of efficient trades can beincreased without limit by choosing an arbitrarily large T h , and that no truthful, dy-namic DA with these properties will be able to execute even a single trade in each ofthese { , . . . , T h − } periods. Moreover, we see that only a vanishingly small fraction ofhigh-patience agents is required for this negative result. The proof only requires that atleast one patient agent is possible in all of the high-valued phases.
6. Experimental Analysis
In this section, we evaluate in simulation each of the
Chain -based DAs introduced inSection 4. We measure the allocative efficiency (total value from the trades), net efficiency(total value discounted for the revenue that flows to the auctioneer), and revenue to theauctioneer. All values are normalized by the total offline value of the optimal matching.For comparison we also implement several other matching schemes: the truthful, surplus-maximizing matching algorithm presented by Blum et al. (2006), an untruthful greedymatching algorithm using truthful bids as input to provide an upper-bound on performance,and an untruthful DA populated with simple adaptive agents that are modeled after theZero-intelligence Plus trading algorithm that has been leveraged in the study of staticDAs (Cliff & Bruten, 1998; Preist & van Tol, 1998). hain: An Online Double Auction
Traders arrive to the market as a Poisson stream to exchange a single commodity at discretemoments. This is a standard model of arrival in dynamic systems, economic or otherwise.Each trader, equally likely to be a buyer or seller, arrives after the previous with an expo-nentially distributed delay, with probability density function (pdf): f ( x ) = λe − λx , x ≥ , (10)where λ > interarrival time , λ , is varied between 0.05 and 1.5; i.e., as the arrival intensity isvaried between 20 and . A single trial continues until at least 5,000 buyers and 5,000 sellershave entered the market. In our experiments we vary the maximal patience K between 2and 10. For the distribution on an agent’s activity period (or patience, d i − a i ), we considerboth a uniform distribution with pdf: f ( x ) = 1 K , x ∈ [0 , K ] , (11)and a truncated exponential distribution with pdf: f ( x ) = αe − αx , x ∈ [0 , K ] , (12)where α = − ln(0 . /K so that 95% of the underlying exponential distribution is less thanthe maximal patience. Both arrival time and activity duration are rounded to the nearestintegral time period. A trader who arrives and departs during the same period is assumedto need an immediate trade and is active for only one period.Each trader’s valuation represents a sample drawn at its arrival from a uniform distri-bution with spread 20% about the current mean valuation. (The value is positive for a bidand negative for an ask.) To simulate market volatility, we run experiments that vary theaverage valuation using Brownian motion, a common model for valuation volatility uponwhich many option pricing models are based (Copeland & Weston, 1992). At every timeperiod, the mean valuation randomly increases or decreases by a constant multiplier, e ± γ , where γ is the approximate volatility and varied between 0 and 0.15 in our experiments.We plot the mean efficiency of 100 runs for each experiment, with the same sets of bidsand asks used across all double auctions. All parameters of an auction rule are reoptimizedfor each market environment; e.g., we can find the optimal fixed price and the optimalsmoothing parameters offline given the ability to sample from the market model. We implement
Chain for the five price-based matching rules (history-clearing, history-median, history-McAfee, history-EWMA, and fixed-price) and the three competition-basedmatching rules (McAfee, active-McAfee, and windowed-McAfee).The price-based implementations keep a fixed-size set of the most recently expired,traded, or priced-out offers, H t . Offers priced-out by their admission prices are insertedinto H t prior to computing p t . The history-clearing metric computes a price to maximize thenumber of trades to agents represented by H t had they all been contemporary. The history-median metric chooses the price to be the median of the absolute valuation of the offers redin, Parkes and Duong in H t . The history-McAfee method computes the “McAfee price” for the scenario whereall agents represented by H t are simultaneously present. The EWMA metric computes anexponentially-weighted average of bids in the order that they expire, trade, or price out.The simulations initialize the price to the average of the mean buy and sell valuations. Iftwo bids expire during the same period, they are included in arbitrary order to the movingaverage.None of the metrics require more than one parameter, which is optimized offline withaccess to the model of the market environment. Parameter optimization proceeds by uni-formly sampling the parameter range, smoothing the result by averaging each result withits immediate neighbors. The optimization repeats twice more over a narrower range aboutthe smoothed maximum, returning the parameter that maximizes (expected) allocative effi-ciency. None of the price-based methods appeared to be sensitive to small ( < H t . With most simulations, the window size was chosen to be about 150 of-fers. For EWMA, the smoothing factor was usually chosen to be around 0.05 or lower. Thewindowed-McAfee matching rule, however, was extremely sensitive to window size for simu-lations with volatile valuations, and the search process frequently converged to suboptimallocal maxima.The admission price in the price-based methods is computed by first determiningwhether Match would check the value of the bid against bid price if the bid had arrived insome earlier period t ′ . Rather than simulate the entire Match procedure, it is sufficient todetermine the probability ρ i of this event. This is determined by checking the constructionof the strong no-trade sets in that earlier period. If SNT t ′ contains non-departing buyers(sellers), then the probability that an additional seller (buyer) would be examined is 1 and ρ i = 1. Otherwise the probability is equal to the ratio of the number of bids (asks) ex-amined not included in SNT t and one more than the total number of bids (asks) present.Finally, with probability ρ i the price the agent would have faced in period t ′ is defined as p t ′ ( − p t ′ for sellers), and otherwise it is −∞ . Here, p t ′ is the history-dependent price definedin period t ′ .The competition-based matching rules price out all non-trading bids at the end of eachperiod in which trade occurs (because of the definition of strong no-trade in that context).The admission prices are calculated by considering the price that a bid (ask) would havefaced in some period t ′ before its reported arrival. In such a period, the price for a bid (ask)is determined by inserting an additional bid (ask) with valuation ∞ (0) and applying thecompetition-based matching rule to that (counterfactual) state. From this we determinewhether the agent would win for its reported value, and if so what price it would face. We use a commercial integer program solver (CPLEX ) to compute the optimal offlinesolution, i.e. with complete knowledge about all offers received over time. In determiningthe offline solution we enforce the constraint that a trade can only be executed if the activityperiods of both buyer, i, and seller, j, overlap,( a i ≤ d j ) ∧ ( a j ≤ d i ) (13) hain: An Online Double Auction An integer-program formulation to maximize total value is:max X ( i,j ) ∈ overlap x ij ( w i + w j ) (14)s.t. 0 ≤ X i :( i,j ) ∈ overlap x ij ≤ , ∀ j ∈ ask ≤ X j :( i,j ) ∈ overlap x ij ≤ , ∀ i ∈ bid x ij ∈ { , } , ∀ i, j, where ( i, j ) ∈ overlap is a bid-ask pair that could potentially trade because they haveoverlapping arrival and departure intervals satisfying Eq. (13). The decision variable x ij ∈{ , } indicates that bid i matches with ask j . This provides the optimal, offline allocativeefficiency. We implement a greedy matching algorithm that immediately matches offers that yield non-negative budget surplus. This is a non-truthful matching rule but provides an additionalcomparison point for the efficiency of the other matching schemes. During each time period,the greedy matching algorithm orders active bids and asks by their valuations, exactly asthe McAfee mechanism does, and matches offers until pairs no longer generate positivesurplus. The algorithm’s performance allows us to infer the number of offers that theoptimal matching defers before matching and the amount of surplus lost by the McAfeemethod due to trade reduction and due to the additional constraint of admission pricing.
Blum et al. (2006) derive a mechanism equivalent to our fixed-price matching mechanism,except that the price used is chosen from the cumulative distribution D ( x ) = 1 rα ln (cid:18) x − w min ( r − w min (cid:19) , (15)where r is the fixed point to the equation r = ln (cid:18) w max − w min ( r − w min (cid:19) , (16)and w min ≥ w max ≥ , ln( w max /w min )) withrespect to the optimal offline solution in an adversarial setting. We were interested to seehow will this performed in practice in our simulations. redin, Parkes and Duong To compare
Chain with the existing literature on continuous double auctions, we implementa DA that in every period sorts all active offers and matches the highest valued bids withthe lowest valued asks so long as the match yields positive net surplus. The DA prices eachtrading pair at the mean of the pair’s declared valuations.
Since the trade price dependson a bidder’s declaration, the market does not support truthful bidding strategies. We musttherefore adopt a method to simulate the behavior of bidding agents within this simple openoutcry market.
For this, we randomly assign each bid to one of several “protocol agents” that each usea modified ZIP trading algorithm, as initially presented by Cliff and Bruten (1998) andimproved upon by Preist and van Tol (1998). The ZIP algorithm is a common benchmarkused to compare learned bidding behavior in a simple double-auction trading environmentin which agents are present at once and adjust their bids in seeking a profitable trade. Weadapt the ZIP algorithm for use in our dynamic environment.In our experiments we consider five of these protocol agents. New offers are assigneduniformly at random to a protocol agent, which remains persistent throughout the simula-tion. Each offer is associated with a patience category, k ∈ { low, medium, high } , definedto evenly partition the range of possible offer patience. Each protocol agent, j , is definedwith parameters ( r j , β j , γ j ) and maintains a profit margin , µ kj , on each patience category k .Parameters ( β j , γ j ) control the adaptivity of the protocol agent in how it adjusts the targetprofit margin on an individual offer, with β j ∼ U (0 . , .
2) defining the offer-level learningrate and γ j ∼ U (0 . , .
8) defining the offer-level damping factor . Parameter r j ∈ [0 ,
1] isthe learning rate adopted for updating the profit margins.The protocol agents are trained over 10 trials and their final performance is measuredin the 11th trial. The learning rate decreases through the training session and depends onthe initial learning rate r j and the adjustment rate r + j . In period t ∈ { , . . . , t k end } of trial k ∈ { , . . . , T + 1 } , where T = 10 is the number of trials used for training and t k end is thenumber of periods in trial k , the learning rate is defined as: r j := 1 − r j + ( k − r + j + (cid:18) tt k end (cid:19) r + j ! (17)where r + j = (1 − r j ) / ( T + 1). We define r j := 0 .
7. The effect of this adjustment rule is that r j is initially 0.3, decreases during training, and trends to 0.0 as t → t end in trial k = 11.Within a given trial, upon assignment of a new offer i in patience category k , the protocolagent managing the offer initializes ( µ i ( t ) , δ i ( t )) := ( µ kj , µ i ( t ) represents the targetprofit margin for the offer and δ i ( t ) represents a profit-margin correction term. The targetprofit margin and the profit margin correction term are adjusted for offer i in subsequentperiods while the bid remains active.The target profit margin is used to define a bid price for the offer in each period whileit remains active: ˆ w i ( t ) := w i (1 + µ i ( t )) . (18) hain: An Online Double Auction At the end of a period in which an offer matches or simply expires, the profit margin µ kj for its patience category is updated as: µ kj := (1 − r j ) µ kj + r j µ i ( t ) , (19)where the amount of adaptivity depends on the learning rate r j . Because the profit marginon an offer decays over its lifetime, this update adjusts towards a small profit margin ifthe offer expires or took many periods to trade, and a larger profit margin otherwise. Thelong-term learning of a protocol agent occurs through the profit margin assigned to eachpatience category.At the start of a period each protocol agent also computes target prices for bids andasks in each patience category. These are used to drive an adjustment in the target profitmargin for each active bid and ask. Target prices τ kb ( t ) and τ ks ( t ) are computed as: τ kb ( t ) := (1 + η ) max i ∈ B k ( t − { ˆ w i ( t − } + ξ , if 0 > max i ∈ S ( t − { ˆ w i ( t − } + max i ∈ B k ( t − { ˆ w i ( t − } (1 − η ) max i ∈ B k ( t − { ˆ w i ( t − } − ξ , otherwise (20)and, τ ks ( t ) := (1 + η ) max i ∈ S k ( t − { ˆ w i ( t − } + ξ , if 0 > max i ∈ B ( t − { ˆ w i ( t − } + max i ∈ S k ( t − { ˆ w i ( t − } (1 − η ) max i ∈ S k ( t − { ˆ w i ( t − } − ξ , otherwise (21)where ξ, η ∼ U (0 , . B ( t −
1) and S ( t −
1) denote the set of active bids and asksin the market in period t − B k ( t −
1) and S k ( t − k . The target price on a bid in category k is set to something slightly greater than the most competitive bid in the previous roundwhen that bid could not trade, and slightly less otherwise. Similarly for the target price onasks, where these prices are negative, so that increasing the target price makes an ask morecompetitive.Target prices are used to adjust the target profit margin at the start of each period onall active offers that arrived in some earlier period, where the influence of target prices isthrough the profit-margin correction term: µ i ( t ) := ( ˆ w i ( t −
1) + δ i ( t )) w i − , (22)and the profit-margin correction term, δ i ( t ), is defined in terms of the target price τ ki ( t )(equal to τ kb ( t ) if i is a bid and τ ks ( t ) otherwise) as, δ i ( t ) := γ j δ i ( t −
1) + (1 − γ j ) β j ( τ ki ( t ) − ˆ w i ( t − , (23)where γ j and β j are the offer-level learning rates and damping factor. The value w i and the“-1” term in Eq. (22) provide normalization. Eq. (23) is the Widrow-Hoff (Hassoun, 1995)rule, designed to minimize the least mean square error in the profit margin and adoptedhere to mimic earlier ZIP designs. redin, Parkes and Duong Our experimental results show that market conditions drive DA choice. We compare al-locative efficiency, revenue, and net efficiency. All results are averaged over 100 trials.In experiments we found only minimal qualitative differences between the use of the twopatience distributions. The uniform patience distribution provides a slight increase in ef-ficiency over result using exponential patience, caused by a larger proportion of patientagents which relaxes somewhat the admission-price constraint in Eq. (3). For this reasonwe choose to report only results for the uniform patience distribution.While the performance of all methods are summarized in Table 3, we omit the perfor-mance of some markets from the plots to keep the presentation of results as clear as possible.We do not plot the price-based results for the median- or clearing-based prices because theperformance was typically around that of the performance of
Chain instantiated on thehistory-EWMA price. We do not plot the windowed-McAfee results because of inconsis-tent performance, and in most cases, upon manual inspection, it was optimal to choose thesmallest possible window size, i.e. including only active bids and making it equivalent toactive-McAfee.Our plots also leave out the performance of the Blum et al. (2006) worst-case optimalmatching scheme because it was dominated by the fixed-price
Chain instantiation and inmany cases failed to yield any substantial surplus. We note here that the modeling assump-tion made by Blum et al. (2006) is quite different than that in our work: they worry aboutperformance in an adversarial environment while we consider probabilistic environments.Our fixed-price
Chain mechanism operates essentially identically to the surplus-maximizingscheme of Blum et al. (2006), except that
Chain can also use additional statistical informa-tion to set the ideal price, rather than drawing the price from a distribution that is used toguarantee worst-case performance against an adversary. We defer the results for the Blumet al. (2006) scheme to Table 3.Figures 5–8 plot results from two sets of experiments, one for high-patience/low-volatilityand one for low-patience/high-volatility, as we vary the inter-arrival time (and thus thearrival intensity), volatility and maximal patience. All plots are for allocative efficiencyexcept Figure 6, where we consider net efficiency. Active-McAfee is included on Figure 5,but not on any other plots because it did not improve upon the McAfee performance inthe other environments. To emphasize: the results for greedy provide an upper-bound onthe best possible performance because this is a non-truthful algorithm, simulated here withtruthful inputs.In Figure 5 (left) we see that from within the truthful DAs, the McAfee-based DA hasthe best efficiency for medium to low arrival intensities. There also is a general decreasein performance, relative to the optimal offline solution, as the arrival intensity falls. Thistrend, also observed with the greedy (non-truthful) DA, occurs because the
Chain schemeis myopic in that it matches as soon as the static DA building block finds a match, whileit is better to be less myopic when arrival intensity is low. The McAfee-based DAs are lesssensitive to this than other methods because they can aggressively update prices using theactive traders. The price-based DAs experience inefficiencies due to the lag in price updatesbecause they use only expired, traded, and priced-out offers to calculate prices. hain: An Online Double Auction A ll o c a t i v e E ff i c i en cy Inter-arrival Time(patience=6, volatility=0.01)greedyzipmcafeeewmaactive-mcafeefixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 A ll o c a t i v e E ff i c i en cy Inter-arrival Time(patience=2, volatility=0.08)greedymcafeeactive-mcafeeewmafixed-pricezip
Figure 5:
Allocative efficiency vs. inter-arrival time (1 / intensity) for several DAs. The leftplot shows high-patience, low-volatility simulations, whereas the right plots results fromlow-patience, high-volatility runs. Both sets of experiments use uniform patience distri-butions. N e t E ff i c i en cy Inter-arrival Time(patience=6, volatility=0.01)greedyzipmcafeeewmaactive-mcafeefixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 N e t E ff i c i en cy Inter-arrival Time(patience=2, volatility=0.08)greedymcafeeactive-mcafeeewmafixed-pricezip
Figure 6:
Net efficiency vs. inter-arrival time (1 / intensity) for several DAs. The left plotshows high-patience, low-volatility simulations, whereas the right plots results from low-patience, high-volatility runs. Both sets of experiments use uniform patience distribu-tions. 167 redin, Parkes and Duong
For very high arrival intensity we see Active-McAfee dominates McAfee. Active-McAfeesmooths the price, which helps to mitigate the impact of fluctuations in cost on the admissionprice via Eq. (3) in return for less responsiveness. This is helpful in “well-behaved” marketswith high arrival intensity and low volatility but was not helpful in most environments westudied, where the additional responsiveness provided by the (vanilla) McAfee scheme paidoff. The ZIP market also has good performance in this high-patience/low-volatility environ-ment. The reason is simple: this is an easy environment for simple learning agents, and theagents quickly learn to be truthful. We emphasize that these ZIP market results should betreated with caution and are certainly optimistic. This is because the ZIP agents are notprogrammed to consider timing-based manipulations. The effect in this environment is thatthe ZIP market tends to operate as if a truthful market, but without the cost of imposingtruthfulness explicitly via market-clearing rules. By comparison the
Chain auctions arefully strategyproof, to both value and temporal manipulations.Compare now with Figure 5 (right), which is for low patience and high volatility. Now wesee that McAfee dominates across the range of arrival intensities. Moreover, the performanceof ZIP is now quite poor because the agents do not have enough time to adjust their bids(patience is low) and high volatility makes this a more difficult environment. With volatilevaluations, the possibility of valuation swings leaves open the possibility of larger profits,luring agents to set wider profit margins, but only after the market changes. The ZIP agentsalso have fewer concurrent competitive offers to use in setting useful price targets duringlearning. As we might expect, high volatility also negatively impacts the efficiency of thefixed-price scheme.In Figure 6 we see that the net efficiency trends are qualitatively similar except that thecompetition-based DAs such as McAfee fare less well in comparison with the price-basedDAs. The auctioneer accrues more revenue for competition-based matching rules such asMcAfee because they often generate buy and sell prices with a spread. Together withthe competition-based schemes being intrinsically more dynamic, this drives an increasedprice spread in
Chain via the admission price constraints. In Figure 6 (left) we see thatthe fixed-price scheme performs well for high arrival intensity while EWMA dominates forintermediate arrival intensities. The McAfee scheme is still dominant for lower patience andhigher volatility (Figure 6, right).To reinforce these observations, in Table 3 we present the the net efficiency, allocativeefficiency and (normalized) revenue across all arrival intensities (i.e. inter-arrival time from0.05 to 1.5) and for both low and high volatility trials. All five price-based methods, allthree competition-based methods, and all three comparison methods are included. Wehighlight the best performing competition-based method, price-based method, as well asthe performance of the ZIP market (skipping over the non-truthful, greedy algorithm). Weomit information about the mean standard error for each measurement because in no casedid this error exceed a tenth of a percent of the mean optimal surplus. From within thetruthful DAs, we see that the McAfee-based scheme dominates overall for both allocativeand net efficiency and both low and high volatility, although EWMA competes with McAfeefor net efficiency in low volatility markets. Notice also the good performance of the ZIP-based market (with the aforementioned caveat about the restricted strategy space) at lowvolatilities. hain: An Online Double Auction scenario low-volt/high-pat high-volt/low-patnet alloc rev net alloc revmcafee active-mcafee 0.24 0.35 0.11 0.32 0.37 0.05windowed-mcafee 0.24 0.26 0.02 0.21 0.23 0.03history-clearing 0.33 0.34 0.01 0.17 0.17 0.01history-ewma history-fixed 0.23 0.23 0.00 0.04 0.04 0.00history-mcafee 0.33 0.34 0.01 0.15 0.16 0.01history-median 0.33 0.34 0.01 0.17 0.18 0.01blum et al. 0.10 0.10 0.00 0.02 0.02 0.00greedy 0.86 0.86 0.00 0.87 0.87 0.00zip
Table 3:
Net efficiency, allocative efficiency and auctioneer revenue (all normalized by the optimalvalue from trade), averaged across all arrival intensities (0.05–1.5) and for low and highvalue volatility. The best performing competition-based, price-based and ‘other’ (ignoringgreedy, which is not truthful) results are highlighted. A ll o c a t i v e E ff i c i en cy Volatility(patience=6, inter-arrival=1.0)greedyzipmcafeeewmafixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 A ll o c a t i v e E ff i c i en cy Volatility(patience=2, inter-arrival=1.0)greedymcafeeewmazipfixed-price
Figure 7:
Allocative efficiency vs. volatility for several DAs for a fairly low arrival intensity. Theleft plot is for large maximal patience and the right plot is for small maximal patience.Both sets of experiments use uniform patience distributions.
Figure 7 plots allocative efficiency versus volatility for high patience (left) and lowpatience (right) and for fairly low arrival intensity. Higher volatility hurts all methods –especially the ZIP agents, which struggle to learn appropriate profit and price targets,probably due to few opportunities to update prices for every individual offer. The McAfeescheme fairs very well, showing good robustness for both large patience and small patienceenvironments. The fixed-price scheme has the best performance when there is zero volatilitybut its efficiency falls off extremely quickly as volatility increases. redin, Parkes and Duong A ll o c a t i v e E ff i c i en cy Maximal Patience(inter-arrival=1.0, volatility=0.01)greedyzipmcafeeewmafixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 A ll o c a t i v e E ff i c i en cy Maximal Patience(inter-arrival=1.0, volatility=0.08)greedyzipmcafeeewmafixed-price
Figure 8:
Allocative efficiency vs. maximal patience for several DAs and fairly low arrival intensity.The left plot is for low volatility and the right plot is for high volatility. Both sets ofexperiments use uniform patience distributions.
We also consider the effect of varying maximal patience. This is shown in Figure 8,with low volatility (left) and high volatility (right). Again, the McAfee scheme is the bestof the truthful DAs based on
Chain . We also see that the performance of ZIP improvesas patience increases due to more opportunities for learning. Perversely, a larger patiencecan negatively affect the truthful DAs. In part this is simply because the performance ofgreedy online schemes, relative to the offline optimal, decreases as patience increases andthe offline optimal matching is able to draw more benefit from its lack of myopia.We also suspected another culprit, however. The possibility of the presence of patientagents requires the truthful DAs to include additional terms in the max operator in Eq. (3)to prevent manipulations, leading to higher admission prices and less admitted offers. Tobetter understand this effect we experimented with delayed market clearing in the McAfeescheme, where the market matches agents only every τ -th period (the “clearing duration”).The idea is to make a tradeoff between using fewer admission prices and the possibility thatwe will miss the opportunity to match some impatient offers.Figure 9 shows allocative efficiency when the matching mechanism clears less frequentlyand for different maximal patience, K . Figure 9 (left) is for low volatility. There wesee that the best clearing duration is roughly 1, 2, 3 and 4 for maximal patience of K ∈{ , , , } and that by optimizing the clearing duration the performance of McAfee remainsapproximately constant as maximal patience increases. In Figure 9 (right) we consider theeffect in a high volatility environment, with these results averaged over 500 trials because theperformance of the DA has higher variance. We see a qualitatively similar trend, althoughhigher maximal patience now hurts overall and cannot be fully compensated for by tuningthe clearing duration.
7. Related Work
Static two-sided market problems have been widely studied (Myerson & Satterthwaite, 1983;Chatterjee & Samuelson, 1987; Satterthwaite & Williams, 1989; Yoon, 2001; Deshmukh hain: An Online Double Auction A ll o c a t i v e E ff i c i en cy Clearing Duration(inter-arrival=1.0 patience=K, volatility=0.01)K=4K=6K=8K=10 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 8 10 A ll o c a t i v e E ff i c i en cy Clearing Duration(inter-arrival=1.0 patience=K, volatility=0.08)K=4K=6K=8K=10
Figure 9:
Allocative efficiency vs. clearing duration in the McAfee-based
Chain auction for fairlylow arrival intensity and as maximal patience is varied from 4 to 10. The left plot isfor low volatility and the right plot is for high volatility. Both sets of experiments useuniform patience distributions. et al., 2002). In a classic result, Myerson and Satterthwaite proved that it is impossible toachieve efficiency with voluntary participation and without running a deficit, even relaxingdominant-strategy equilibrium to a Bayesian-Nash equilibrium. Some truthful DAs areknown for static problems (McAfee, 1992; Huang et al., 2002; Babaioff & Nisan, 2004;Babaioff & Walsh, 2005). For instance, McAfee introduced a DA that sometimes forfeitstrade in return for achieving truthfulness. McAfee’s auction achieves asymptotic efficiencyas the number of buyers and sellers increases. Huang et al. extend McAfee’s mechanismto handle agents exchanging multiple units of a single commodity. Babaioff and colleagueshave considered extensions of this work to supply-chain and spatially distributed markets.Our problem is also similar to a traditional continuous double auction (CDA), wherebuyers and sellers may at any time submit offers to a market that pairs an offer as soonas a matching offer is submitted. Early work considered market efficiency of CDAs withhuman experiments in labs (Smith, 1962), while recent work investigates the use of softwareagents to execute trades (Rust et al., 1994; Cliff & Bruten, 1998; Gjerstad & Dickhaut,1998; Tesauro & Bredin, 2002). While these markets have no dominant strategy equilibria,populations of software trading agents can learn to extract virtually all available surplus,and even simple automated trading strategies outperform human traders (Das et al., 2001).However, these studies of CDAs assume that all traders share a known deadline by whichtrades must be executed. This is quite different from our setting, in which we have dynamicarrival and departure.Truthful one-sided online auctions, in which agents arrive and depart across time, havereceived some recent attention (Lavi & Nisan, 2000; Hajiaghayi et al., 2004, 2005; Porter,2004; Lavi & Nisan, 2005). We adopt and extend the monotonicity-based truthful charac-terization in the work of Hajiaghayi et al. (2005) in developing our framework for truthfulDAs. Our model of DAs must also address some of the same constraints on timing thatoccur in Porter, Hajiaghayi, and Lavi and Nisan’s work. In these previous works, the items redin, Parkes and Duong were reusable or expiring and could only be allocated in particular periods. In our work weprovide limited allowance to the match-maker, allowing it to hold onto a seller’s item untila matched buyer is ready to depart (perhaps after the seller has departed).The closest work in the literature is due to Blum et al. (2006), who present online match-ing algorithms for the same dynamic DA model. The main focus in their paper is on thedesign of matching algorithms with good worst-case performance in an adversarial setting,i.e. within the framework of competitive analysis. Issues related to incentive compatibilityreceive less attention. One way in which their work is more general is that they also studygoals of profit and maximizing the number of trades , in addition to the goal of maximizingsocial welfare that we consider in our work. However, the only algorithmic result that theypresent that is truthful in our model (where agents can misreport arrival and departure) isfor the goal of social welfare. The DA that they describe is an instance of Chain in whicha fixed price is drawn from a distribution at the start of time, and used as the matchingprice in every period. Perhaps unsurprisingly, given their worst-case approach, we observethat their auction performs significantly worse than
Chain defined for a fixed price that ispicked to optimize welfare given distributional information about the domain.
8. Conclusions
We presented a general framework to construct algorithms to match buyers with sellers inonline markets where both valuation and activity-period information are private to agents.These algorithms guarantee truthful dominant strategies by first imposing a minimum ad-mission price for each offer and then pricing and pairing the offer at the first opportunity.At the heart of the
Chain framework lies a pricing algorithm that must for each offer eitherdetermine a price independent of any information describing the offer or choose to discardthe offer. The pricing algorithm should be chosen to match market conditions. We presentseveral examples of suitable pricing schemes, including fixed-price, moving-average, andMcAfee-based schemes.More often than not, we find that the competition-based scheme that employs a McAfee-based rule to truthfully price the market delivers the best allocative efficiency. For excep-tionally low volatility and high arrival intensity, we find that adaptive price-based schemessuch as an exponentially-weighted moving average (EWMA) and even fixed price schemesperform well. We see qualitatively similar results for net efficiency, where the revenue thataccrues to the auctioneer is discounted, albeit that the price-based rules such as EWMAhave improved performance because they have no price spread. The observations are rootedin simulations comparing the market efficiency under each mechanism with the optimal of-fline solution.Additionally, we compare the efficiency of our truthful markets with a fixed-price worst-case optimal scheme presented by Blum et al. (2006), a market of strategic agents usinga variant on the ZIP price update algorithm developed by Cliff and Bruten (1998) forcontinuous double auctions, and a non-truthful, greedy matching algorithm to provide anupper-bound on performance. The best of our schemes yield around 33% net efficiencyin low volatility, high patience environments and 40% net efficiency in high volatility, lowpatience environments, while the greedy bound suggests that as much as 86% efficiencyis possible with non-strategic agents. We note that the Blum et al.scheme, designed foradversarial settings, fairs poorly in our simulations ( < hain: An Online Double Auction One can argue, we think convincingly, that truthfulness brings benefits in itself in that itavoids the waste of costly counterspeculation and promotes fairness in markets (Sandholm,2000; Abdulkadiroˇglu et al., 2006). On the other hand, it is certainly of interest thatthe gap between the efficiency of greedy matching with non-truthful matching and that ofour truthful auctions is so large. Here, we observe that the ZIP-populated (non-truthful)markets achieve around 82% efficiency in low volatility environments but collapse to around23% efficiency in high volatility environments. Based on this, one might conjecture thatdesigning for truthfulness is especially important in badly behaved, highly volatile (“thin”)environments but less important in well behaved, less volatile (“thick”) environments.Formalizing this tradeoff between providing absolute truthfulness and approximatetruthfulness, and while considering the nature of the environment, is an interesting di-rection for future work (see paper by Parkes et al., 2001). Given that reporting of marketstatistics can be incorporated within our framework (see Section 5.1), and given that mar-kets also play a role in information aggregation and value discovery, future research shouldalso consider this additional aspect of market design. Perhaps there is an interesting tradeoffbetween efficiency, truthful value revelation, and the process of information aggregation.While the general
Chain framework achieves good efficiency, further tuning seems pos-sible. One direction is to adopt a meta -pricing scheme that chooses, or blends, prices fromcompeting algorithms. Another direction is to consider richer temporal models; e.g., thevalue of goods to agents might decay or grow over time to better account for the timevalue of assets. A richer temporal model might also consider the possibility of agents or thematch-maker taking short positions (including short-term cash deficits) to increase trade.It is also interesting to extend our work to markets with non-identical goods and morecomplex valuation models such as bundle trades (Chu & Shen, 2007; Babaioff & Walsh,2005; Gonen et al., 2007), and to dynamic matching problems without prices, such as anonline variation of the classic “marriage” problem (Gusfield & Irving, 1989).
Acknowledgments
An earlier version of this paper appeared in the Proceedings of the 21st Conference on Uncer-tainty in Artificial Intelligence, 2005. This paper further characterizes necessary conditionsfor truthful online trade; truthfully matches offers using a generalized framework basedupon an arbitrary truthful static pricing rule; and compares the efficiency of our truthfulframework to that achieved in non-truthful markets populated with strategic trading agentsand with that of worst-case optimal double auctions.Parkes is supported in part by NSF grant IIS-0238147 and an Alfred P. Sloan Fellowshipand Bredin would like to thank the Harvard School of Engineering and Applied Sciencesfor hosting his sabbatical during which much this work was completed. Thanks also tothe three anonymous reviewers, who provided excellent suggestions in improving an earlierdraft of this paper.
Appendix: Proofs
Lemma 1
Procedure
Match defines a valid strong no-trade construction. redin, Parkes and Duong
Proof:
In all cases, SNT t ⊆ NT t . The set NT t is correctly constructed: equal to allremaining bids b t when ( j = 0) in Case I, all remaining bids s t when ( i = 0) in Case II,and all remaining bids and asks otherwise. In each case, no bid (or ask) in NT t could havetraded at any price because there was no available bid or ask on the opposite of the marketgiven its order.In verifying strong no-trade (SNT) conditions (a) and (b), we proceed by case analysis. Case I. ( i = 0) and ( j = 0). NT t := b t .(I-1) ∀ k ∈ s t · ( ˆ d k = t ) and SNT t := b t . For SNT-a, consider l ∈ NT t with ˆ d l > t . If l deviates and i changes but we remain in Case I then NT t is unchanged and stillcontains l . If l deviates and i → t := b t ∪ s t and stillcontains l . For SNT-b, consider l ∈ SNT t that deviates with ˆ d l > t . Again, either weremain in this case and SNT t is unchanged or i → t still contains all b t and is therefore unchanged for all agents with ˆ d k > t .(I-2) Buyer k ∈ b t with ˆ d k = t and b k ≥ p t and SNT t := b t . For SNT-a, consider l ∈ NT t with ˆ d l > t . We remain in this case for any deviation by buyer l because buyer k will ensure i = 0, and so SNT t remains unchanged and still contains l . For SNT-b, if l ∈ SNT t with ˆ d l > t deviates we again remain in this case and SNT t is unchanged.(I-3) Some seller with ˆ d k > t and no buyer with ˆ d k ′ = t willing to accept the price.SNT t := b t \ checked B . For SNT-a, consider l ∈ NT t with ˆ d l > t . First, suppose l ∈ checked B and i = l . If l deviates but still has ˆ d l > t, then even if i := l then weremain in this case and l does not enter SNT t . Second, suppose l ∈ checked B and( i = l ). If l deviates but still has ˆ d l > t, then even if ( i = 0) and ( j = 0), we goto Case III and SNT t = ∅ and l does not enter SNT t . Third, suppose l / ∈ checked B and ˆ d l > t . Deviating while ˆ d l > t has no effect and we remain in this case and l remains in SNT t . For SNT-b, consider l ∈ SNT t with ˆ d l > t , i.e. with l / ∈ checked B .If l deviates but ˆ d l > t, then this has no effect and we remain in this case and SNT t remains unchanged. Case II . ( j = 0) and ( i = 0). NT t := s t . Symmetric with Case I. Case III . ( i = 0) and ( j = 0). NT t := b t ∪ s t .(III-1) ∀ k ∈ b t · ( ˆ d k = t ) but ∃ k ′ ∈ s t · ( ˆ d k ′ > t ) and SNT t := b t ∪ s t . For SNT-a, consider l ∈ NT t with ˆ d l > t . This must be an ask. If l deviates but we remain in this case,then l remains in SNT t . If j := l, then we go to Case II and SNT t := s t and l remainsin SNT t . For SNT-b, consider l ∈ SNT t with ˆ d l > t , which must be an ask. If l deviates but we remain in this case, SNT t is unchanged. If l deviates and j := l, thenwe go to Case II, SNT t := s t , and buyers b t are removed from SNT t . But this is OKbecause all buyers depart in period t anyway.(III-2) ∀ k ∈ s t cdot ( ˆ d k = t ) but ∃ k ′ ∈ b t · ( ˆ d k ′ > t ) and SNT t := b t ∪ s t . Symmetric to CaseIII-1.(III-3) ∀ k ∈ b t · ( ˆ d k = t ) and ∀ k ∈ s t cdot ( ˆ d k = t ). SNT t := b t ∪ s t . SNT-a and SNT-b aretrivially met because no bids or asks have departure past the current period. hain: An Online Double Auction (III-4) ∃ k ∈ b t · ( ˆ d k > t ) and ∃ k ′ ∈ s t · ( ˆ d k ′ > t ) and SNT t := ∅ . For SNT-a, consider l ∈ NT t with ˆ d l > t . Assume that l is a bid. If l deviates and ˆ d l > t and i = 0 then we remainin this case and l is not in SNT t . If l deviates and ˆ d l > t but i := l, then we go toCase I and we are necessarily in Sub-case (I-a) because ˆ d l > t and there can be noother bid willing to accept the price (else i = 0 in the first place). Thus, we wouldhave SNT t := b t \ checked B and l would not be in SNT t . For SNT-b, this is triviallysatisfied because there are no agents l ∈ SNT t . (cid:3) Lemma 5
The set of active agents (other than i ) in period t in Chain is independent of i ’s report while agent i remains active, and would be unchanged if i ’s arrival is later thanperiod t . Proof:
Fix some arrival period ˆ a i . Show for any ˆ a i ≥ a i , the set of active agents in period t ≥ ˆ a i while i is active is the same as A t without agent i ’s arrival until some a ′ i > t . Proceedby induction on the number of periods that t is after ˆ a i . For period t = ˆ a i this is trivial.Now consider some period ˆ a i + r , for some r ≥ a i + r −
1. Since i is still active then, i ∈ SNT t ′ for t ′ = ˆ a i + r −
1, and therefore the otheragents in SNT t ′ that survive into this period are independent of agent i ’s report by strongno-trade condition (b). This completes the proof. (cid:3) Lemma 6
The price constructed from admission price ˇ q and post-arrival price ˇ p is value-independent and monotonic-increasing when the matching rule in Chain is well-defined,the strong no-trade construction is valid, and agent patience is bounded by K . Proof:
First fix ˆ a i , ˆ d i and θ − i . To show value-independence (B1), first note that ˇ q isvalue-independent, since whether or not i ∈ SNT t in some pre-arrival period t is value-independent by strong no-trade condition (a) and price z i ( H t , A t \ i, ω ) in such a periodis agent-independent by definition. Term ˇ p is also value-independent: the decision period t ∗ to agent i , if any, is independent of ˆ w i since the other agents that remain active areindependent of agent i while it is active by Lemma 5, and whether or not i ∈ SNT t isvalue-independent by strong no-trade (a); and the price in t ∗ is value-independent whenthe set of other active agents are value-independent.Now fix θ − i and show the price is monotonically-increasing in a tighter arrival-departureinterval (B2). First note that ˇ q is monotonic-increasing in [ˆ a i , ˆ d i ] ⊂ [ a i , d i ] because an earlierˆ d i and later ˆ a i increases the domain t ∈ [ ˆ d i − K, ˆ a i −
1] on which ˇ q is defined. Fix someˆ a i ≥ a i . Argue the price increases with earlier d ′ i ≤ ˆ d i , for any ˆ d i > ˆ a i . To see this, note thateither ˆ d i < t ∗ and so p i (ˆ a i , d ′ i , θ − i , ω ) = ∞ for all d ′ i ≤ ˆ d i , or ˆ d i ≥ t ∗ and the price is constantuntil ˆ d i < t ∗ at which point it becomes ∞ . Fix some ˆ d i ≥ a i . Argue the price increases withlater a ′ i ≥ ˆ a i , where ˆ a i ≥ ˆ d i − K . First, while a ′ i ≤ t ∗ , then ˇ p is unchanged by Lemma 5.The interesting case is when a ′ i > t ∗ , especially when ˇ q (ˆ a i , ˆ d i , θ − i , ω ) < ˇ p (ˆ a i , ˆ d i , θ − i , ω ).By reporting a later arrival, the agent can delay its decision period and perhaps hope toachieve a lower price. But, note that in this case t ∗ ∈ [ ˆ d i − K, a ′ i −
1] since ˆ d i − K ≤ ˆ a i and t ∗ ∈ [ˆ a i , a ′ i −
1] and so ˇ q ( a ′ i , ˆ d i , θ − i , ω ) ≥ ˇ p ( H t ∗ , A t ∗ \ i, ω ) because ˇ q now includes the pricein period t ∗ since i / ∈ SNT t ∗ in that pre-arrival period by Lemma 5. Overall, we see thatalthough ˇ p may decrease, max(ˇ q, ˇ p ) cannot decrease. (cid:3) redin, Parkes and Duong Lemma 7
A strongly truthful, canonical dynamic DA must define price p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ z i ( H t ∗ , A t ∗ \ i, ω ) where t ∗ is the decision period for bid i (if it exists). Moreover, the bidmust be priced-out in period t ∗ if it is not matched. Proof: (a) First, suppose z i ( H t ∗ , A t ∗ \ i, ω ) > ˆ w i but bid i is not priced-out and insteadsurvives as an active bid into the next period. But with i / ∈ SNT t ∗ , the set of active bids inperiod t ∗ +1 need not be independent of agent i ’s bid and the price z i ( H t ∗ +1 , A t ∗ +1 \ i, ω )need not be agent-independent. Yet, canonical rule (iii) requires that this price be usedto determine whether or not the agent matches, and so the dynamic DA need not betruthful. (b) Now assume for contradiction that p i (ˆ a i , ˆ d i , θ − i , ω ) < z i ( H t ∗ , A t ∗ \ i, ω ). First,if z i ( H t ∗ , A t ∗ \ i, ω ) < ∞ , then an agent with value p i (ˆ a i , ˆ d i , θ − i , ω ) < w i < z i ( H t ∗ , A t ∗ \ i, ω )will report ˆ w i = z i ( H t ∗ , A t ∗ \ i, ω ) + ǫ and trade now for a final payment less than its truevalue (whereas it would be priced-out if it reported its true value). If z i ( H t ∗ , A t ∗ \ i, ω ) = ∞ ,then p i (ˆ a i , ˆ d i , θ − i , ω ) < z i ( H t ∗ , A t ∗ \ i, ω ) implies that some bids will survive this period eventhough they are priced-out by the matching rule and not in the strong no-trade set. Thiscompromises the truthfulness of the dynamic DA, as discussed in part (a). (cid:3) Lemma 8
A strongly truthful, canonical and individual-rational dynamic DA must defineprice p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , and a bid with ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) must be priced-out upon admission. Proof:
Suppose ˆ d i < ˆ a i + K so that [ ˆ d i − K, ˆ a i −
1] is non-empty. For ˆ d i = ˆ a i + K − t = ˆ d i − K is a decision period (and i / ∈ SNT t ), we have p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ p i ( ˆ d i − K, ˆ d i , θ − i , ω ) ≥ z i ( H t , A t \ i, ω ) , (24)where the first inequality is by monotonicity (B2) and the second follows from Lemma 7since ˆ d i − K is a decision period, and would remain one with report θ ′ i = ( ˆ d i − K, ˆ d i , w ′ i )by Lemma 5. This establishes p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) for ˆ d i = ˆ a i + K −
1. Whenˆ d i = ˆ a i + K −
2, then we need Eq. (24), and also when t = ˆ d i − K + 1 is a decision period(and i / ∈ SNT t ) we have, p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ p i ( ˆ d i − K + 1 , ˆ d i , θ − i , ω ) ≥ z i ( H t , A t \ i, ω ) , (25)by the same reasoning as above. This generalizes to d i = a i + K − r for r ∈ { , . . . , K } to establish p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) for the general case. To see the bid must bepriced-out when ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , note that if it were to remain active it could match inthe matching rule and by canonical (iii) need to trade, and thus fail individual-rationalitysince the payment collected would be more than the value. (cid:3) References
Abdulkadiroˇglu, A., Pathak, P. A., Roth, A. E., & S¨onmez, T. (2006). Changing the Bostonschool choice mechanism. Tech. rep., National Bureau of Economic Research WorkingPaper No. 11965.Archer, A., & Tardos, E. (2001). Truthful mechanisms for one-parameter agents. In
Proceed-ings of the 42nd IEEE Symposium on Foundations of Computer Science , pp. 482–491. hain: An Online Double Auction
Babaioff, M., & Nisan, N. (2004). Concurrent auctions across the supply chain.
Journal ofArtificial Intelligence Research , , 595–629.Babaioff, M., Nisan, N., & Pavlov, E. (2001). Mechanisms for a spatially distributed market.In Proceedings of the 5th ACM Conference on Electronic Commerce , pp. 9–20.Babaioff, M., & Walsh, W. E. (2005). Incentive-compatible, budget-balanced, yet highlyefficient auctions for supply chain formation.
Decision Support Systems , , 123–149.Blum, A., Sandholm, T., & Zinkevich, M. (2006). Online algorithms for market clearing. Journal of the ACM , , 845–879.Chatterjee, K., & Samuelson, L. (1987). Bargaining with two-sided incomplete information:An infinite horizon model with alternating offers. Review of Economic Studies , ,175–192.Chu, L. Y., & Shen, Z. M. (2007). Truthful double auction mechanisms for e-marketplace. Operations Research . To appear.Cliff, D., & Bruten, J. (1998). Simple bargaining agents for decentralized market-basedcontrol.. In
Proceedings of the European Simulation Multiconference – Simulation -Past, Present and Future , pp. 478–485, Manchester, UK.Copeland, T. E., & Weston, J. F. (1992).
Financial Theory and Corporate Policy (Thirdedition). Addison-Wesley, Reading, MA.Das, R., Hanson, J. E., Kephart, J. O., & Tesauro, G. (2001). Agent-human interactionsin the continuous double auction. In
Proceedings of the 17th International JointConference on Artificial Intelligence , pp. 1169–1187.Deshmukh, K., Goldberg, A. V., Hartline, J. D., & Karlin, A. R. (2002). Truthful and com-petitive double auctions. In
Proceedings of the European Symposium on Algorithms ,pp. 361–373.Gerkey, B. P., & Mataric, M. J. (2002). Sold!: Auction methods for multirobot coordination.
IEEE Transactions on Robotics and Automation , (5), 758–768.Gjerstad, S., & Dickhaut, J. (1998). Price formation in double auctions. Games andEconomic Behavior , (1), 1–29.Goldberg, A., & Hartline, J. (2003). Envy-free auctions for digital goods. In Proceedings ofthe 4th ACM Conference on Electronic Commerce , pp. 29–35.Gonen, M., Gonen, R., & Pavlov, E. (2007). Generalized trade reduction mechanisms. In
Proceedings of the 8th ACM Conference on Electronic Commerce , pp. 20–29.Gusfield, D., & Irving, R. W. (1989).
The Stable Marriage Problem: Structure and Algo-rithms . MIT Press, Cambridge, MA.Hajiaghayi, M. T., Kleinberg, R., Mahdian, M., & Parkes, D. C. (2005). Online auctions withre-usable goods. In
Proceedings of the 6th ACM Conference on Electronic Commerce ,pp. 165–174.Hajiaghayi, M. T., Kleinberg, R., & Parkes, D. C. (2004). Adaptive limited-supply onlineauctions. In
Proceedings of the 5th ACM Conference on Electronic Commerce , pp.71–80. redin, Parkes and Duong
Hassoun, M. H. (1995).
Fundamentals of Artificial Neural Networks . MIT Press, Cambridge,MA.Huang, P., Scheller-Wolf, A., & Sycara, K. (2002). Design of a multi-unit double auctione-market.
Computational Intelligence , , 596–617.Lagoudakis, M., Markakis, V., Kempe, D., Keskinocak, P., Koenig, S., Kleywegt, A., Tovey,C., Meyerson, A., & Jain, S. (2005). Auction-based multi-robot routing. In Proceedingsof the Robotics Science and Systems Conference , pp. 343–350.Lavi, R., & Nisan, N. (2005). Online ascending auctions for gradually expiring goods. In
Proceedings of the ACM-SIAM Symposium on Discrete Algorithms , pp. 1146–1155.Lavi, R., & Nisan, N. (2000). Competitive analysis of incentive compatible on-line auctions.In
Proceedings of the 2nd ACM Conference on Electronic Commerce , pp. 233–241.Lin, L., & Zheng, Z. (2005). Combinatorial bids based multi-robot task allocation method.In
Proceedings of the 2005 IEEE International Conference on Robotics and Automa-tion , pp. 1145–1150.McAfee, R. P. (1992). A dominant strategy double auction.
Journal of Economic Theory , (2), 434–450.Myerson, R. B., & Satterthwaite, M. A. (1983). Efficient mechanisms for bilateral trading. Journal of Economic Theory , , 265–281.Pai, M., & Vohra, R. (2006). Optimal dynamic auctions. Tech. rep., Kellogg School ofManagement, Northwestern University.Parkes, D. C. (2007). Online mechanisms. In Nisan, N., Roughgarden, T., Tardos, E., &Vazirani, V. (Eds.), Algorithmic Game Theory , chap. 16. Cambridge University Press.Parkes, D. C., Kalagnanam, J. R., & Eso, M. (2001). Achieving budget-balance withVickrey-based payment schemes in exchanges. In
Proceedings of the 17th Interna-tional Joint Conference on Artificial Intelligence , pp. 1161–1168.Porter, R. (2004). Mechanism design for online real-time scheduling. In
Proceedings of the5th ACM Conference on Electronic Commerce , pp. 61–70.Preist, C., & van Tol, M. (1998). Adaptive agents in a persistent shout double auction. In
InProceedings of the First International Conference on Information and ComputationEconomies , pp. 11–18.Rust, J., Miller, J., & Palmer, R. (1994). Characterizing effective trading strategies: Insightsfrom the computerized double auction tournament.
Journal of Economic Dynamicsand Control , , 61–96.Sandholm, T. (2000). Issues in computational Vickrey auctions. International Journal ofElectronic Commerce , (3), 107–129.Satterthwaite, M. A., & Williams, S. R. (1989). Bilateral trade with the sealed bid k -doubleauction: Existence and efficiency. Journal of Economic Theory , , 107–133.Smith, V. L. (1962). An experimental study of competitive market behavior. Journal ofPolitical Economy , , 111–137. hain: An Online Double Auction Tesauro, G., & Bredin, J. (2002). Strategic sequential bidding in auctions using dynamic pro-gramming. In
Proceedings of the First International Joint Conference on AutonomousAgents and Multiagent Systems , pp. 591–598, Bologna, Italy.Yoon, K. (2001). The Modified Vickrey Double Auction.
Journal of Economic Theory , ,572–584. r X i v : . [ c s . G T ] O c t Journal of Artificial Intelligence Research 30 (2007) 133–179 Submitted 3/07; published 9/07
Chain: A Dynamic Double Auction Framework for MatchingPatient Agents
Jonathan Bredin [email protected]
Dept. of Mathematics and Computer Science, Colorado CollegeColorado Springs, CO 80903, USA
David C. Parkes [email protected]
Quang Duong [email protected]
School of Engineering and Applied Sciences, Harvard UniversityCambridge, MA 02138, USA
Abstract
In this paper we present and evaluate a general framework for the design of truthfulauctions for matching agents in a dynamic, two-sided market. A single commodity, suchas a resource or a task, is bought and sold by multiple buyers and sellers that arriveand depart over time. Our algorithm,
Chain , provides the first framework that allows atruthful dynamic double auction (DA) to be constructed from a truthful, single-period (i.e.static) double-auction rule. The pricing and matching method of the
Chain constructionis unique amongst dynamic-auction rules that adopt the same building block. We examineexperimentally the allocative efficiency of
Chain when instantiated on various single-periodrules, including the canonical McAfee double-auction rule. For a baseline we also considernon-truthful double auctions populated with “zero-intelligence plus”-style learning agents.
Chain -based auctions perform well in comparison with other schemes, especially as arrivalintensity falls and agent valuations become more volatile.
1. Introduction
Electronic markets are increasingly popular as a method to facilitate increased efficiencyin the supply chain, with firms using markets to procure goods and services. Two-sidedmarkets facilitate trade between many buyers and many sellers and find application totrading diverse resources, including bandwidth, securities and pollution rights. Recent yearshave also brought increased attention to resource allocation in the context of on-demandcomputing and grid computing. Even within settings of cooperative coordination, such asthose of multiple robots, researchers have turned to auctions as methods for task allocationand joint exploration (?, ?, ?).In this paper we consider a dynamic two-sided market for a single commodity, for in-stance a unit of a resource (e.g. time on a computer, some quantity of memory chips) ora task to perform (e.g. a standard database query to execute, a location to visit). Eachagent, whether buyer or seller, arrives dynamically and needs to be matched within a timeinterval. Cast as a task-allocation problem, a seller can perform the task when allocatedwithin some time interval and incurs a cost when assigned. A buyer has positive value forthe task being assigned (to any seller) within some time interval. The arrival time, accept-able time interval, and value (negative for a seller) for a trade are all private information c (cid:13) redin, Parkes and Duong to an agent. Agents are self-interested and can choose to misrepresent all and any of thisinformation to the market in order to obtain a more desirable price.The matching problem combines elements of online algorithms and sequential decisionmaking with considerations from mechanism design. Unlike traditional sequential decisionmaking, a protocol for this problem must provide incentives for agents to report truthfulinformation to a match-maker. Unlike traditional mechanism design, this is a dynamicproblem with agents that arrive and leave over time. We model this problem as a dynamicdouble auction (DA) for identical items. The match-maker becomes the auctioneer. Eachseller brings a task to be performed during a time window and each buyer brings thecapability to perform a single task. The double-auction setting also is of interest in its ownright as a protocol for matching in a dynamic business-to-business exchange.Uncertainty about the future coupled with the two-sided nature of the market leads toan interesting mechanism design problem. For example, consider the scenario where theauctioneer must decide how (and whether) to match a seller with reported cost of $6 at theend of its time interval with a present and unmatched buyer, one of which has a reportedvalue of $8 and one a reported value of $9. Should the auctioneer pair the higher bidderwith the seller? What happens if a seller, willing to sell for $4, arrives after the auctioneeracts upon the matching decision? How should the matching algorithm be designed so thatno agent can benefit from misstating its earliest arrival, latest departure, or value for atrade? Chain provides a general framework that allows a truthful dynamic double auction tobe constructed from a truthful, single-period (i.e. static) double-auction rule. The auctionsconstructed by
Chain are truthful, in the sense that the dominant strategy for an agent,whatever the future auction dynamics and bids from other agents, is to report its truevalue for a trade (negative if selling) and true patience (maximal tolerance for trade delay)immediately upon arrival into the market. We also allow for randomized mechanisms and,in this case, require strong truthfulness: the DA should be truthful for all possible randomcoin flips of the mechanism. One of the DAs in the class of auctions implied by
Chain is a dynamic generalization of McAfee’s (?) canonical truthful, no-deficit auction for asingle period. Thus, we provide the first examples of truthful, dynamic DAs that allow fordynamic price competition between buyers and sellers. The main technical challenge presented by dynamic DAs is to provide truthfulness with-out incurring a budget deficit, while handling uncertainty about future trade opportunities.Of particular concern is to ensure that an agent does not indirectly affect its price throughthe effect of its bid on the prices faced by other agents and thus other supply and demandin the market. We need to preclude this because the availability of trades depends on theprice faced by other agents. For example, a buyer that is required to pay $4 in the DA totrade might like to decrease the price that a potentially matching seller will receive from $6to $3 to allow for trade.
Chain is a modular approach to auction design, which takes as a building block a single-period matching rule and provides a method to invoke the rule in each of multiple periodswhile also providing for truthfulness. We characterize properties that a well-defined single-
1. The closest work in the literature is due to Blum et al. (?), who present a truthful, dynamic DA for ourmodel that matches bids and asks based on a price sampled from some bid-independent distribution.We compare the performance of our schemes with this scheme in Section 6. hain: An Online Double Auction period matching rule must satisfy in order for
Chain to be truthful. We further identify thetechnical property of strong no-trade , with which we can isolate agents that fail to trade inthe current period but can nevertheless survive and be eligible to trade in a future period.An auction designer defines the strong no-trade predicate, in addition to providing a well-defined single-period matching rule. Instances within this class include those constructedin terms of both “price-based” matching rules and “competition-based” matching rules.Both can depend on history and be adaptive, but only the competition-based rules use theactive bids and asks to determine the prices in the current period, facilitating a more directcompetitive processes.In proving that
Chain , when combined with a well-defined matching rule and a validstrong no-trade predicate, is truthful we leverage a recent price-based characterization fortruthful online mechanisms (?). We also show that the pricing and matching rules definedby
Chain are unique amongst the family of mechanisms that are constructed with a single-period matching rule as a building block. Throughout our work we assume that a constantlimits every buyer and seller’s patience. To motivate this assumption we provide a simpleenvironment in which no truthful, no-deficit DA can implement some constant fraction ofthe number of the efficient trades, for any constant.We adopt allocative efficiency as our design objective, which is to say auction protocolsthat maximize the expected total value from the sequence of trades. We also consider netefficiency , wherein any net outflow of payments to the marketmaker is also accounted for inconsidering the quality of a design. Experimental results explore the allocative efficiency of Chain when instantiated to various single-period matching rules and for a range of differentassumptions about market volatility and maximal patience. For a baseline we consider theefficiency of a standard (non-truthful) open outcry DA populated with simple adaptivetrading agents modeled after “zero-intelligence plus” (ZIP) agents (?, ?). We also comparethe efficiency of
Chain with that of a truthful online DA due to Blum et al. (?), whichselects a fixed trading price to guarantee competitiveness in an adversarial model.From within the truthful mechanisms we find that adaptive, price-based instantiationsof
Chain are the most effective for high arrival intensity and low volatility. Even defininga single, well-chosen price that is optimized for the market conditions can be reasonablyeffective in promoting efficient trades in low volatility environments. On the other hand,for medium to low arrival intensity and medium to high volatility we find that the
Chain -based DAs that allow for dynamic price competition, such as the McAfee-based rule, aremost efficient. The same qualitative observations hold whether one is interested in allocativeefficiency or net efficiency, although the adaptive, price-based methods have better perfor-mance in terms of net efficiency. The Blum et al. (?) rule fairs poorly in our tests, which isperhaps unsurprising given that it is optimized for worst-case performance in an adversar-ial setting. When populated with ZIP agents, we find that non-truthful DAs can providevery good efficiency in low volatility environments but poor performance in high volatilityenvironments. The good performance of the ZIP-based market occurs when agents learnto bid approximately truthfully; i.e., when the market operates as if truthful, but withoutincurring the stringent cost (e.g., through trading constraints) of imposing truthfulness ex-plicitly. An equilibrium analysis is available only for the truthful DAs; we have no way ofknowing how close the ZIP agents are to playing an equilibrium, and note that the ZIPagents do not even consider performing time-based manipulations. redin, Parkes and Duong
Section 2 introduces the dynamic DA model, including our assumptions, and presentsdesiderata for online DAs and a price-based characterization for the design of truthfuldynamic auctions. Section 3 defines the
Chain algorithm together with the building blockof a well-defined, single-period matching rule and the strong no-trade predicate. Section 4gives a number of instantiations to both price-based and competition-based matching rules,including a general method to define the strong no-trade predicate given a price-based in-stantiation. Section 5 proves truthfulness, no-deficit and feasibility of the
Chain auctionsand also establishes their uniqueness amongst auctions constructed from the same single-period matching-rule building block. The importance of the assumption about maximalagent patience is established. Section 6 presents our empirical analysis, including a descrip-tion of the simple adaptive agents that we use to populate a non-truthful open-outcry DAand provide a benchmark. Section 7 gives related work. In Section 8 we conclude with adiscussion about the merits of truthfulness in markets and present possible extensions.
2. Preliminaries: Basic Definitions
Consider a dynamic auction model with discrete, possibly infinite, time periods T = { , , . . . } , indexed by t . The double auction (DA) provides a market for a single commodity.Agents are either buyers or sellers interested in trading a single unit of the commodity. Anagent’s type, θ i = ( a i , d i , w i ) ∈ Θ i , where Θ i is the set of possible types for agent i , definesan arrival a i , departure d i , and value w i ∈ R for trade. If the agent is a buyer, then w i > w i ≤
0. We assume a maximal patience K , so that d i ≤ a i + K for all agents.The arrival time models the first time at which an agent learns about the market orlearns about its value for a trade. Thus, information about its type is not available beforeperiod a i (not even to agent i ) and the agent cannot engage in trade before period a i . Thedeparture time, d i , models the final period in which a buyer has positive value for a trade,or the final period in which a seller is willing to engage in trade. We model risk-neutralagents with quasi-linear utility, w i − p when a trade occurs in t ∈ [ a i , d i ] and payment p is collected (with p < bid to refer, generically, to a claim that an agent –either a buyer or a seller – makes to a DA about its type. In addition, when we need to bespecific about the distinction between claims made by buyers and claims made by sellerswe refer to the bid from a buyer and the ask from a seller. Consider the following naive generalization of the (static) trade-reduction DA (?, ?) tothis dynamic environment. A bid from an agent is a claim about its type ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ),necessarily made in period t = ˆ a i . Bids are active while t ∈ [ˆ a i , ˆ d i ] and no trade hasoccurred.Then in each period t , use the trade-reduction DA to determine which (if any) of theactive bids trade and at what price. These trades occur immediately. The trade-reduction hain: An Online Double Auction DA (tr-DA) works as follows: Let B denote the set of bids and S denote the set of asks.Insert a dummy bid with value + ∞ into B and a dummy ask with value 0 into S . When | B | ≥ | S | ≥ B and S in order of decreasing value. Let ˆ w b ≥ ˆ w b ≥ . . . andˆ w s ≥ ˆ w s ≥ . . . denote the bid and ask values with ( b , s ) denoting the dummy bid-askpair. Let m ≥ w b m + ˆ w s m ≥ w b m +1 + ˆ w s m +1 <
0. When m ≥ { b , . . . , b m − } and asks { s , . . . , s m − } trade and payment ˆ w b m is collected from each winning buyer and payment − ˆ w s m is made to each winning seller.First consider a static tr-DA with the following bids and asks: B S i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b ∗ s ∗ -2 b s -2 s -5The line indicates that bids (1–4) and asks (1–4) could be matched for efficient trade.By the rules of the tr-DA, bids (1–3) and asks (1–3) trade, with payments $3 collected fromwinning buyers and payment $2 made to winning sellers. The auctioneer earns a profit of$3. The asterisk notation indicates the bids and asks that trade. The tr-DA is truthful , inthe sense that it is a dominant-strategy for every agent to report its true value whateverthe reports of other agents. For intuition, consider the buy-side. The payment made bywinners is independent of their bid price while the losing bidder could only win by biddingmore than $4, at which point his payment would be $4 and more than his true value.Now consider a dynamic variation with buyer types { (1 , , , (1 , , , (1 , , , (2 , , } and seller types { (1 , , − , (2 , , − , (1 , , − , (2 , , − , (1 , , − } . When agents are truth-ful, the dynamic tr-DA plays out as follows:period 1 period 2 B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b s -2 b s -2 b s -5 b s -5In period 1, buyer 1 and seller 1 trade at payments of $10 and $2 respectively. Inperiod 2, buyer 2 and seller 2 trade at payments of $4 and $2 respectively. But now we canconstruct two kinds of manipulation to show that this dynamic DA is not truthful. First,buyer 1 can do better by delaying his reported arrival until period 2:period 1 period 2 B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b s -2 b s -2 s -5 b s -5 redin, Parkes and Duong Now, buyer 2 trades in period 1 and does not set the price to buyer 1 in period 2. Instead,buyer 1 now trades in period 2 and makes payment $4.Second, buyer 3 can do better by increasing his reported value:period 1 period 2
B S B S i ˆ w i i ˆ w i i ˆ w i i ˆ w i b ∗ s ∗ -1 b ∗ s ∗ -1 b ∗ s ∗ -2 b s -2 b s -5 s -5Now, buyers 1 and 2 both trade in period 1 and this allows buyer 3 to win (at a pricebelow his true value) in period 2. This is a particularly interesting manipulation becausethe agent’s manipulation is by increasing its bid above its true value. By doing so, it allowsmore trades to occur and makes the auction less competitive in the next period. We consider only direct-revelation, dynamic DAs that restrict the message that an agentcan send to the auctioneer to a single, direct claim about its type. We also consider “closed”auctions so that an agent receives no feedback before reporting its type and cannot conditionits strategy on the report of another agent. Given this, let θ t denote the set of agent types reported in period t , θ = ( θ , θ , . . . , θ t , . . . , )denote a complete type profile (perhaps unbounded), and θ ≤ t denote the type profile re-stricted to agents with (reported) arrival no later than period t . A report ˆ θ i = (ˆ a i , ˆ d i , ˆ w i )represents a commitment to buy (sell) one unit of the commodity in any period t ∈ [ˆ a i , ˆ d i ]for a payment of at most ˆ w i . Thus, if a seller reports a departure time ˆ d i > d i , it mustcommit to complete a trade that occurs after her true departure and even though a selleris modeled as having no utility for payments received after her true departure.A dynamic DA, M = ( π, x ), defines an allocation policy π = { π t } t ∈ T and paymentpolicy x = { x t } t ∈ T , where π ti ( θ ≤ t ) ∈ { , } indicates whether or not agent i trades in period t given reports θ ≤ t , and x ti ( θ ≤ t ) ∈ R indicates a payment made by agent i , negative if this isa payment received by the agent. The auction rules can also be stochastic , so that π ti ( θ ≤ t )and x ti ( θ ≤ t ) are random variables. For a dynamic DA to be well defined, it must hold that π ti ( θ ≤ t ) = 1 in at most one period t ∈ [ a i , d i ] and zero otherwise, and the payment collectedfrom agent i is zero except in periods t ∈ [ a i , d i ].In formalizing the desiderata for dynamic DAs, it will be convenient to adopt ( π ( θ ) , x ( θ ))to denote the complete sequence of allocation decisions given reports θ , with shorthand π i ( θ ) ∈ { , } and x i ( θ ) ∈ R to indicate whether agent i trades during its reported arrival-departure interval, and the total payment made by agent i , respectively. By a slight abuseof notation, we write i ∈ θ ≤ t to denote that agent i reported a type no later than period t .Let B denote the set of buyers and S denote the set of sellers.
2. The restriction to direct-revelation, online mechanisms is without loss of generality when combined witha simple heart-beat message from an agent to indicate its presence in any period t during its reportedarrival-departure interval. See the work of Pai and Vohra (?) and Parkes (?). hain: An Online Double Auction We shall require that the dynamic DA satisfies no-deficit , feasibility , individual-rationality and truthfulness . No-deficit ensures that the auctioneer has a cash surplus in every period: Definition 1 (no-deficit)
A dynamic DA, M = ( π, x ) is no-deficit if: X i ∈ θ ≤ t X t ′ ∈ [ a i , min( t,d i )] x t ′ i ( θ ≤ t ′ ) ≥ , ∀ t, ∀ θ (1)Feasibility ensures that the auctioneer does not need to take a short position in thecommodity traded in the market in any period: Definition 2 (feasible trade)
A dynamic DA, M = ( π, x ) is feasible if: X i ∈ θ ≤ t ,i ∈ S X t ′ ∈ [ a i , min( t,d i )] π t ′ i ( θ ≤ t ′ ) − X i ∈ θ ≤ t ,i ∈ B X t ′ ∈ [ a i , min( t,d i )] π t ′ i ( θ ≤ t ′ ) ≥ , ∀ t, ∀ θ (2)This definition of feasible trade assumes that the auctioneer can “hold” an item thatis matched between a seller-buyer pair, for instance only releasing it to the buyer uponhis reported departure. See the remark concluding this section for a discussion of thisassumption.Let v i ( θ i , π ( θ ′ i , θ − i )) ∈ R denote the value of an agent with type θ i for the allocationdecision made by policy π given report ( θ ′ i , θ − i ), i.e. v i ( θ i , π ( θ ′ i , θ − i )) = w i if the agenttrades in period t ∈ [ a i , d i ] and 0 if it trades outside of this interval and is a buyer, or −∞ ifit trades outside of this interval and is a seller. Individual-rationality requires that agent i ’sutility is non-negative when it reports its true type, whatever the reports of other agents: Definition 3 (individual-rational)
A dynamic DA, M = ( π, x ) is individual-rational(IR) if v i ( θ i , π ( θ )) − x i ( θ ) ≥ for all i , all θ . In order to define truthfulness, we introduce notation C ( θ i ) ⊆ Θ i for θ i ∈ Θ i to denotethe set of available misreports to an agent with true type θ i . In the standard model adoptedin offline mechanism design, it is typical to assume C ( θ i ) = Θ i with all misreports available.Here, we shall assume no early-arrival misreports, with C ( θ i ) = { ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ) : a i ≤ ˆ a i ≤ ˆ d i } . This assumption of limited misreports is adopted in earlier work on online mechanismdesign (?), and is well-motivated when the arrival time is the first period in which a buyerfirst decides to acquire an item or the period in which a seller first decides to sell an item. Definition 4 (truthful)
Dynamic DA, M = ( π, x ) , is dominant-strategy incentive-compatible, or truthful, given limited misreports C if: v i ( θ i , π ( θ i , θ ′− i )) − x i ( θ i , θ ′− i ) ≥ v i ( θ i , π (ˆ θ i , θ ′− i )) − x i (ˆ θ i , θ ′− i ) . for all ˆ θ i ∈ C ( θ i ) , all θ i , all θ ′− i ∈ C ( θ − i ) , all θ − i ∈ Θ − i . This is a robust equilibrium concept: an agent maximizes its utility by reporting itstrue type whatever the reports of other agents. Truthfulness is useful because it simplifiesthe decision problem facing bidders: an agent can determine its optimal bidding strategywithout a model of either the auction dynamics or the other agents. In the case thatthe allocation and payment policy is stochastic , then we adopt the requirement of strongtruthfulness so that an agent maximizes its utility whatever the random sequence of coinflips within the auction. redin, Parkes and Duong
Remark.
The flexible definition of feasibility, in which the auctioneer is able to take along position in the commodity, allows the auctioneer to time trades by receiving the unitsold by a seller in one period but only releasing it to a buyer in a later period. Thisallows for truthfulness in environments in which bidders can overstate their departure pe-riod. In some settings this is an unreasonable requirement, however, for instance whenthe commodity represents a task that is performed, or because a physical good is beingtraded in an electronic market. In these cases, the definition of feasibility strengthenedto require exact trade-balance in every period. The tradeoff is that available misreportsmust be further restricted, with agents limited to reporting no late-departures in additionto no early-arrivals (?, ?). For the rest of the paper we work in the “relaxed feasibility, noearly-arrival” model. The
Chain framework can be immediately extended to the “strong-feasibility, no early-arrival and no late-departure” model by executing trades immediatelyrather than delaying the trade until a buyer’s departure.
3. Chain: A Framework for Truthful Dynamic DAs
Chain provides a general algorithmic framework with which to construct truthful dynamicDAs from well-defined single-period matching rules, such as the tr-DA rules described inthe earlier section.Before introducing
Chain we need a few more definitions: Bids reported to
Chain are active while t ≤ ˆ d i (for reported departure period ˆ d i ), and while the bid is unmatchedand still eligible to be matched. In each period, a single-period matching rule is used todetermine whether any of the active bids will trade and also which (if any) of the bids thatdo not match will remain active in the next period.Now we define the building blocks, well-defined single-period matching rules, and intro-duce the important concept of a strong no-trade predicate, which is defined for a single-period matching rule. In defining a matching rule, it is helpful to adopt b t ∈ R m> and s t ∈ R n ≤ to denote theactive bids and active asks in period t , where there are m ≥ n ≥ history in period t , denoted H t ∈ R h where h ≥ matching rule ), M mr = ( π mr , x mr ) definesan allocation rule π mr ( H t , b t , s t , ω ) ∈ { , } ( m + n ) and a payment rule x mr ( H t , b t , s t , ω ) ∈ R ( m + n ) . Here, we include random event ω ∈ Ω to allow explicitly for stochastic matchingand allocation rules.
Definition 5 (well-defined matching rule)
A matching rule M mr = ( π mr , x mr ) is well-defined when it is strongly truthful, no-deficit, individual-rational, and strong-feasible. Here, the properties of truthfulness, no-deficit, and individual-rationality are exactlythe single-period specializations of those defined in the previous section. For instance,
3. Note that if the task is a computational task, then tasks can be handled within this model by requiringthat the seller performs the task when it is matched but with a commitment to hold onto the result untilthe matched buyer is ready to depart. hain: An Online Double Auction function
SimpleMatch ( H t , b t , s t )matched := ∅ p t := mean( | H t | ) while ( b t = ∅ )&( s t = ∅ ) do i := 0, b i := − ǫ , j := 0, s j := −∞ while ( b i < p t )&( b t = ∅ ) do i := random( b t ), b t := b t \ { i } end whilewhile ( s j < − p t )&( s t = ∅ ) do j := random( s t ), s t := s t \ { j } end whileif ( i = 0)&( j = 0) then matched := matched ∪ { ( i, j ) } end ifend whileend function Figure 1:
A well-defined matching rule defined in terms of the mean bid price in the history. a matching rule is truthful in this sense when the dominant strategy for an agent in aDA defined with this rule, and in a static environment, is to bid truthfully and for allpossible random events ω . Similarly for individual-rationality. No-deficit requires that thetotal payments are always non-negative. Strong-feasibility requires that exactly the samenumber of asks are accepted as bids, again for all random events.One example of a well-defined matching rule is the tr-DA, which is invariant to thehistory of bids and asks. For an example of a well-defined, adaptive (history-dependent)and price-based matching rule, consider procedure SimpleMatch in Figure 1. The
Sim-pleMatch matching rule computes the mean of the absolute value of the bids and asks inthe history H t and adopts this as the clearing price in the current period. It is a stochasticmatching rule because bids and asks are picked from the sets b t and s t at random andoffered the price. We can reason about the properties of SimpleMatch as follows:(a) truthful: the price p t is independent of the bids and the probability that a bid (orask) is matched is independent of its bid (or ask) price(b) no-deficit: payment p t is collected from each matched buyer and made to eachmatched seller(c) individual-rational: only bids b i ≥ p t and asks s j ≥ − p t are accepted.(d) feasible: bids and asks are introduced to the “matched” set in balanced pairs In addition to defining a matching rule M mr , we allow a designer to (optionally) designatea subset of losing bids that satisfy a property of strong no-trade. Bids that satisfy strongno-trade are losing bids for which trade was not possible at any bid price (c.f. ask pricefor asks), and moreover for which additional independence conditions hold between bidsprovided with this designation.We first define the weaker concept of no-trade. In the following, notation π mr,i ( H t , b t , s t , ω | ˆ w i ) indicates the allocation decision made for bid (or ask) i when its bid(ask) price is replaced with ˆ w i : redin, Parkes and Duong Definition 6 (no-trade)
Given matching rule M mr = ( π mr , x mr ) then the set of agents, NT t , for which no trade is possible in period t and given random events ω are those forwhich π mr,i ( H t , b t , s t , ω | ˆ w i ) = 0 , for every ˆ w i ∈ R > when i ∈ b t and for every ˆ w i ∈ R ≤ when i ∈ s t . It can easily happen that no trade is possible, for instance when the agent is a buyerand there are no sellers on the other side of the market. Let SNT t ⊆ NT t denote the setof agents designated with the property of strong no-trade . Unlike the no-trade property,strong no-trade need not be uniquely defined for a matching rule. To be valid, however, theconstruction offered by a designer for strong no-trade must satisfy the following: Definition 7 (strong no-trade)
A construction for strong no-trade,
SNT t ⊆ NT t , is valid for a matching rule when:(a) ∀ i ∈ NT t with ˆ d i > t , whether or not i ∈ SNT t is unchanged for all alternate reports θ ′ i = ( a ′ i , d ′ i , w ′ i ) = ˆ θ i while d ′ i > t ,(b) ∀ i ∈ SNT t with ˆ d i > t , the set { j : j ∈ SNT t , j = i, ˆ d j > t } is unchanged for allreports θ ′ i = ( a ′ i , d ′ i , w ′ i ) = ˆ θ i while d ′ i > t , and independent even of whether or not agent i ispresent in the market. The strong no-trade conditions must be checked only for agents with a reported depar-ture later than the current period. Condition (a) requires that such an agent in NT t cannotaffect whether or not it satisfies the strong no-trade predicate as long as it continues toreport a departure later than the current period. Condition (b) is defined recursively, andrequires that if such an agent is identified as satisfying strong no-trade, then its own reportmust not affect the designation of strong no-trade to other agents, with reported departurelater than the current period, while it continues to report a departure later than the currentperiod – even if it delays its reported arrival until a later period.Strong no-trade allows for flexibility in determining whether or not a bid is eligible formatching. Specifically, only those bids that satisfy strong no-trade amongst those that losein the current period can remain as a candidate for trade in a future period. The propertyis defined to ensure that such a “surviving” agent does not, and could not, affect the set ofother agents against which it competes in future periods. Example 1
Consider the tr-DA matching rule defined earlier with bids and asks hain: An Online Double Auction
B S i ˆ w i i ˆ w i b ∗ s ∗ − b s − b s − Bid 1 and ask 1 trade at price and − respectively. NT t = ∅ because bids 2 and 3 couldeach trade if they had (unilaterally) submitted a bid price of greater than 10. Similarly forasks 2 and 3. Now consider the order book B S i ˆ w i i ˆ w i b s − b s − b s − No trade occurs. In this case, NT t = { b , b , b , s } . No trade is possible for any bids, evenbids 2 and 3, because ˆ w b + ˆ w s = 8 − < . But, trade is possible for asks 2 and 3, because ˆ w b + ˆ w s = 7 − ≥ and either ask could trade by submitting a low enough ask price. Example 2
Consider the tr-DA matching rule and explore possible alternative construc-tions for strong no-trade.(i) Dictatorial: in each period t , identify an agent that could be present in the period ina way that is oblivious to all agent reports. Let i denote the index of this agent. If i ∈ NT t ,then include SNT t = { i } . Strong no-trade condition (a) is satisfied because whether or not i is selected as the “dictator” is agent-independent, and given that it is selected, then whetheror not trade is possible is agent-independent. Condition (b) is trivially satisfied because | SNT t | = 1 and there is no cross-agent coupling to consider.(ii) SNT t := NT t . Consider the order book B S i ˆ w i i ˆ w i b s − b s − b s − Suppose all bids and asks remain in the market for at least one more period. Clearly, NT t = { b , b , b , s , s , s } . Consider the candidate construction SNT t = NT t . Strong no-trade condition (a) is satisfied because whether or not i is in set NT t is agent-independent.Condition (b) is not satisfied, however. Consider bid 2. If bid 2’s report had been insteadof 2 then trade would be possible for bids 1 and 3, and SNT t = NT t = { b , s , s , s } . Thus,whether or not bids 1 and 3 satisfy the strong no-trade predicate depends on the value of bid2. This is not a valid construction for strong no-trade for the tr-DA matching rule.(iii) SNT t = NT t if | b t | < or | s t | < , and SNT t = ∅ otherwise. As above, strongno-trade condition (a) is immediately satisfied. Moreover, condition (b) is now satisfiedbecause trade is not possible for any bid or ask irrespective of bid values because there aresimply not enough bids or asks to allow for trade with tr-DA (which needs at least 2 bidsand at least 2 asks). redin, Parkes and Duong Figure 2:
The decision process in
Chain upon arrival of a new bid. If admitted, then the bidparticipates in a sequence of matching events while it remains unmatched and in thestrong no-trade set. The bid matches at the first available opportunity and is pricedimmediately.
Example 3
Consider a variant of the
SimpleMatch matching rule, defined with fixedprice 9. We can again ask whether
SNT t := NT t is a valid construction for strong no-trade.Throughout this example suppose all bids and asks remain in the market for at least onemore period. First consider a bid with ˆ w b = 8 and two asks with values ˆ w s = − and ˆ w s = − . Here, NT t = { s , s } because the asks cannot trade whatever their price sincethe bid is not high enough to meet the fixed trading price of 9. Moreover, SNT t = { s , s } is a valid construction; strong no-trade condition (a) is satisfied as above and condition (b)is satisfied because whether or not ask 2 is in NT t (and thus SNT t ) is independent of theprice on ask 1, and vice versa. But consider instead a bid with ˆ w b = 8 and an ask with ˆ w s = − . Now, NT t = { b , s } and SNT t = { b , s } is our candidate strong no-trade set.However if bid 1 had declared value 10 instead of 8 then NT t = { b } and ask 1 drops out of SNT t . Thus, strong no-trade condition (b) is not satisfied. We see from the above examples that it can be quite delicate to provide a valid, non-trivial construction of strong no-trade. Note, however, that SNT t = ∅ is a (trivial) validconstruction for any matching rule. Note also that the strong no-trade conditions (a) and(b) require information about the reported departure period of a bid. Thus, while thematching rules do not use temporal information about bids, this information is used in theconstruction for strong no-trade. The control flow in
Chain is illustrated in Figure 2. Upon arrival of a new bid, an admissiondecision is made and bid i is admitted if its value ˆ w i is at least its admission price q i . Anadmitted bid competes in a sequence of matching events , where a matching event simplyapplies the matching rule to the set of active bids and asks. If a bid fails to match in someperiod and is not in the strong no-trade set ( i / ∈ SNT t ), then it is priced out and leaves themarket without trading. Otherwise, if it is still before its departure time ( t ≤ ˆ d i ) , then it isavailable for matching in the next period.Each bid is always in one of three states: active , matched or priced-out . Bids are activeif they are admitted to the market until t ≤ ˆ d i , or until they are matched or priced-out. An hain: An Online Double Auction active bid becomes matched in the first period (if any) when it trades in the single-periodmatching rule. An active bid is marked as priced-out in the first period in which it losesbut is not in the strong no-trade set. As soon as a bid is no longer active, it enters thehistory, H t , and the information about its bid price can be used in defining matching rulesfor future periods.Let E t denote the set of bids that will expire in the current period. A well-definedmatching rule, when coupled with a valid strong no-trade construction, must provide Chain with the following information, given history H t , active bids b t and active asks s t , andexpiration set E t in period t :(a) for each bid or ask, whether it wins or loses(b) for each winning bid or ask, the payment collected (negative for an ask)(c) for each losing bid or ask, whether or not it satisfies the strong no-trade conditionNote that the expiration set E t is only used for the strong no-trade construction. Thisinformation is not made available to the matching rule. The following table summarizes theuse of this information within Chain . Note that a winning bid cannot be in set SNT t : ¬ SNT t SNT t Lose priced-out surviveWin matched n/aWe describe
Chain by defining the events that occur for a bid upon its arrival into themarket, and then in each period in which it remains active:
Upon arrival : Consider all possible earlier arrival periods t ′ ∈ [ ˆ d i − K, ˆ a i −
1] consistentwith the reported type. There are no such periods to consider if the bid is maximallypatient. If the bid would lose and not be in SNT t ′ for any one of these arrival periods t ′ , then it is not admitted. Otherwise, the bid would win in all periods t ′ for which i / ∈ SNT t ′ , and define the admission price as: q (ˆ a i , ˆ d i , θ − i , ω ) := max t ′ ∈ [ ˆ d i − K, ˆ a i − ,i/ ∈ SNT t ′ [ p t ′ i , −∞ ] , (3)where p t ′ i is the payment the agent would have made (negative for a seller) in arrivalperiod t ′ (as determined by running the myopic matching rule in that period). Whenthe agent would lose in all earlier arrival periods t ′ (and so i ∈ SNT t ′ for all t ′ ), orthe bid is maximally patient, then the admission price defaults to −∞ and the bid isadmitted. While active : Consider period t ∈ [ˆ a i , ˆ d i ]. If the bid is selected to trade by the myopicmatching rule, then mark it as matched and define final payment: x ti ( θ ≤ t ) = max( q (ˆ a i , ˆ d i , θ − i , ω ) , p ti ) , (4)where p ti is the price (negative for a seller) determined by the myopic matching rule inthe current period. If this is a buyer, then collect the payment but delay transferringthe item until period ˆ d i . If this is a seller, then collect the item but delay makingthe payment until the reported departure period. If the bid loses and is not in SNT t , then mark the bid as priced-out . redin, Parkes and Duong We illustrate
Chain by instantiating it to various matching rules in the next section.In Section 5 we prove that
Chain is strongly truthful and no-deficit when coupled with awell-defined matching rule and a valid strong no-trade construction. We will see that thedelay in buyer delivery and seller payment ensures truthful revelation of a trader’s departureinformation. For instance, in the absence of this delay, a buyer might be able to do better byover-reporting departure information, still trading early enough but now for a lower price.
We choose not to allow the single-period matching rules to use the reported arrival anddeparture associated with active bids and asks. This maintains a clean separation betweennon-temporal considerations (in the matching rules) and temporal considerations (in thewider framework of
Chain ). This is also for simplicity. The single-period matching rulescan be allowed to depend on the reported arrival-departure interval, as long as the (single-period) rules are monotonic in tighter arrival-departure intervals, in the sense that an agentthat wins for some ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ) continues to win and for an improved price if it insteadreports ( a ′ i , d ′ i , ˆ w i ) with [ a ′ i , d ′ i ] ⊂ [ˆ a i , ˆ d i ]. However, whether or not trade is possible mustbe independent of the reported arrival-departure interval and similarly for strong no-trade.Determinations such as these would need to be made with respect to the most patient type( ˆ d i − K, ˆ d i , ˆ w i ) given report ˆ θ i = (ˆ a i , ˆ d i , ˆ w i ).
4. Practical Instantiations: Price-Based and Competition-Based Rules
In this section we offer a number of instantiations of the
Chain online DA framework. Wepresent two different classes of well-defined matching rules: those that are price-based andcompute simple price statistics based on the history which are then used for matching, andthose that we refer to as competition-based and leverage the history but also consider directcompetition between the active bids and asks in any period. In each case, we establish thatthe matching rules are well-defined and provide a valid strong no-trade construction.
Each one of these rules constructs a single price, p t , in period t based on the history H t ofearlier bids and asks that are traded or expired. For this purpose we define variations on areal valued statistic, ξ ( H t ), that is used to define this price given the history. Generalizingthe SimpleMatch procedure, as introduced in Section 3.1, the price p t is used to determinethe trades in period t . We also provide a construction for strong no-trade in this context.The main concern in setting prices is that they may be too volatile, with price updatesdriving the admission price higher (via the max operator in the admission rule of Chain )and having the effect of pricing bids and asks out of the market. We describe various formsof smoothing and windowing, all designed to provide adaptivity while dampening short-term variations. In each case, the parameters (e.g. the smoothing factor , or the windowsize ) can be determined empirically through off-line tuning.We experiment with five price variants:
History-EWMA:
Exponentially-weighted moving average. The bid history, H t , is usedto define price p t in period t , computed as p t := λ ξ ( H t ) + (1 − λ ) p t − , where λ ∈ (0 ,
1] is a hain: An Online Double Auction smoothing constant and ξ ( H t ) is a statistic defined for bids and asks that enter the historyin period t . Experimentally we find that the mean statistic, ξ mean ( H t ) , of the absolutevalues of bids and asks that enter the history performs well with λ of 0.05 or lower for mostscenarios that we test. For cases in which ξ ( H t ) is not well-defined because of too few (orzero) new bids or asks, then we set p t := p t − . History-median:
Compute price p t from a statistic over a fixed-size window of the mostrecent history, p t := ξ ( H t , ∆) where ∆ is the window-size, i.e. defining bids introducedto history H t in periods [ t − ∆ , . . . , t ]. Experimentally, we find that the median statistic, ξ median ( H t , ∆) , of the absolute bid and ask values performs well for the scenarios we test,with the window size depending inversely with the volatility of agents’ valuations. Typically,we observe optimal window sizes of 20 and 150, depending on volatility. For cases in which ξ ( H t , ∆) is not well-defined because of too few (or zero) new bids or asks, then we set p t := p t − . History-clearing:
Identical to the history-median rule except the statistic ξ ( H t , ∆) isdefined as ( b m − s m ) / b m and s m are the lowest value pair of trades that would beexecuted in the efficient (value-maximizing) trade given all bids and asks to enter history H t in periods [ t − ∆ , . . . , t ]. Empirically, we find similar optimal window sizes for history-clearing as for history-median. History-McAfee:
Define the statistic ξ ( H t , ∆) to represent the McAfee price, defined inSection 4.2, for the bids in H t had they all simultaneously arrived. Fixed price:
This simple rule computes a single fixed price p t := p ∗ for all trading periods,with the price optimized offline to maximize the average-case efficiency of the dynamic DAgiven Chain and the associated single-period matching rule that leverages price p ∗ as thecandidate trading price.For each pricing variant, procedure Match (see Figures 3–4) is used to determine whichbids win (at price p t ), which lose, and, of those that lose, which satisfy the strong no-trade predicate. The subroutine used to determine the current price is referred to as determineprice in Match . We provide as input to
Match the set E t in addition to( H t , b t , s t ) because Match also constructs the strong no-trade set, and E t is used exclu-sively for this purpose.The proof of the following lemma is technical and is postponed until the Appendix. Lemma 1
Procedure
Match defines a valid strong no-trade construction.
Theorem 1
Procedure
Match defines a well-defined matching rule and a valid strong no-trade construction.
Proof:
No-deficit, feasibility, and individual-rationality are immediate by the constructionof
Match since bids and asks are added to matched in pairs, with the same payment, andonly if the payment is less than or equal to their value. Truthfulness is also easy to see: theorder with which a bid (or ask) is selected is independent of its bid price, and the price itfaces, when selected, is independent of its bid. If the price is less than or equal to its bid,then whether or not it trades depends only on its order. The rest of the claim follows fromLemma 1. (cid:3) redin, Parkes and Duong function
Match ( H t , b t , s t , E t )matched := ∅ , lose := ∅ , NT t := ∅ , SNT t := ∅ stop := false p t := determineprice( H t ) while ¬ stop do i := 0, j := 0, checked B := ∅ , checked S := ∅ while ((checked B ⊂ b t )&( i =0)) ∨ ((checked S ⊂ s t )&( j =0)) doif ( i = 0)&( j = 0) then k := random( b t \ checked B S s t \ checked S ) else if ( i = 0) then k := random( b t \ checked B ) else if ( j = 0) then k := random( s t \ checked S ) end ifif ( k ∈ b t ) then checked B := checked B ∪ { k } if ( b k ≥ p t ) then i := k end ifelse checked S := checked S ∪ { k } if ( s k ≥ − p t ) then j := k end ifend ifend whileif ( i = 0)&( j = 0) then matched := matched S { ( i, j ) } lose := lose S (checked B \ { i } ) S (checked S \ { j } ) b t := b t \ checked B , s t := s t \ checked S else stop := true end ifend whileend function Figure 3: The procedure used for single-period matching in applying
Chain to the price-based rules. The algorithm continues in Figure 4. hain: An Online Double Auction function
Match (continued)( H t , b t , s t , E t ) if ( i = 0)&( j = 0) then ⊲ Ilose := lose S s t , NT t := b t if ( ∃ k ∈ b t · (( b k ≥ p t )&( ˆ d k = t ))) ∨ ( ∀ k ∈ s t · ( ˆ d k = t )) then SNT t := b t else ⊲ I-aSNT t := b t \ checked B end ifelse if ( j = 0)&( i = 0) then ⊲ IIlose := lose S b t , NT t := s t if ( ∃ k ∈ s t · (( s k ≥ − p t )&( ˆ d k = t ))) ∨ ( ∀ k ∈ b t · ( ˆ d k = t )) then SNT t := s t else SNT t := s t \ checked S end ifelse if ( i = 0)&( j = 0) then ⊲ IIINT t := b t S s t if ( ∀ k ∈ b t · ( ˆ d k = t )) ∨ ( ∀ k ∈ s t · ( ˆ d k = t )) then SNT t := b t S s t end ifend ifend function Figure 4: Continuing procedure from Figure 3 for single-period matching in applying
Chain to the price-based rules.
Example 4 (i) Bid b t = { } , ask s t = {− } , indexed { , } and price p t = 9 . The outer while loop in Figure 3 terminates with j = 2 and i = 0 in Case II. The bid is marked asa loser while NT t = { } . If the bid will depart immediately, then SNT t = { } , otherwise SNT t = ∅ .(ii) Bid b t = { } , asks s t = {− , − } , indexed { , , } , and price p t = 9 . Suppose thatask 2 is selected before ask 3 in the outer while loop. Then the loop terminates with j = 2 and i = 0 in Case II and NT t = { , } . Suppose the bid and asks leave the market laterthan this period. Then SNT t = { } because checked S = { } .(iii) Bid b t = { } and ask s t = {− } , indexed { , } , price p t = 9 and both the bid andthe ask is patient. The outer while loop terminates with i = 0 and j = 0 in Case III sothat NT t = { , } . However, SNT t = ∅ . Each one of these rules determines which bids match in the current period through pricecompetition between the active bids. We present three variations:
McAfee, Windowed-McAfee and Active-McAfee . The latter two rules are hybrid rules in that they leverage redin, Parkes and Duong history of past offers, in smoothing prices generated by the competition-based matchingrules.
McAfee:
Use the static DA protocol due to McAfee as the matching rule. Let B denotethe set of bids and S denote the set of asks. If min( | B | , | S | ) < , then there is no trade.Otherwise, first insert two dummy bids with value {∞ , } and two dummy asks with value { , −∞} into the set of bids and asks. Let b ≥ b ≥ . . . ≥ b m and s ≥ s ≥ . . . ≥ s n . . . denote the bid and ask values with ( b , s ) denoting dummy pair ( ∞ ,
0) and ( b m , s n )denoting dummy pair (0 , −∞ ) and ties otherwise broken at random. Let m ≥ b m + s m ≥ b m +1 + s m +1 <
0. When m ≥ , consider the following two cases: • (Case I) If price p m +1 = b m +1 − s m +1 ≤ b m and − p m +1 ≤ s m then the first m bids andasks trade and payment p m +1 is collected from each winning buyer and made to eachwinning seller. • (Case II) Otherwise, the first m − b m is collectedfrom each winning buyer and payment − s m is made to each winning seller.To define NT t , replace a bid that does not trade with a bid reporting a very large valueand see whether this bid trades. To determine whether trade is possible for an ask thatdoes not trade: replace the ask with an ask reporting value ǫ >
0, some small ǫ . Saythat there is a quorum if and only if there are at least two bids and at least two asks, i.e.min( | b t | , | s t | ) ≥
2. Define strong no-trade as follows: set SNT t := NT t = b t ∪ s t whenthere is no quorum and SNT t := ∅ otherwise. Lemma 2
For any bid b i in the McAfee matching rule, then for any other bid (or ask) j there is some bid ˆ b i that will make trade possible for bid (or ask) j when there is a quorum. Proof:
Without loss of generality, suppose there are three bids and three asks. Label thebids ( a, c, e ) and the asks ( b, d, f ), both ordered from highest to lowest so that ( a, b ) is themost competitive bid-ask pair. Proceed by case analysis on bids. The analysis is symmetricfor asks and omitted. Let tp ( i ) ∈ { , } denote whether or not trade is possible for bid i , sothat i ∈ NT t ⇔ tp ( i ) = 0. For bid a : when b ≥ − ( a − d ) / tp ( c ) = tp ( e ) = 1 and thisinequality can always be satisfied for a large enough a; when a ≥ ( c − d ) / tp ( b ) = 1and when a ≥ ( c − b ) / tp ( d ) = tp ( f ) = 1, and both of these inequalities are satisfiedfor a large enough a . For bid c : when b ≥ − ( c − d ) / tp ( a ) = 1 and when, in addition, c > a , then tp ( e ) = 1 and each one of these inequalities are satisfied for a large enough c ;similarly when c ≥ ( a − d ) / tp ( b ) = 1 and when c ≥ ( a − b ) / tp ( d ) = tp ( f ) = 1.Analysis for bid e follows from that for bid c . (cid:3) Lemma 3
The construction for strong no-trade is valid and there is no valid strong no-trade construction that includes more than one losing bid or ask that will not depart in thecurrent period for any period in which there is a quorum.
Proof:
To see that this is a valid construction, notice that strong no-trade condition (a)holds since any bid (or ask) is always in both NT t and SNT t . Similarly, condition (b)trivially holds (with the other bids and asks remaining in SNT t even if any bid is not hain: An Online Double Auction present in the market). To see that this definition is essentially maximal, consider now thatmin( | b t | , | s t | ) ≥
2. For contradiction, suppose that two losing bids { i, j } with departureafter the current period are designated as strong no-trade. But, strong no-trade condition(b) fails because of Lemma 2 because either bid could have submitted an alternate bid pricethat would remove the other bid from NT t and thus necessarily also from SNT t . (cid:3) The construction offered for SNT t cannot be extended even to include one agent selectedat random from the set i ∈ NT t that will not depart immediately, in the case of a quorum.Such a construction would fail strong no-trade condition (b) when the set NT t containsmore than one bid (or ask) that does not depart in the current period, because bid i ’sabsence from the market would cause some other agent to be (randomly) selected as SNT t . Windowed-McAfee:
This myopic matching rule is parameterized on window size ∆.Augment the active bids and asks with the bids and asks introduced to the history H t inperiods t ′ ∈ { t − ∆ + 1 , . . . , t } . Run McAfee with this augmented set of bids and asks anddetermine which of these bids and asks would trade. Denote this candidate set C . Someactive agents identified as matching in C may not be able to trade in this period because C can also contain non-active agents.Let B ′ and S ′ denote, respectively, the active bids and active asks in set C . Windowed-McAfee then proceeds by picking a random subset of min( | B ′ | , | S ′ | ) bids and asks to trade.When | B ′ | 6 = | S ′ | , then some bids and asks will not trade.Define strong no-trade for this matching rule as:(i) if there are no active asks but active bids, then SNT t := b t (ii) if there are no active bids but active asks, then SNT t := s t (iii) if there are fewer than 2 asks or fewer than 2 bids in the augmented bid set, thenSNT t := b t ∪ s t ,and otherwise set SNT t := ∅ . In all cases it should be clear that SNT t ⊆ NT t . Lemma 4
The strong no-trade construction for windowed-McAfee is valid.
Proof:
That this is a valid SNT criteria in case (iii) follows immediately from the validityof the SNT criteria for the standard McAfee matching rule. Consider case (i). Case (ii)is symmetric and omitted. For strong no-trade condition (a), we see that all bids i ∈ NT t and also i ∈ SNT t , and whether or not they are designated strong no-trade is independentof their own bid price but simply because there are no active asks. Similarly, for strongno-trade condition (b), we see that all bids (and never any asks) are in SNT t whatever thebid price of any particular bid (and even whether or not it is present). (cid:3) Empirically, we find that the efficiency of Windowed-McAfee is sensitive to the size of H t , but that frequently the best choice is a small window size that includes only the activebids. Active-McAfee:
Active-McAfee augments the active bids and asks to include all un-expired but traded or priced-out offers. It proceeds as in Windowed-McAfee given thisaugmented bid set. redin, Parkes and Duong
We next provide two stylized examples to demonstrate the matching performed by
Chain using both a price-based and a competition-based matching rule. For both examples, weassume a maximal patience of K = 2. Moreover, while we describe when Chain determinesthat a bid or an ask trades, remember that a winning buyer is not allocated the good untilits reported departure and a winning seller does not receive payment until its reporteddeparture.
Example 5
Consider
Chain using an adaptive, price-based matching rule. The particulardetails of how prices are determined are not relevant. Assume that the prices in periods 1and 2 are { p , p } = { , } and the maximal patience is three periods. Now consider period3 and suppose that the order book is empty at the end of period 2 and that the bids and asksin Table 1 arrive in period 3. B S i ˆ w i ˆ d i ˆ d i − K q i p i SNT? i ˆ w i ˆ d i ˆ d i − K q i p i SNT? b * 15 4 2 7 7 N s -1 4 2 -7 n/a Y b * 10 3 1 8 8 N s * -3 5 3 −∞ -6.5 N b s -4 3 1 -7 n/a Y b −∞ n/a N s * -5 4 2 -7 -6.5 N s -10 5 3 −∞ n/a YTable 1: Bids and asks that arrive in period 3. Bids { b , b } match with asks { s , s } (as indicatedwith a *). Bid b is priced-out upon admission because q b > ˆ w b (indicated with a strike-through). The admission price is q i and the payment made by an agent that trades is p i . Column ‘SNT?’ indicates whether or not the bid or ask satisfies the strong no-tradepredicate. Asks { s , s } survive into the next period because they are in SNT and have d i > Bids { b , b , b } and asks { s , .., s } are admitted. Bid b is priced out because q b =max( p , p , −∞ ) = max(8 , , −∞ ) = 8 > ˆ w b = 7 by Eq. (3). Note that b and s areadmitted despite low bids (asks) because they have maximal patience and their admissionprices are −∞ . Now, suppose that p := 6 . is defined by the matching rule and considerapplying Match to the admitted bids and asks.Suppose that the bids are randomly ordered as ( b , b , b ) and the asks as ( s , s , s , s , s ) . Bid b is picked first but priced-out because ˆ w b = 6 < p = 6 . . Bid b is tentatively accepted ( ˆ w b = 10 ≥ p = 6 . ) and then ask s is accepted ( w s = − ≥ p = − . ). Bid b is matched with ask s , with payment max( q b , p ) = max(8 , .
5) = 8 for b by Eq. (4) and payment max( q s , p ) = max( −∞ , − .
5) = − . for s . Bid b is thententatively accepted ( ≥ . ) and then matched with ask s , which is accepted because − ≥ − . . The payments are max(7 , .
5) = 7 for b and max( −∞ , − .
5) = − . for s .Ask s expires but asks s and s survive and are marked i ∈ SNT in this period becausethey were never offered the chance to match with any bid. These asks will be active in period4. Note the role of the admission price in truthfulness. Had bid b delayed arrival untilperiod 4, its admission price would be max( p , p , −∞ ) = max(7 , .
5) = 7 and its payment hain: An Online Double Auction in period 4 (if it matches) at least 7. Similarly, had ask s delayed arrival, then its admissionprice would be max( − , − . , −∞ ) = − . and the maximal payment it can receive in period4 is 6.5. Example 6
Consider
Chain using the McAfee-based matching rule with K = 3 and withthe same bids and asks arriving in period 3. Suppose that the prices in periods 1 and 2 thatwould have been faced by a buyer are { p b , p b } = { , } and { p s , p s } = {− , − } for a seller.These prices are determined by inserting an additional bid (with value ∞ ) or an additionalask (with value 0) into the order books in each of periods 1 and 2. We will illustrate thisfor period 3. Consider now the bids and asks in period 3 in Table 2. B S i w i d i d i − K q i p i SNT? i w i d i d i − K q i p i SNT? b * 15 4 2 7 7 N s * -1 4 2 -6 -4 N b * 10 3 1 8 8 N s * -3 5 3 −∞ -4 N b s -4 3 1 -6 n/a N b −∞ n/a N s -5 4 2 -6 n/a N s -10 5 3 −∞ n/a NTable 2: Bids and asks that arrive in period 3. Bids { b , b } match with asks { s , s } (as indicatedwith a *). Bid b is priced-out upon admission because q b > ˆ w b . The admission price is q i and the payment made by an agent that trades is p i . Column ‘SNT?’ indicates whetheror not the bid or ask satisfies the strong no-trade predicate. No asks or bids survive intothe next period. As before bid b is not admitted. The myopic matching rule now runs the (static) McAfeeauction rule on bids { b , b , b } and asks { s , .., s } . Consider bids and asks in decreasingorder of value, the last efficient trade is indexed m = 3 with ˆ w b + ˆ w s = 6 − ≥ . But p m +1 = (0 − ( − / . (inserting a dummy bid with value 0 as described in Section 4.2).Price − p m +1 = − . > s = − and this trade cannot be executed by McAfee. Instead,buyers { b , b } trade and face price p bm = ˆ w b = 6 and sellers { s , s } trade and face price p sm = ˆ w s = − . Bids b and asks { s , s , s } are priced-out and do not survive into thenext round. Ultimately, payment max( q b , p bm ) = max(7 ,
6) = 7 is collected from buyer b and payment max( q b , p bm ) = max(8 ,
6) = 8 is collected from buyer b . For sellers, payment max( − , −
4) = − and max( −∞ , −
4) = − for s and s respectively.The prices p b and p s that are used in Eq. (3) to define the admission price for bids andasks with arrivals in periods 4 and 5 are determined as follows. For the buy-side price, weintroduce an additional bid with bid-price ∞ . With this the bid values considered by McAfeewould be ( ∞ , , , , and the ask values would be ( − , − , − , − , − , where a dummybid with value 0 is included on the buy-side. The last efficient pair to trade is m = 4 with − ≥ and p m +1 = (0 − ( − / , which satisfies this bid-ask pair. Therefore thebuy-side price, p b := 5 . On the sell-side, we introduce an additional ask with ask-price so that the bid values considered by McAfee are (15 , , , (again, with a dummy bidincluded) and the ask values are (0 , − , − , − , − , − . This time m = 3 and the lastefficient pair to trade is − ≥ . Now p m +1 = (0 − ( − / and this price does not redin, Parkes and Duong satisfy s , with − p m +1 > s and price p sm +1 = s = − is adopted. Therefore the sell-sideprice, p s := − .Again, we can see that bidder 1 cannot improve its price by delaying its entry until period4. The admission price for the bidder would be max( p b , p b ) = max(7 , p b ) ≥ and thus itspayment in period 4, if it matches, will be at least 7. Similarly for ask s , which would faceadmission price max { p s , p s } = max {− , − } = − and can receive a payment of at most 4in period 4. We leave it as an exercise for the reader to verify that p s = − if ask s delaysits arrival until period 4 (in comparison, p s = − when ask s is truthful). Because the McAfee-based pricing scheme computes a price and clears the order bookfollowing every period in which there are at least two bids and two asks, the bid activityperiods tend to be short in comparison to the adaptive, price-based rules where orders canbe kept active longer when there is an asymmetry in the number of bids and asks in themarket. In fact, one interesting artifact that occurs with adaptive, price-based matchingrules is that the admission-price and SNT can perpetuate this kind of bid-ask asymmetry.Once the market has more asks than bids, SNT becomes likely for future asks, but not bids.Therefore, bids are much more likely than asks to be immediately priced out of the marketby failing to meet the admission price constraint.
5. Theoretical Analysis: Truthfulness, Uniqueness, and JustifyingBounded-Patience
In this section we prove that
Chain combined with a well-defined matching rule and a validstrong no-trade construction generates a truthful, no-deficit, feasible and individual-rationaldynamic DA. In Section 5.2, we establish that uniqueness of
Chain amongst dynamic DAsthat are constructed from single-period matching rules as building blocks. In Section 5.3,we establish the importance of the existence of a maximal bound on bidder patience bypresenting a simple environment in which no truthful, no-deficit DA can implement even asingle trade despite the number of efficient trades can be increased without bound.
It will be helpful to adopt a price-based interpretation of a valid single-period matching rule.Given rule M mr , define an agent-independent price , z i ( H t , A t \ i, ω ) ∈ R where A t = b t ∪ s t ,such that for all i , all bids b t , all asks s t , all history H t , and all random events ω ∈ Ω. Wehave: (A1) ˆ w i − z i ( H t , A t \ i, ω ) > ⇒ π mr,i ( H t , b t , s t , ω ) = 1, and ˆ w i − z i ( H t , A t \ i, ω ) < ⇒ π mr,i ( H t , b t , s t , ω ) = 0(A2) payment x mr,i ( H t , b t , s t , ω ) = z i ( H t , A t \ i, ω ) if π mr,i ( H t , b t , s t , ω ) = 1 and x mr,i ( H t , b t , s t , ω ) = 0 otherwise
4. We can check that p b := 6 in this case. Suppose that bidder 1 were not present in period 3. Now considerintroducing an additional bid with value ∞ so that the bids values are {∞ , , , } (with a dummy bid)with ask values {− , − , − , − , − } . Then m = 3 and p m +1 = (0 − ( − / .
5, which does notsupport the trade between bid b and ask s . Instead, p bm = ˆ w b = 6 is adopted, and we would have p b := 6. Of course, this is exactly the price determined by McAfee for bid b in period 3 when the bidderis truthful. hain: An Online Double Auction The interpretation is that there is an agent-independent price, z i ( H t , A t \ i, ω ), that isat least ˆ w i when the agent loses and no greater than ˆ w i otherwise. In particular, z i ( H t , A t \ i, ω ) = ∞ when i ∈ NT t . Although an agent’s price is only explicit in a matching rule whenthe agent trades, it is well known that such a price exists for any truthful, single-parametermechanism; e.g., see works by Archer and Tardos (?) and Goldberg and Hartline (?). Moving forward we adopt price z i to characterize the matching rule used as a buildingblock for Chain , and assume without loss of generality properties (A1) and (A2).Given this, we will now establish the truthfulness of
Chain by appeal to a price-basedcharacterization due to Hajiaghayi et al. (?) for truthful, dynamic mechanisms. We state(without proof) a variant on the characterization result that holds for stochastic policies( π, x ) and strong -truthfulness. The theorem that we state is also specialized to our DAenvironment. We continue to adopt ω ∈ Ω to capture the realization of stochastic eventsinternal to the mechanism:
Theorem 2 (?) A dynamic DA M = ( π, x ) , perhaps stochastic, is strongly truthful formisreports limited to no early-arrivals if and only if, for every agent i , all ˆ θ i , all θ − i , andall random events ω ∈ Ω , there exists a price p i (ˆ a i , ˆ d i , θ − i , ω ) such that:(B1) the price is independent of agent i ’s reported value(B2) the price is monotonic-increasing in tighter [ a ′ i , d ′ i ] ⊂ [ˆ a i , ˆ d i ] (B3) trade π i (ˆ θ i , θ − i ) = 1 whenever p i (ˆ a i , ˆ d i , θ − i , ω ) < ˆ w i and π i (ˆ θ i , θ − i ) = 0 whenever p i (ˆ a i , ˆ d i , θ − i , ω ) > ˆ w i , and the trade is performed for a buyer upon its departure period ˆ d i .(B4) the agent’s payment is x i (ˆ θ i , θ − i ) = p i (ˆ a i , ˆ d i , θ − i , ω ) when π i (ˆ θ i , θ − i ) = 1 , with x i (ˆ θ i , θ − i ) = 0 otherwise, and the payment is made to a seller upon its departure, ˆ d i .where random event ω is independent of the report of agent i in as much as it affects theprice to agent i . Just as for the single-period, price-based characterization, the price p i ( a i , d i , θ − i , ω ) neednot always be explicit in Chain . Rather, the theorem states that given any truthful dynamicDA, such as
Chain , there exists a well-defined price function with these properties of value-independence (B1) and arrival-departure monotonicity (B2), and such that they define thetrade (B3) and the payment (B4).To establish the truthfulness of
Chain , we prove that it is well-defined with respect tothe following price function: p i (ˆ a i , ˆ d i , θ − i , ω ) = max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p i (ˆ a i , ˆ d i , θ − i , ω )) , (5)where ˇ q (ˆ a i , ˆ d i , θ − i , ω ) = max t ∈ [ ˆ d i − K, ˆ a i − ,i/ ∈ SNT t ( z i ( H t , A t \ i, ω ) , −∞ ) (6)
5. A single-parameter mechanism is one in which the private information of an agent is limited to onenumber. This fits the single-period matching problem because the arrival and departure informationis discarded. Moreover, although there are both buyers and sellers, the problem is effectively single-parameter because no buyer can usefully pretend to be a seller and vice versa . redin, Parkes and Duong and ˇ p (ˆ a i , ˆ d i , θ − i , ω ) = (cid:26) z i ( H t ∗ , A t ∗ \ i, ω ) , if decision ( i ) = 1+ ∞ , otherwise (7)where decision ( i ) = 0 indicates that i ∈ SNT t for all t ∈ [ˆ a i , ˆ d i ] and decision ( i ) = 1otherwise, and where t ∗ ∈ [ˆ a i , ˆ d i ] is the first period in which i / ∈ SNT t . We refer to thisas the decision period . Term ˇ q (ˆ a i , ˆ d i , θ − i , ω ) denotes the admission price, and is defined onperiods t before the agent arrives for which i / ∈ SNT t had it arrived in that period. Notecarefully that the rules of Chain are implicit in defining this price function. For instance,whether or not i ∈ SNT t in some period t depends, for example, on the other bids thatremain active in that period.We now establish conditions (B1)–(B4). The proofs of the technical lemmas are deferreduntil the Appendix. The following lemma is helpful and gets to the heart of the strong no-trade concept. Lemma 5
The set of active agents (other than i ) in period t in Chain is independent of i ’s report while agent i remains active, and would be unchanged if i ’s arrival is later thanperiod t . The following result establishes properties (B1) and (B2).
Lemma 6
The price constructed from admission price ˇ q and post-arrival price ˇ p is value-independent and monotonic-increasing when the matching rule in Chain is well-defined,the strong no-trade construction is valid, and agent patience is bounded by K . Having established properties (B1) and (B2) for price function p i (ˆ a i , ˆ d i , θ − i , ω ), we justneed to establish (B3) and (B4) to show truthfulness. The timing aspect of (B3) and (B4),which requires that the buyer receives an item and the seller receives its payment uponreported departure, is already clear from the definition of Chain . Theorem 3
The online DA
Chain is strongly truthful, no-deficit, feasible and individual-rational when the matching rule is well-defined, the strong no-trade construction is valid,and agent patience is bounded by K . Proof:
Properties (B1) and (B2) follow from Lemma 6. The timing aspects of (B3) and(B4) are immediate. To complete the proof, we first consider (B3). If ˇ q > ˆ w i , then agent i is priced out at admission by Chain because this reflects that z i ( H t , A t \ i, ω ) > ˆ w i insome t ∈ [ ˆ d i − K, ˆ a i −
1] with i / ∈ SNT t , and thus the bid would lose if it arrived in thatperiod (either because it could trade, but for a payment greater than its reported value, orbecause i ∈ NT t ). Also, if there is no decision period, then ˇ p = ∞ , which is consistent with Chain , because there is no bid price at which a bid will trade when i ∈ SNT t for all periods t ∈ [ˆ a i , ˆ d i ]. Suppose now that there is a decision period t ∗ and ˇ q < ˆ w i . If ˇ p > ˆ w i , then thereshould be no trade. This is the case in Chain , because the price z i ( H t ∗ , A t ∗ \ i, ω ) in t ∗ isgreater than ˆ w i and thus the agent is priced-out. If ˇ p < ˆ w i then the bid should trade andindeed it does, again because the price z i in that period satisfies (A1) and (A2) with respectto the matching rule. Turning to (B4), it is immediate that the payments collected in Chain hain: An Online Double Auction are equal to price p i (ˆ a i , ˆ d i , θ − i , ω ), because if bid i trades then p i (ˆ a i , ˆ d i , θ − i , ω ) ≤ ˆ w i andthus ˇ q ≤ ˆ w i and ˇ p ≤ ˆ w i . The admission price q (ˆ a i , ˆ d i , θ − i , ω ) = ˇ q (ˆ a i , ˆ d i , θ − i , ω ) when ˇ q ≤ ˆ w i because price z i is well-defined by properties (A1) and (A2). Similarly, the payment p t ∗ defined by the matching rule in Chain in the decision period is equal to ˇ p .That Chain is individual-rational and feasible follows from inspection.
Chain is no-deficit because the payment collected from every agent (whether a buyer or a seller) is atleast that defined by a valid matching rule in the decision period t ∗ (it can be higher whenthe admission price is higher than this matching price), the matching rules are themselvesno-deficit, and because the auctioneer delays making a payment to a seller until its reporteddeparture but collects payment from a buyer immediately upon a match. (cid:3) We remark that information can be reported to bidders that are not currently partici-pating in the market, for instance to assist in their valuation process. If this informationis delayed by at least the maximal patience of a bidder, so that the bid of a current biddercannot influence the other bids and asks that it faces, then this is without any strategicconsequences. Of course, without this constraint, or with bidders that participate in themarket multiple times, the effect of such feedback would require careful analysis and bringus outside of the private values framework.
In what follows, we establish that
Chain is unique amongst all truthful, dynamic DAs thatadopt well-defined, myopic matching rules as simple building blocks. For this, we definethe class of canonical, dynamic DAs , which take a well-defined single period matching rulecoupled with a valid strong no-trade construction, and satisfy the following requirements:(i) agents are active until they are matched or priced-out,(ii) agents participate in the single-period matching rule while active(iii) agents are matched if and only if they trade in the single-period matching rule.We think that these restrictions capture the essence of what it means to construct adynamic DA from single-period matching rules. Notice that a number of design elements areleft undefined, including the payment collected from matched bids, when to mark an activebid as priced-out, what rule to use upon admission, and how to use the strong no-tradeinformation within the dynamic DA. In establishing a uniqueness result, we leverage thenecessary and sufficient price-based characterization in Theorem 2, and exactly determinethe price function p i (ˆ a i , ˆ d i , θ − i , ω ) to that defined in Eq. (4) and associated with Chain .The proofs for the two technical lemmas are deferred until the Appendix.
Lemma 7
A strongly truthful, canonical dynamic DA must define price p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ z i ( H t ∗ , A t ∗ \ i, ω ) where t ∗ is the decision period for bid i (if it exists). Moreover, the bidmust be priced-out in period t ∗ if it is not matched. Lemma 8
A strongly truthful, canonical and individual-rational dynamic DA must defineprice p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , and a bid with ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) must be priced-out upon admission. redin, Parkes and Duong Theorem 4
The dynamic DA algorithm
Chain uniquely defines a strongly truthful,individual-rational auction among canonical dynamic DAs that only designate bids as priced-out when necessary.
Proof:
If there is no decision period, then we must have p i (ˆ a i , ˆ d i , θ − i , ω ) = ∞ , by canonical(iii) coupled with (B3). Combining this with Lemmas 7 and 8, we have p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )). We have also established that a bid must be priced-out if its bid value is less than the admission price, or it fails to match in its decisionperiod. Left to show is that the price is exactly as in Chain , and that a bid is admittedwhen its value ˆ w i ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) and retained as active when it is in the strong no-trade set. The last two control aspects are determined once we choose a rule that “onlydesignates bids as priced-out when necessary.” We prefer to allow a bid to remain activewhen this does not compromise truthfulness or individual-rationality. Finally, supposefor contradiction that p ′ = p i (ˆ a i , ˆ d i , θ − i , ω ) > max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )). Thenan agent with max(ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )) < w i < p ′ would prefer to bid ˆ w i =ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , ˇ p (ˆ a i , ˆ d i , θ − i , ω )) − ǫ and avoid winning – otherwise its payment would begreater than its value. (cid:3) Chain depends on a maximal bound on patience used to calculate the admission price facedby a bidder on entering the market with Eq. (3). To motivate this assumption about theexistence of a maximal patience, we construct a simple environment in which the number oftrades implemented by a truthful, no-deficit DA can be made an arbitrarily small fractionof the number of efficient trades with even a small number of bidders having potentially un-bounded patience. This illustrates that a bound on bidder patience is required for dynamicDAs with reasonable performance.In achieving this negative result, we impose the additional requirement of anonymity ,This anonymity property is already satisfied by
Chain , when coupled with matching rulesthat satisfy anonymity, as is the case with all the rules presented in Section 4. In defininganonymity, extend the earlier definition of a dynamic DA, M = ( π, x ), so that allocationpolicy π = { π t } t ∈ T defines the probability π ti ( θ ≤ t ) ∈ [0 ,
1] that agent i trades in period t given reports θ ≤ t . Payment, x = { x t } t ∈ T , continues to define the payment x ti ( θ ≤ t ) by agent i in period t , and is a random variable when the mechanism is stochastic. Definition 8 (anonymity)
A dynamic DA, M = ( π, x ) is anonymous if allocation policy π = { π t } t ∈ T defines probability of trade π ti ( θ ≤ t ) in each period t that is independent ofidentity i and invariant to a permutation of ( θ ≤ t \ i ) and if the payment x ti ( θ ≤ t ) , contingenton trade by agent i , is independent of identity i and invariant to a permutation of ( θ ≤ t \ i ) . We now consider the following simple environment. Informally, there will be a randomnumber of high-valued phases in which bids and asks have high value and there might be asingle bidder with patience that exceeds that of the other bids and asks in the phase. Thesehigh-valued phases are then followed by some number, perhaps zero, of low-valued phaseswith bounded-patience bids and asks. Formally, there are T h ≥ high-valued phases(a random variable, unknown to the auction), each of duration L ≥ k ∈ { , , . . . , T h − } and each with: hain: An Online Double Auction • N or N − kL, ( k + 1) L, v H ), • kL, d, αv H ) for some mark-up parameter, α > d ∈ T , • N asks with type (1 + kL, ( k + 1) L, − ( v H − ǫ )),followed by some number (perhaps zero) of low-valued phases, also of duration L , andindexed k ∈ { T h , . . . , ∞} , with: • N or N − kL, ( k + 1) L, v L ) • N asks with type (1 + kL, ( k + 1) L, − ( v L − ǫ )),where N ≥
1, 0 < v L < v H , and bid-spread parameter ǫ >
0. Note that any phase canbe the last phase, with no additional bids or asks arriving in the future.
Definition 9 (reasonable DA)
A dynamic DA is reasonable in this simple environmentif there is some parameterization of new bids, N ≥ , and periods-per-phase, L ≥ , forwhich it will execute at least one trade between new bids and new asks in each phase ,for any choice of high value v H , low value v L < v H , bid-spread ǫ > , mark-up α > , highpatience d . All of the dynamic DAs presented in Section 4 can be parameterized to make themreasonable for a suitably large N ≥ L ≥
1, and without the possibility of a bid withan unbounded patience.
Theorem 5
No strongly truthful, individual-rational, no-deficit, feasible, anonymous dy-namic DA can be reasonable when a bidder’s patience can be unbounded.
Proof:
Fix any N ≥ L ≥
1, and for the number of high-valued phases, T h ≥
1, set thedeparture of a high-patience agent to d = ( T h + 1) L . Keep v H > v L > ǫ >
0, and α > k = 0 and with N − , L, v H ), N of type (1 , L, − ( v H − ǫ )) and 1 agent of patient type,(1 , ( T h + 1) L, αv H ). If the patient bid deviates to (1 , L, v H ) , then the bids are all identical,and with probability at least 1 /N the bid would win by anonymity and reasonableness.Also, by anonymity, individual-rationality and no-deficit we have that the payment madeby any winning bid is the same, and must be p ′ ∈ [ v H − ǫ, v H ]. (If the payment had been lessthan this, the DA would run at a deficit since the sellers require at least this much paymentfor individual-rationality.) Condition now on the case that the patient bid would win if itdeviates and reports (1 , L, v H ). Suppose the bidder is truthful, reports (1 , ( T h + 1) L, αv H )but does not trade in this phase. But, if phase k = 0 is the last phase with new bids andasks, then the bid will not be able to trade in the future and for strong-truthfulness theDA would need to make a payment of at least αv H − v H = ( α − v H in a later phase toprevent the bid having a useful deviation to (1 , L, v H ) and winning in phase k = 0. But, if: N ǫ < ( α − v H , (8) redin, Parkes and Duong then the DA cannot make this payment without failing no-deficit (because N ǫ is an upper-bound on the surplus the auctioneer could extract from bidders in this phase withoutviolating individual-rationality). We will later pick values of α, ǫ and v H , to satisfy Eq. (8).So, the bid must trade when it reports (1 , ( T h + 1) L, αv H ), in the event that it would winwith report (1 , L, v H ), as “insurance” against this being the last phase with new bids andasks. Moreover, it should trade for payment, p ′ ∈ [ v H − ǫ, v H ], to ensure an agent with truetype (1 , L, v H ) cannot benefit by reporting (1 , ( T h + 1) L, αv H ).Now suppose that this was not the last phase with new bids and asks, and T h > k = 0 deviated and reported(1 + T h L, ( T h + 1) L, v L ). As before, this bid would win with probability at least 1 /N byanonymity and reasonableness, but now with some payment p ′′ ∈ [ v L − ǫ, v L ]. Conditionnow on the case that the patient bid would win, both with a report of (1 , L, v H ) and with areport of (1 + T h L, ( T h + 1) L, v L ). When truthful, it trades in phase k = 0 with payment atleast v H − ǫ . If it had reported (1 + T h L, ( T h + 1) L, v L ) , it would trade in phase k = T h forpayment at most v L . For strong truthfulness, the DA must make an additional payment tothe patient agent of at least ( v H − v L ) − ( v H − ( v H − ǫ )) = v H − v L − ǫ . But, suppose thatthe high and low values are such that,( T h + 1) N ǫ < v H − v L − ǫ. (9)Making this payment in this case would violate no-deficit, because ( T h +1) N ǫ is an upper-bound on the surplus the auctioneer can extract from bidders across all phases, includingthe current phase, without violating individual-rationality. But now we can fix any v L > ǫ < v L and choose v H > ( T h + 1) N ǫ + v L + ǫ to satisfy Eq. (9) and α > ( N ǫ/v H ) + 1to satisfy Eq. (8). Thus, we have proved that no truthful dynamic DA can choose a bid-ask pair to trade in period k = 0. The proof can be readily extended to show a similarproblem with choosing a bid-ask pair in any period k < T h , by considering truthful type of(1 + kL, ( T h + 1) L, αv H ). (cid:3) To drive home the negative result: notice that the number of efficient trades can beincreased without limit by choosing an arbitrarily large T h , and that no truthful, dy-namic DA with these properties will be able to execute even a single trade in each ofthese { , . . . , T h − } periods. Moreover, we see that only a vanishingly small fraction ofhigh-patience agents is required for this negative result. The proof only requires that atleast one patient agent is possible in all of the high-valued phases.
6. Experimental Analysis
In this section, we evaluate in simulation each of the
Chain -based DAs introduced inSection 4. We measure the allocative efficiency (total value from the trades), net efficiency(total value discounted for the revenue that flows to the auctioneer), and revenue to theauctioneer. All values are normalized by the total offline value of the optimal matching.For comparison we also implement several other matching schemes: the truthful, surplus-maximizing matching algorithm presented by Blum et al. (?), an untruthful greedy matchingalgorithm using truthful bids as input to provide an upper-bound on performance, and anuntruthful DA populated with simple adaptive agents that are modeled after the Zero-intelligence Plus trading algorithm that has been leveraged in the study of static DAs (?,?). hain: An Online Double Auction
Traders arrive to the market as a Poisson stream to exchange a single commodity at discretemoments. This is a standard model of arrival in dynamic systems, economic or otherwise.Each trader, equally likely to be a buyer or seller, arrives after the previous with an expo-nentially distributed delay, with probability density function (pdf): f ( x ) = λe − λx , x ≥ , (10)where λ > interarrival time , λ , is varied between 0.05 and 1.5; i.e., as the arrival intensity isvaried between 20 and . A single trial continues until at least 5,000 buyers and 5,000 sellershave entered the market. In our experiments we vary the maximal patience K between 2and 10. For the distribution on an agent’s activity period (or patience, d i − a i ), we considerboth a uniform distribution with pdf: f ( x ) = 1 K , x ∈ [0 , K ] , (11)and a truncated exponential distribution with pdf: f ( x ) = αe − αx , x ∈ [0 , K ] , (12)where α = − ln(0 . /K so that 95% of the underlying exponential distribution is less thanthe maximal patience. Both arrival time and activity duration are rounded to the nearestintegral time period. A trader who arrives and departs during the same period is assumedto need an immediate trade and is active for only one period.Each trader’s valuation represents a sample drawn at its arrival from a uniform distri-bution with spread 20% about the current mean valuation. (The value is positive for a bidand negative for an ask.) To simulate market volatility, we run experiments that vary theaverage valuation using Brownian motion, a common model for valuation volatility uponwhich many option pricing models are based (?). At every time period, the mean valuationrandomly increases or decreases by a constant multiplier, e ± γ , where γ is the approximatevolatility and varied between 0 and 0.15 in our experiments.We plot the mean efficiency of 100 runs for each experiment, with the same sets of bidsand asks used across all double auctions. All parameters of an auction rule are reoptimizedfor each market environment; e.g., we can find the optimal fixed price and the optimalsmoothing parameters offline given the ability to sample from the market model. We implement
Chain for the five price-based matching rules (history-clearing, history-median, history-McAfee, history-EWMA, and fixed-price) and the three competition-basedmatching rules (McAfee, active-McAfee, and windowed-McAfee).The price-based implementations keep a fixed-size set of the most recently expired,traded, or priced-out offers, H t . Offers priced-out by their admission prices are insertedinto H t prior to computing p t . The history-clearing metric computes a price to maximize thenumber of trades to agents represented by H t had they all been contemporary. The history-median metric chooses the price to be the median of the absolute valuation of the offers redin, Parkes and Duong in H t . The history-McAfee method computes the “McAfee price” for the scenario whereall agents represented by H t are simultaneously present. The EWMA metric computes anexponentially-weighted average of bids in the order that they expire, trade, or price out.The simulations initialize the price to the average of the mean buy and sell valuations. Iftwo bids expire during the same period, they are included in arbitrary order to the movingaverage.None of the metrics require more than one parameter, which is optimized offline withaccess to the model of the market environment. Parameter optimization proceeds by uni-formly sampling the parameter range, smoothing the result by averaging each result withits immediate neighbors. The optimization repeats twice more over a narrower range aboutthe smoothed maximum, returning the parameter that maximizes (expected) allocative effi-ciency. None of the price-based methods appeared to be sensitive to small ( < H t . With most simulations, the window size was chosen to be about 150 of-fers. For EWMA, the smoothing factor was usually chosen to be around 0.05 or lower. Thewindowed-McAfee matching rule, however, was extremely sensitive to window size for simu-lations with volatile valuations, and the search process frequently converged to suboptimallocal maxima.The admission price in the price-based methods is computed by first determiningwhether Match would check the value of the bid against bid price if the bid had arrived insome earlier period t ′ . Rather than simulate the entire Match procedure, it is sufficient todetermine the probability ρ i of this event. This is determined by checking the constructionof the strong no-trade sets in that earlier period. If SNT t ′ contains non-departing buyers(sellers), then the probability that an additional seller (buyer) would be examined is 1 and ρ i = 1. Otherwise the probability is equal to the ratio of the number of bids (asks) ex-amined not included in SNT t and one more than the total number of bids (asks) present.Finally, with probability ρ i the price the agent would have faced in period t ′ is defined as p t ′ ( − p t ′ for sellers), and otherwise it is −∞ . Here, p t ′ is the history-dependent price definedin period t ′ .The competition-based matching rules price out all non-trading bids at the end of eachperiod in which trade occurs (because of the definition of strong no-trade in that context).The admission prices are calculated by considering the price that a bid (ask) would havefaced in some period t ′ before its reported arrival. In such a period, the price for a bid (ask)is determined by inserting an additional bid (ask) with valuation ∞ (0) and applying thecompetition-based matching rule to that (counterfactual) state. From this we determinewhether the agent would win for its reported value, and if so what price it would face. We use a commercial integer program solver (CPLEX ) to compute the optimal offlinesolution, i.e. with complete knowledge about all offers received over time. In determiningthe offline solution we enforce the constraint that a trade can only be executed if the activityperiods of both buyer, i, and seller, j, overlap,( a i ≤ d j ) ∧ ( a j ≤ d i ) (13) hain: An Online Double Auction An integer-program formulation to maximize total value is:max X ( i,j ) ∈ overlap x ij ( w i + w j ) (14)s.t. 0 ≤ X i :( i,j ) ∈ overlap x ij ≤ , ∀ j ∈ ask ≤ X j :( i,j ) ∈ overlap x ij ≤ , ∀ i ∈ bid x ij ∈ { , } , ∀ i, j, where ( i, j ) ∈ overlap is a bid-ask pair that could potentially trade because they haveoverlapping arrival and departure intervals satisfying Eq. (13). The decision variable x ij ∈{ , } indicates that bid i matches with ask j . This provides the optimal, offline allocativeefficiency. We implement a greedy matching algorithm that immediately matches offers that yield non-negative budget surplus. This is a non-truthful matching rule but provides an additionalcomparison point for the efficiency of the other matching schemes. During each time period,the greedy matching algorithm orders active bids and asks by their valuations, exactly asthe McAfee mechanism does, and matches offers until pairs no longer generate positivesurplus. The algorithm’s performance allows us to infer the number of offers that theoptimal matching defers before matching and the amount of surplus lost by the McAfeemethod due to trade reduction and due to the additional constraint of admission pricing.
Blum et al. (?) derive a mechanism equivalent to our fixed-price matching mechanism,except that the price used is chosen from the cumulative distribution D ( x ) = 1 rα ln (cid:18) x − w min ( r − w min (cid:19) , (15)where r is the fixed point to the equation r = ln (cid:18) w max − w min ( r − w min (cid:19) , (16)and w min ≥ w max ≥ , ln( w max /w min )) withrespect to the optimal offline solution in an adversarial setting. We were interested to seehow will this performed in practice in our simulations. redin, Parkes and Duong To compare
Chain with the existing literature on continuous double auctions, we implementa DA that in every period sorts all active offers and matches the highest valued bids withthe lowest valued asks so long as the match yields positive net surplus. The DA prices eachtrading pair at the mean of the pair’s declared valuations.
Since the trade price dependson a bidder’s declaration, the market does not support truthful bidding strategies. We musttherefore adopt a method to simulate the behavior of bidding agents within this simple openoutcry market.
For this, we randomly assign each bid to one of several “protocol agents” that eachuse a modified ZIP trading algorithm, as initially presented by Cliff and Bruten (?) andimproved upon by Preist and van Tol (?). The ZIP algorithm is a common benchmarkused to compare learned bidding behavior in a simple double-auction trading environmentin which agents are present at once and adjust their bids in seeking a profitable trade. Weadapt the ZIP algorithm for use in our dynamic environment.In our experiments we consider five of these protocol agents. New offers are assigneduniformly at random to a protocol agent, which remains persistent throughout the simula-tion. Each offer is associated with a patience category, k ∈ { low, medium, high } , definedto evenly partition the range of possible offer patience. Each protocol agent, j , is definedwith parameters ( r j , β j , γ j ) and maintains a profit margin , µ kj , on each patience category k .Parameters ( β j , γ j ) control the adaptivity of the protocol agent in how it adjusts the targetprofit margin on an individual offer, with β j ∼ U (0 . , .
2) defining the offer-level learningrate and γ j ∼ U (0 . , .
8) defining the offer-level damping factor . Parameter r j ∈ [0 ,
1] isthe learning rate adopted for updating the profit margins.The protocol agents are trained over 10 trials and their final performance is measuredin the 11th trial. The learning rate decreases through the training session and depends onthe initial learning rate r j and the adjustment rate r + j . In period t ∈ { , . . . , t k end } of trial k ∈ { , . . . , T + 1 } , where T = 10 is the number of trials used for training and t k end is thenumber of periods in trial k , the learning rate is defined as: r j := 1 − r j + ( k − r + j + (cid:18) tt k end (cid:19) r + j ! (17)where r + j = (1 − r j ) / ( T + 1). We define r j := 0 .
7. The effect of this adjustment rule is that r j is initially 0.3, decreases during training, and trends to 0.0 as t → t end in trial k = 11.Within a given trial, upon assignment of a new offer i in patience category k , the protocolagent managing the offer initializes ( µ i ( t ) , δ i ( t )) := ( µ kj , µ i ( t ) represents the targetprofit margin for the offer and δ i ( t ) represents a profit-margin correction term. The targetprofit margin and the profit margin correction term are adjusted for offer i in subsequentperiods while the bid remains active.The target profit margin is used to define a bid price for the offer in each period whileit remains active: ˆ w i ( t ) := w i (1 + µ i ( t )) . (18) hain: An Online Double Auction At the end of a period in which an offer matches or simply expires, the profit margin µ kj for its patience category is updated as: µ kj := (1 − r j ) µ kj + r j µ i ( t ) , (19)where the amount of adaptivity depends on the learning rate r j . Because the profit marginon an offer decays over its lifetime, this update adjusts towards a small profit margin ifthe offer expires or took many periods to trade, and a larger profit margin otherwise. Thelong-term learning of a protocol agent occurs through the profit margin assigned to eachpatience category.At the start of a period each protocol agent also computes target prices for bids andasks in each patience category. These are used to drive an adjustment in the target profitmargin for each active bid and ask. Target prices τ kb ( t ) and τ ks ( t ) are computed as: τ kb ( t ) := (1 + η ) max i ∈ B k ( t − { ˆ w i ( t − } + ξ , if 0 > max i ∈ S ( t − { ˆ w i ( t − } + max i ∈ B k ( t − { ˆ w i ( t − } (1 − η ) max i ∈ B k ( t − { ˆ w i ( t − } − ξ , otherwise (20)and, τ ks ( t ) := (1 + η ) max i ∈ S k ( t − { ˆ w i ( t − } + ξ , if 0 > max i ∈ B ( t − { ˆ w i ( t − } + max i ∈ S k ( t − { ˆ w i ( t − } (1 − η ) max i ∈ S k ( t − { ˆ w i ( t − } − ξ , otherwise (21)where ξ, η ∼ U (0 , . B ( t −
1) and S ( t −
1) denote the set of active bids and asksin the market in period t − B k ( t −
1) and S k ( t − k . The target price on a bid in category k is set to something slightly greater than the most competitive bid in the previous roundwhen that bid could not trade, and slightly less otherwise. Similarly for the target price onasks, where these prices are negative, so that increasing the target price makes an ask morecompetitive.Target prices are used to adjust the target profit margin at the start of each period onall active offers that arrived in some earlier period, where the influence of target prices isthrough the profit-margin correction term: µ i ( t ) := ( ˆ w i ( t −
1) + δ i ( t )) w i − , (22)and the profit-margin correction term, δ i ( t ), is defined in terms of the target price τ ki ( t )(equal to τ kb ( t ) if i is a bid and τ ks ( t ) otherwise) as, δ i ( t ) := γ j δ i ( t −
1) + (1 − γ j ) β j ( τ ki ( t ) − ˆ w i ( t − , (23)where γ j and β j are the offer-level learning rates and damping factor. The value w i and the“-1” term in Eq. (22) provide normalization. Eq. (23) is the Widrow-Hoff (?) rule, designedto minimize the least mean square error in the profit margin and adopted here to mimicearlier ZIP designs. redin, Parkes and Duong Our experimental results show that market conditions drive DA choice. We compare al-locative efficiency, revenue, and net efficiency. All results are averaged over 100 trials.In experiments we found only minimal qualitative differences between the use of the twopatience distributions. The uniform patience distribution provides a slight increase in ef-ficiency over result using exponential patience, caused by a larger proportion of patientagents which relaxes somewhat the admission-price constraint in Eq. (3). For this reasonwe choose to report only results for the uniform patience distribution.While the performance of all methods are summarized in Table 3, we omit the perfor-mance of some markets from the plots to keep the presentation of results as clear as possible.We do not plot the price-based results for the median- or clearing-based prices because theperformance was typically around that of the performance of
Chain instantiated on thehistory-EWMA price. We do not plot the windowed-McAfee results because of inconsis-tent performance, and in most cases, upon manual inspection, it was optimal to choose thesmallest possible window size, i.e. including only active bids and making it equivalent toactive-McAfee.Our plots also leave out the performance of the Blum et al. (?) worst-case optimalmatching scheme because it was dominated by the fixed-price
Chain instantiation and inmany cases failed to yield any substantial surplus. We note here that the modeling assump-tion made by Blum et al. (?) is quite different than that in our work: they worry aboutperformance in an adversarial environment while we consider probabilistic environments.Our fixed-price
Chain mechanism operates essentially identically to the surplus-maximizingscheme of Blum et al. (?), except that
Chain can also use additional statistical informationto set the ideal price, rather than drawing the price from a distribution that is used toguarantee worst-case performance against an adversary. We defer the results for the Blumet al. (?) scheme to Table 3.Figures 5–8 plot results from two sets of experiments, one for high-patience/low-volatilityand one for low-patience/high-volatility, as we vary the inter-arrival time (and thus thearrival intensity), volatility and maximal patience. All plots are for allocative efficiencyexcept Figure 6, where we consider net efficiency. Active-McAfee is included on Figure 5,but not on any other plots because it did not improve upon the McAfee performance inthe other environments. To emphasize: the results for greedy provide an upper-bound onthe best possible performance because this is a non-truthful algorithm, simulated here withtruthful inputs.In Figure 5 (left) we see that from within the truthful DAs, the McAfee-based DA hasthe best efficiency for medium to low arrival intensities. There also is a general decreasein performance, relative to the optimal offline solution, as the arrival intensity falls. Thistrend, also observed with the greedy (non-truthful) DA, occurs because the
Chain schemeis myopic in that it matches as soon as the static DA building block finds a match, whileit is better to be less myopic when arrival intensity is low. The McAfee-based DAs are lesssensitive to this than other methods because they can aggressively update prices using theactive traders. The price-based DAs experience inefficiencies due to the lag in price updatesbecause they use only expired, traded, and priced-out offers to calculate prices. hain: An Online Double Auction A ll o c a t i v e E ff i c i en cy Inter-arrival Time(patience=6, volatility=0.01)greedyzipmcafeeewmaactive-mcafeefixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 A ll o c a t i v e E ff i c i en cy Inter-arrival Time(patience=2, volatility=0.08)greedymcafeeactive-mcafeeewmafixed-pricezip
Figure 5:
Allocative efficiency vs. inter-arrival time (1 / intensity) for several DAs. The leftplot shows high-patience, low-volatility simulations, whereas the right plots results fromlow-patience, high-volatility runs. Both sets of experiments use uniform patience distri-butions. N e t E ff i c i en cy Inter-arrival Time(patience=6, volatility=0.01)greedyzipmcafeeewmaactive-mcafeefixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 N e t E ff i c i en cy Inter-arrival Time(patience=2, volatility=0.08)greedymcafeeactive-mcafeeewmafixed-pricezip
Figure 6:
Net efficiency vs. inter-arrival time (1 / intensity) for several DAs. The left plotshows high-patience, low-volatility simulations, whereas the right plots results from low-patience, high-volatility runs. Both sets of experiments use uniform patience distribu-tions. 167 redin, Parkes and Duong
For very high arrival intensity we see Active-McAfee dominates McAfee. Active-McAfeesmooths the price, which helps to mitigate the impact of fluctuations in cost on the admissionprice via Eq. (3) in return for less responsiveness. This is helpful in “well-behaved” marketswith high arrival intensity and low volatility but was not helpful in most environments westudied, where the additional responsiveness provided by the (vanilla) McAfee scheme paidoff. The ZIP market also has good performance in this high-patience/low-volatility environ-ment. The reason is simple: this is an easy environment for simple learning agents, and theagents quickly learn to be truthful. We emphasize that these ZIP market results should betreated with caution and are certainly optimistic. This is because the ZIP agents are notprogrammed to consider timing-based manipulations. The effect in this environment is thatthe ZIP market tends to operate as if a truthful market, but without the cost of imposingtruthfulness explicitly via market-clearing rules. By comparison the
Chain auctions arefully strategyproof, to both value and temporal manipulations.Compare now with Figure 5 (right), which is for low patience and high volatility. Now wesee that McAfee dominates across the range of arrival intensities. Moreover, the performanceof ZIP is now quite poor because the agents do not have enough time to adjust their bids(patience is low) and high volatility makes this a more difficult environment. With volatilevaluations, the possibility of valuation swings leaves open the possibility of larger profits,luring agents to set wider profit margins, but only after the market changes. The ZIP agentsalso have fewer concurrent competitive offers to use in setting useful price targets duringlearning. As we might expect, high volatility also negatively impacts the efficiency of thefixed-price scheme.In Figure 6 we see that the net efficiency trends are qualitatively similar except that thecompetition-based DAs such as McAfee fare less well in comparison with the price-basedDAs. The auctioneer accrues more revenue for competition-based matching rules such asMcAfee because they often generate buy and sell prices with a spread. Together withthe competition-based schemes being intrinsically more dynamic, this drives an increasedprice spread in
Chain via the admission price constraints. In Figure 6 (left) we see thatthe fixed-price scheme performs well for high arrival intensity while EWMA dominates forintermediate arrival intensities. The McAfee scheme is still dominant for lower patience andhigher volatility (Figure 6, right).To reinforce these observations, in Table 3 we present the the net efficiency, allocativeefficiency and (normalized) revenue across all arrival intensities (i.e. inter-arrival time from0.05 to 1.5) and for both low and high volatility trials. All five price-based methods, allthree competition-based methods, and all three comparison methods are included. Wehighlight the best performing competition-based method, price-based method, as well asthe performance of the ZIP market (skipping over the non-truthful, greedy algorithm). Weomit information about the mean standard error for each measurement because in no casedid this error exceed a tenth of a percent of the mean optimal surplus. From within thetruthful DAs, we see that the McAfee-based scheme dominates overall for both allocativeand net efficiency and both low and high volatility, although EWMA competes with McAfeefor net efficiency in low volatility markets. Notice also the good performance of the ZIP-based market (with the aforementioned caveat about the restricted strategy space) at lowvolatilities. hain: An Online Double Auction scenario low-volt/high-pat high-volt/low-patnet alloc rev net alloc revmcafee active-mcafee 0.24 0.35 0.11 0.32 0.37 0.05windowed-mcafee 0.24 0.26 0.02 0.21 0.23 0.03history-clearing 0.33 0.34 0.01 0.17 0.17 0.01history-ewma history-fixed 0.23 0.23 0.00 0.04 0.04 0.00history-mcafee 0.33 0.34 0.01 0.15 0.16 0.01history-median 0.33 0.34 0.01 0.17 0.18 0.01blum et al. 0.10 0.10 0.00 0.02 0.02 0.00greedy 0.86 0.86 0.00 0.87 0.87 0.00zip
Table 3:
Net efficiency, allocative efficiency and auctioneer revenue (all normalized by the optimalvalue from trade), averaged across all arrival intensities (0.05–1.5) and for low and highvalue volatility. The best performing competition-based, price-based and ‘other’ (ignoringgreedy, which is not truthful) results are highlighted. A ll o c a t i v e E ff i c i en cy Volatility(patience=6, inter-arrival=1.0)greedyzipmcafeeewmafixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 A ll o c a t i v e E ff i c i en cy Volatility(patience=2, inter-arrival=1.0)greedymcafeeewmazipfixed-price
Figure 7:
Allocative efficiency vs. volatility for several DAs for a fairly low arrival intensity. Theleft plot is for large maximal patience and the right plot is for small maximal patience.Both sets of experiments use uniform patience distributions.
Figure 7 plots allocative efficiency versus volatility for high patience (left) and lowpatience (right) and for fairly low arrival intensity. Higher volatility hurts all methods –especially the ZIP agents, which struggle to learn appropriate profit and price targets,probably due to few opportunities to update prices for every individual offer. The McAfeescheme fairs very well, showing good robustness for both large patience and small patienceenvironments. The fixed-price scheme has the best performance when there is zero volatilitybut its efficiency falls off extremely quickly as volatility increases. redin, Parkes and Duong A ll o c a t i v e E ff i c i en cy Maximal Patience(inter-arrival=1.0, volatility=0.01)greedyzipmcafeeewmafixed-price 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 A ll o c a t i v e E ff i c i en cy Maximal Patience(inter-arrival=1.0, volatility=0.08)greedyzipmcafeeewmafixed-price
Figure 8:
Allocative efficiency vs. maximal patience for several DAs and fairly low arrival intensity.The left plot is for low volatility and the right plot is for high volatility. Both sets ofexperiments use uniform patience distributions.
We also consider the effect of varying maximal patience. This is shown in Figure 8,with low volatility (left) and high volatility (right). Again, the McAfee scheme is the bestof the truthful DAs based on
Chain . We also see that the performance of ZIP improvesas patience increases due to more opportunities for learning. Perversely, a larger patiencecan negatively affect the truthful DAs. In part this is simply because the performance ofgreedy online schemes, relative to the offline optimal, decreases as patience increases andthe offline optimal matching is able to draw more benefit from its lack of myopia.We also suspected another culprit, however. The possibility of the presence of patientagents requires the truthful DAs to include additional terms in the max operator in Eq. (3)to prevent manipulations, leading to higher admission prices and less admitted offers. Tobetter understand this effect we experimented with delayed market clearing in the McAfeescheme, where the market matches agents only every τ -th period (the “clearing duration”).The idea is to make a tradeoff between using fewer admission prices and the possibility thatwe will miss the opportunity to match some impatient offers.Figure 9 shows allocative efficiency when the matching mechanism clears less frequentlyand for different maximal patience, K . Figure 9 (left) is for low volatility. There wesee that the best clearing duration is roughly 1, 2, 3 and 4 for maximal patience of K ∈{ , , , } and that by optimizing the clearing duration the performance of McAfee remainsapproximately constant as maximal patience increases. In Figure 9 (right) we consider theeffect in a high volatility environment, with these results averaged over 500 trials because theperformance of the DA has higher variance. We see a qualitatively similar trend, althoughhigher maximal patience now hurts overall and cannot be fully compensated for by tuningthe clearing duration.
7. Related Work
Static two-sided market problems have been widely studied (?, ?, ?, ?, ?). In a classic result,Myerson and Satterthwaite proved that it is impossible to achieve efficiency with voluntary hain: An Online Double Auction A ll o c a t i v e E ff i c i en cy Clearing Duration(inter-arrival=1.0 patience=K, volatility=0.01)K=4K=6K=8K=10 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 8 10 A ll o c a t i v e E ff i c i en cy Clearing Duration(inter-arrival=1.0 patience=K, volatility=0.08)K=4K=6K=8K=10
Figure 9:
Allocative efficiency vs. clearing duration in the McAfee-based
Chain auction for fairlylow arrival intensity and as maximal patience is varied from 4 to 10. The left plot isfor low volatility and the right plot is for high volatility. Both sets of experiments useuniform patience distributions. participation and without running a deficit, even relaxing dominant-strategy equilibrium toa Bayesian-Nash equilibrium. Some truthful DAs are known for static problems (?, ?, ?, ?).For instance, McAfee introduced a DA that sometimes forfeits trade in return for achievingtruthfulness. McAfee’s auction achieves asymptotic efficiency as the number of buyers andsellers increases. Huang et al. extend McAfee’s mechanism to handle agents exchangingmultiple units of a single commodity. Babaioff and colleagues have considered extensionsof this work to supply-chain and spatially distributed markets.Our problem is also similar to a traditional continuous double auction (CDA), wherebuyers and sellers may at any time submit offers to a market that pairs an offer as soonas a matching offer is submitted. Early work considered market efficiency of CDAs withhuman experiments in labs (?), while recent work investigates the use of software agentsto execute trades (?, ?, ?, ?). While these markets have no dominant strategy equilibria,populations of software trading agents can learn to extract virtually all available surplus,and even simple automated trading strategies outperform human traders (?). However,these studies of CDAs assume that all traders share a known deadline by which trades mustbe executed. This is quite different from our setting, in which we have dynamic arrival anddeparture.Truthful one-sided online auctions, in which agents arrive and depart across time, havereceived some recent attention (?, ?, ?, ?, ?). We adopt and extend the monotonicity-basedtruthful characterization in the work of Hajiaghayi et al. (?) in developing our frameworkfor truthful DAs. Our model of DAs must also address some of the same constraints ontiming that occur in Porter, Hajiaghayi, and Lavi and Nisan’s work. In these previousworks, the items were reusable or expiring and could only be allocated in particular periods.In our work we provide limited allowance to the match-maker, allowing it to hold onto aseller’s item until a matched buyer is ready to depart (perhaps after the seller has departed). redin, Parkes and Duong The closest work in the literature is due to Blum et al. (?), who present online matchingalgorithms for the same dynamic DA model. The main focus in their paper is on thedesign of matching algorithms with good worst-case performance in an adversarial setting,i.e. within the framework of competitive analysis. Issues related to incentive compatibilityreceive less attention. One way in which their work is more general is that they also studygoals of profit and maximizing the number of trades , in addition to the goal of maximizingsocial welfare that we consider in our work. However, the only algorithmic result that theypresent that is truthful in our model (where agents can misreport arrival and departure) isfor the goal of social welfare. The DA that they describe is an instance of
Chain in whicha fixed price is drawn from a distribution at the start of time, and used as the matchingprice in every period. Perhaps unsurprisingly, given their worst-case approach, we observethat their auction performs significantly worse than
Chain defined for a fixed price that ispicked to optimize welfare given distributional information about the domain.
8. Conclusions
We presented a general framework to construct algorithms to match buyers with sellers inonline markets where both valuation and activity-period information are private to agents.These algorithms guarantee truthful dominant strategies by first imposing a minimum ad-mission price for each offer and then pricing and pairing the offer at the first opportunity.At the heart of the
Chain framework lies a pricing algorithm that must for each offer eitherdetermine a price independent of any information describing the offer or choose to discardthe offer. The pricing algorithm should be chosen to match market conditions. We presentseveral examples of suitable pricing schemes, including fixed-price, moving-average, andMcAfee-based schemes.More often than not, we find that the competition-based scheme that employs a McAfee-based rule to truthfully price the market delivers the best allocative efficiency. For excep-tionally low volatility and high arrival intensity, we find that adaptive price-based schemessuch as an exponentially-weighted moving average (EWMA) and even fixed price schemesperform well. We see qualitatively similar results for net efficiency, where the revenue thataccrues to the auctioneer is discounted, albeit that the price-based rules such as EWMAhave improved performance because they have no price spread. The observations are rootedin simulations comparing the market efficiency under each mechanism with the optimal of-fline solution.Additionally, we compare the efficiency of our truthful markets with a fixed-price worst-case optimal scheme presented by Blum et al. (?), a market of strategic agents using a varianton the ZIP price update algorithm developed by Cliff and Bruten (?) for continuous doubleauctions, and a non-truthful, greedy matching algorithm to provide an upper-bound onperformance. The best of our schemes yield around 33% net efficiency in low volatility, highpatience environments and 40% net efficiency in high volatility, low patience environments,while the greedy bound suggests that as much as 86% efficiency is possible with non-strategicagents. We note that the Blum et al.scheme, designed for adversarial settings, fairs poorlyin our simulations ( < in itself in thatit avoids the waste of costly counterspeculation and promotes fairness in markets (?, ?).On the other hand, it is certainly of interest that the gap between the efficiency of greedy hain: An Online Double Auction matching with non-truthful matching and that of our truthful auctions is so large. Here, weobserve that the ZIP-populated (non-truthful) markets achieve around 82% efficiency in lowvolatility environments but collapse to around 23% efficiency in high volatility environments.Based on this, one might conjecture that designing for truthfulness is especially importantin badly behaved, highly volatile (“thin”) environments but less important in well behaved,less volatile (“thick”) environments.Formalizing this tradeoff between providing absolute truthfulness and approximatetruthfulness, and while considering the nature of the environment, is an interesting di-rection for future work (see paper by ?). Given that reporting of market statistics can beincorporated within our framework (see Section 5.1), and given that markets also play arole in information aggregation and value discovery, future research should also considerthis additional aspect of market design. Perhaps there is an interesting tradeoff betweenefficiency, truthful value revelation, and the process of information aggregation.While the general Chain framework achieves good efficiency, further tuning seems pos-sible. One direction is to adopt a meta -pricing scheme that chooses, or blends, prices fromcompeting algorithms. Another direction is to consider richer temporal models; e.g., thevalue of goods to agents might decay or grow over time to better account for the timevalue of assets. A richer temporal model might also consider the possibility of agents or thematch-maker taking short positions (including short-term cash deficits) to increase trade.It is also interesting to extend our work to markets with non-identical goods and more com-plex valuation models such as bundle trades (?, ?, ?), and to dynamic matching problemswithout prices, such as an online variation of the classic “marriage” problem (?).
Acknowledgments
An earlier version of this paper appeared in the Proceedings of the 21st Conference on Uncer-tainty in Artificial Intelligence, 2005. This paper further characterizes necessary conditionsfor truthful online trade; truthfully matches offers using a generalized framework basedupon an arbitrary truthful static pricing rule; and compares the efficiency of our truthfulframework to that achieved in non-truthful markets populated with strategic trading agentsand with that of worst-case optimal double auctions.Parkes is supported in part by NSF grant IIS-0238147 and an Alfred P. Sloan Fellowshipand Bredin would like to thank the Harvard School of Engineering and Applied Sciencesfor hosting his sabbatical during which much this work was completed. Thanks also tothe three anonymous reviewers, who provided excellent suggestions in improving an earlierdraft of this paper.
Appendix: Proofs
Lemma 1
Procedure
Match defines a valid strong no-trade construction.
Proof:
In all cases, SNT t ⊆ NT t . The set NT t is correctly constructed: equal to allremaining bids b t when ( j = 0) in Case I, all remaining bids s t when ( i = 0) in Case II,and all remaining bids and asks otherwise. In each case, no bid (or ask) in NT t could havetraded at any price because there was no available bid or ask on the opposite of the marketgiven its order. redin, Parkes and Duong In verifying strong no-trade (SNT) conditions (a) and (b), we proceed by case analysis.
Case I. ( i = 0) and ( j = 0). NT t := b t .(I-1) ∀ k ∈ s t · ( ˆ d k = t ) and SNT t := b t . For SNT-a, consider l ∈ NT t with ˆ d l > t . If l deviates and i changes but we remain in Case I then NT t is unchanged and stillcontains l . If l deviates and i → t := b t ∪ s t and stillcontains l . For SNT-b, consider l ∈ SNT t that deviates with ˆ d l > t . Again, either weremain in this case and SNT t is unchanged or i → t still contains all b t and is therefore unchanged for all agents with ˆ d k > t .(I-2) Buyer k ∈ b t with ˆ d k = t and b k ≥ p t and SNT t := b t . For SNT-a, consider l ∈ NT t with ˆ d l > t . We remain in this case for any deviation by buyer l because buyer k will ensure i = 0, and so SNT t remains unchanged and still contains l . For SNT-b, if l ∈ SNT t with ˆ d l > t deviates we again remain in this case and SNT t is unchanged.(I-3) Some seller with ˆ d k > t and no buyer with ˆ d k ′ = t willing to accept the price.SNT t := b t \ checked B . For SNT-a, consider l ∈ NT t with ˆ d l > t . First, suppose l ∈ checked B and i = l . If l deviates but still has ˆ d l > t, then even if i := l then weremain in this case and l does not enter SNT t . Second, suppose l ∈ checked B and( i = l ). If l deviates but still has ˆ d l > t, then even if ( i = 0) and ( j = 0), we goto Case III and SNT t = ∅ and l does not enter SNT t . Third, suppose l / ∈ checked B and ˆ d l > t . Deviating while ˆ d l > t has no effect and we remain in this case and l remains in SNT t . For SNT-b, consider l ∈ SNT t with ˆ d l > t , i.e. with l / ∈ checked B .If l deviates but ˆ d l > t, then this has no effect and we remain in this case and SNT t remains unchanged. Case II . ( j = 0) and ( i = 0). NT t := s t . Symmetric with Case I. Case III . ( i = 0) and ( j = 0). NT t := b t ∪ s t .(III-1) ∀ k ∈ b t · ( ˆ d k = t ) but ∃ k ′ ∈ s t · ( ˆ d k ′ > t ) and SNT t := b t ∪ s t . For SNT-a, consider l ∈ NT t with ˆ d l > t . This must be an ask. If l deviates but we remain in this case,then l remains in SNT t . If j := l, then we go to Case II and SNT t := s t and l remainsin SNT t . For SNT-b, consider l ∈ SNT t with ˆ d l > t , which must be an ask. If l deviates but we remain in this case, SNT t is unchanged. If l deviates and j := l, thenwe go to Case II, SNT t := s t , and buyers b t are removed from SNT t . But this is OKbecause all buyers depart in period t anyway.(III-2) ∀ k ∈ s t cdot ( ˆ d k = t ) but ∃ k ′ ∈ b t · ( ˆ d k ′ > t ) and SNT t := b t ∪ s t . Symmetric to CaseIII-1.(III-3) ∀ k ∈ b t · ( ˆ d k = t ) and ∀ k ∈ s t cdot ( ˆ d k = t ). SNT t := b t ∪ s t . SNT-a and SNT-b aretrivially met because no bids or asks have departure past the current period.(III-4) ∃ k ∈ b t · ( ˆ d k > t ) and ∃ k ′ ∈ s t · ( ˆ d k ′ > t ) and SNT t := ∅ . For SNT-a, consider l ∈ NT t with ˆ d l > t . Assume that l is a bid. If l deviates and ˆ d l > t and i = 0 then we remainin this case and l is not in SNT t . If l deviates and ˆ d l > t but i := l, then we go toCase I and we are necessarily in Sub-case (I-a) because ˆ d l > t and there can be noother bid willing to accept the price (else i = 0 in the first place). Thus, we would hain: An Online Double Auction have SNT t := b t \ checked B and l would not be in SNT t . For SNT-b, this is triviallysatisfied because there are no agents l ∈ SNT t . (cid:3) Lemma 5
The set of active agents (other than i ) in period t in Chain is independent of i ’s report while agent i remains active, and would be unchanged if i ’s arrival is later thanperiod t . Proof:
Fix some arrival period ˆ a i . Show for any ˆ a i ≥ a i , the set of active agents in period t ≥ ˆ a i while i is active is the same as A t without agent i ’s arrival until some a ′ i > t . Proceedby induction on the number of periods that t is after ˆ a i . For period t = ˆ a i this is trivial.Now consider some period ˆ a i + r , for some r ≥ a i + r −
1. Since i is still active then, i ∈ SNT t ′ for t ′ = ˆ a i + r −
1, and therefore the otheragents in SNT t ′ that survive into this period are independent of agent i ’s report by strongno-trade condition (b). This completes the proof. (cid:3) Lemma 6
The price constructed from admission price ˇ q and post-arrival price ˇ p is value-independent and monotonic-increasing when the matching rule in Chain is well-defined,the strong no-trade construction is valid, and agent patience is bounded by K . Proof:
First fix ˆ a i , ˆ d i and θ − i . To show value-independence (B1), first note that ˇ q isvalue-independent, since whether or not i ∈ SNT t in some pre-arrival period t is value-independent by strong no-trade condition (a) and price z i ( H t , A t \ i, ω ) in such a periodis agent-independent by definition. Term ˇ p is also value-independent: the decision period t ∗ to agent i , if any, is independent of ˆ w i since the other agents that remain active areindependent of agent i while it is active by Lemma 5, and whether or not i ∈ SNT t isvalue-independent by strong no-trade (a); and the price in t ∗ is value-independent whenthe set of other active agents are value-independent.Now fix θ − i and show the price is monotonically-increasing in a tighter arrival-departureinterval (B2). First note that ˇ q is monotonic-increasing in [ˆ a i , ˆ d i ] ⊂ [ a i , d i ] because an earlierˆ d i and later ˆ a i increases the domain t ∈ [ ˆ d i − K, ˆ a i −
1] on which ˇ q is defined. Fix someˆ a i ≥ a i . Argue the price increases with earlier d ′ i ≤ ˆ d i , for any ˆ d i > ˆ a i . To see this, note thateither ˆ d i < t ∗ and so p i (ˆ a i , d ′ i , θ − i , ω ) = ∞ for all d ′ i ≤ ˆ d i , or ˆ d i ≥ t ∗ and the price is constantuntil ˆ d i < t ∗ at which point it becomes ∞ . Fix some ˆ d i ≥ a i . Argue the price increases withlater a ′ i ≥ ˆ a i , where ˆ a i ≥ ˆ d i − K . First, while a ′ i ≤ t ∗ , then ˇ p is unchanged by Lemma 5.The interesting case is when a ′ i > t ∗ , especially when ˇ q (ˆ a i , ˆ d i , θ − i , ω ) < ˇ p (ˆ a i , ˆ d i , θ − i , ω ).By reporting a later arrival, the agent can delay its decision period and perhaps hope toachieve a lower price. But, note that in this case t ∗ ∈ [ ˆ d i − K, a ′ i −
1] since ˆ d i − K ≤ ˆ a i and t ∗ ∈ [ˆ a i , a ′ i −
1] and so ˇ q ( a ′ i , ˆ d i , θ − i , ω ) ≥ ˇ p ( H t ∗ , A t ∗ \ i, ω ) because ˇ q now includes the pricein period t ∗ since i / ∈ SNT t ∗ in that pre-arrival period by Lemma 5. Overall, we see thatalthough ˇ p may decrease, max(ˇ q, ˇ p ) cannot decrease. (cid:3) Lemma 7
A strongly truthful, canonical dynamic DA must define price p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ z i ( H t ∗ , A t ∗ \ i, ω ) where t ∗ is the decision period for bid i (if it exists). Moreover, the bidmust be priced-out in period t ∗ if it is not matched. Proof: (a) First, suppose z i ( H t ∗ , A t ∗ \ i, ω ) > ˆ w i but bid i is not priced-out and insteadsurvives as an active bid into the next period. But with i / ∈ SNT t ∗ , the set of active bids in redin, Parkes and Duong period t ∗ +1 need not be independent of agent i ’s bid and the price z i ( H t ∗ +1 , A t ∗ +1 \ i, ω )need not be agent-independent. Yet, canonical rule (iii) requires that this price be usedto determine whether or not the agent matches, and so the dynamic DA need not betruthful. (b) Now assume for contradiction that p i (ˆ a i , ˆ d i , θ − i , ω ) < z i ( H t ∗ , A t ∗ \ i, ω ). First,if z i ( H t ∗ , A t ∗ \ i, ω ) < ∞ , then an agent with value p i (ˆ a i , ˆ d i , θ − i , ω ) < w i < z i ( H t ∗ , A t ∗ \ i, ω )will report ˆ w i = z i ( H t ∗ , A t ∗ \ i, ω ) + ǫ and trade now for a final payment less than its truevalue (whereas it would be priced-out if it reported its true value). If z i ( H t ∗ , A t ∗ \ i, ω ) = ∞ ,then p i (ˆ a i , ˆ d i , θ − i , ω ) < z i ( H t ∗ , A t ∗ \ i, ω ) implies that some bids will survive this period eventhough they are priced-out by the matching rule and not in the strong no-trade set. Thiscompromises the truthfulness of the dynamic DA, as discussed in part (a). (cid:3) Lemma 8
A strongly truthful, canonical and individual-rational dynamic DA must defineprice p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , and a bid with ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) must be priced-out upon admission. Proof:
Suppose ˆ d i < ˆ a i + K so that [ ˆ d i − K, ˆ a i −
1] is non-empty. For ˆ d i = ˆ a i + K − t = ˆ d i − K is a decision period (and i / ∈ SNT t ), we have p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ p i ( ˆ d i − K, ˆ d i , θ − i , ω ) ≥ z i ( H t , A t \ i, ω ) , (24)where the first inequality is by monotonicity (B2) and the second follows from Lemma 7since ˆ d i − K is a decision period, and would remain one with report θ ′ i = ( ˆ d i − K, ˆ d i , w ′ i )by Lemma 5. This establishes p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) for ˆ d i = ˆ a i + K −
1. Whenˆ d i = ˆ a i + K −
2, then we need Eq. (24), and also when t = ˆ d i − K + 1 is a decision period(and i / ∈ SNT t ) we have, p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ p i ( ˆ d i − K + 1 , ˆ d i , θ − i , ω ) ≥ z i ( H t , A t \ i, ω ) , (25)by the same reasoning as above. This generalizes to d i = a i + K − r for r ∈ { , . . . , K } to establish p i (ˆ a i , ˆ d i , θ − i , ω ) ≥ ˇ q (ˆ a i , ˆ d i , θ − i , ω ) for the general case. To see the bid must bepriced-out when ˆ w i < ˇ q (ˆ a i , ˆ d i , θ − i , ω ) , note that if it were to remain active it could match inthe matching rule and by canonical (iii) need to trade, and thus fail individual-rationalitysince the payment collected would be more than the value. (cid:3)(cid:3)