Limits of Efficiency in Sequential Auctions
aa r X i v : . [ c s . G T ] S e p Limits of Efficiency in Sequential Auctions
Michal Feldman , Brendan Lucier , and Vasilis Syrgkanis Hebrew Univsersity [email protected] Microsoft Research New England [email protected] Dept of Computer Science, Cornell University [email protected]
Abstract.
We study the efficiency of sequential first-price item auc-tions at (subgame perfect) equilibrium. This auction format has recentlyattracted much attention, with previous work establishing positive re-sults for unit-demand valuations and negative results for submodularvaluations. This leaves a large gap in our understanding between thesevaluation classes. In this work we resolve this gap on the negative side.In particular, we show that even in the very restricted case in which eachbidder has either an additive valuation or a unit-demand valuation, thereexist instances in which the inefficiency at equilibrium grows linearly withthe minimum of the number of items and the number of bidders. More-over, these inefficient equilibria persist even under iterated elimination ofweakly dominated strategies. Our main result implies linear inefficiencyfor many natural settings, including auctions with gross substitute valu-ations, capacitated valuations, budget-additive valuations, and additivevaluations with hard budget constraints on the payments. Another im-plication is that the inefficiency in sequential auctions is driven by themaximum number of items contained in any player’s optimal set, andthis is tight. For capacitated valuations, our results imply a lower boundthat equals the maximum capacity of any bidder, which is tight followingthe upper-bound technique established by Paes Leme et al. [19].
Consider the following natural auction setting. An auction house has a numberof items that are offered for sale in an auction on a particular day. To orchestratethis, the auction house publishes a list of the items to be sold and the order inwhich they will be auctioned off. The items are then sold one at a time in thegiven order. A group of bidders attends this session of auctions, with each bidderbeing allowed to participate in any or all of the single-item auctions that will berun throughout the day. Since the auctions are run one at a time, in sequence,this format is referred to as a sequential auction.This way of auctioning multiple items is prevalent in practice, due to itsrelative simplicity and transparency. It also arises naturally in electronic markets,such as eBay, due to the asynchronous nature of the multiple single-item auctions
Michal Feldman, Brendan Lucier, and Vasilis Syrgkanis that are executed on the platform. A natural question, then, is how well such asequential auction performs in practice. Note that while the auction of a singleitem is relatively simple, equilibria of the larger game may be significantly morecomplex. For instance, a bidder who views two of the items as substitutes mightprefer to win whichever sells at the lower price, and hence when bidding on thefirst item he must look ahead to the anticipated outcome of the second auction.What’s more, the sequential nature of the mechanism implies that the outcomeof one auction can influence the behavior of bidders in subsequent auctions. Thisgives rise to complex reasoning about the value of individual outcomes, with thepotential to undermine the efficiency of the overall auction.In this work we study the efficiency of sequential single-item first-price auc-tions, where items are sold sequentially using some predefined order and eachitem is sold by means of a first-price auction. We study the efficiency of outcomesat subgame perfect equilibrium, which is the natural solution concept for a dy-namic, sequential game. Theoretical properties of these sequential auctions havebeen long studied in the economics literature starting from the seminal work ofWeber [23]. However, most of the prior literature has focused on very restrictedsettings, such as unit-demand valuations, identical items, and symmetricallydistributed player valuations. The few exceptions that have attempted to studyequilibria when bidders have more complex valuations tend to have other restric-tions, such as a very limited number of players or items [11,20,3,2]. Much of thedifficulty in studying these auctions under complex environments and/or valua-tions stems from the inherent complexity of the equilibrium structure, which (asalluded to above) can involve complex reasoning about future auction outcomes.Paes Leme et al. [19] and Syrgkanis and Tardos [21] circumvented this diffi-culty by performing an indirect analysis on efficiency using the price-of-anarchyframework. They showed that when bidders have unit-demand valuations (UD),items are heterogeneous, and bidders’ valuations are arbitrarily asymmetricallydistributed, then the social welfare at every equilibrium is a constant fraction ofthe optimal welfare. Syrgkanis and Tardos [22] extended this result to no-regretlearning outcomes and to settings with budget constraints. On the negative side,Paes Leme et al. [19] showed that this result does not extend to submodularvaluations (SM): there exists an instance with submodular valuations where theunique “natural” subgame perfect equilibrium leads to inefficiency that increaseslinearly with the number of items, even for a constant number of bidders.The above results leave a large gap between the positive regime (unit-demandbidders) and the negative (submodular bidders). Many natural and heavily-studied classes of valuations fall in the range between UD and SM valuations.Among them are the following, arranged roughly from most to least general: – Gross-substitutes valuations (GS):
A valuation satisfies the gross-substitutesvaluation property if, whenever the cost of one item increases, this cannotreduce the demand for another item whose price did not increase. – k -capacitated valuations ( k -CAP): Each player i has a capacity k i ≤ k anda value for each item; the value for a set of items is then the value of the k i highest-valued items in the set. imits of Efficiency in Sequential Auctions 3 – Budget-additive valuations (BA):
The value of a player i is additive up to aplayer-specific budget B i and then remains constant.The class of GS valuations is motivated by the fact that it is (in a certain sense)the largest class of valuations for which a Walrasian equilibrium is guaranteedto exist [12], and a Walrasian equilibrium, if exists, is always efficient (see, e.g.,[6]). It is known that every k -capacitated valuation satisfies gross substitutes [9].Moreover, every gross substitutes valuation is submodular [15], and it is easyto see that unit-demand valuations are precisely 1-capacitated valuations. Wetherefore have UD ⊂ k -CAP ⊂ GS ⊂ SM. The set of budget-additive valuationsis incomparable to UD, k -CAP, and GS, but it is known that BA ⊂ SM.We ask: for which of the above classes does the sequential first-price auctionobtain a constant fraction of the optimal social welfare at equilibrium?
In thiswork we show that the answer to the above question is none of them .Specifically, we show that for the case of gross substitutes valuations and forbudget additive valuations, the inefficiency of equilibrium can grow linearly withthe number of items and the number of players. Thus, even for settings in whicha Walrasian equilibrium is guaranteed to exist, an auction that handles itemssequentially cannot find an approximately optimal outcome at equilibrium. Forthe case of k -capacitated valuations, we show that the inefficiency can be as highas k . This bound of k is tight, following the upper bound established by [19].To prove these lower bounds we consider a different, conceptually more re-strictive, class of valuations: the union of unit-demand and additive valuations.We construct an instance in which every bidder has either a unit-demand valu-ation or an additive valuation, then show that the unique “natural” equilibriumfor this instance has extremely poor social efficiency. We then adapt this con-struction to provide a lower bound for the valuation classes described above.We also extend our lower bound to apply to one other setting: additive valua-tions when players have hard budget-constraints on their payments. This settingfalls outside the quasi-linear regime, but is very relevant in the sequential auctionsetting: for instance, each bidder may arrive at an auction session with only acertain fixed amount of money to spend. Note that this is different from the BAvaluation class, since it does not restrict the value of a player for a set of items,but rather limits the total payment that a player can make. For this setting,it is known that maximizing welfare is not an achievable goal in most auctionsettings, as a participant with low budget is necessarily ineffective at maximizingthe value of the item(s) she obtains. Instead, the natural notion of social effi-ciency is the “effective welfare,” in which the contribution of each participant tothe welfare is capped by her budget [22]. We show that, even comparing againstthe benchmark of effective welfare, our negative result also applies to this set-ting: for additive valuations with hard budget constraints, the inefficiency cangrow linearly with the number of items or players. This is in stark contrast to thesetting of simultaneous first-price auctions, where it is known that a constantfraction of the optimal effective social welfare occurs at equilibrium for bidderswith hard budget constraints, even when valuations are fractionally subadditive[22] (where this class falls between submodular and subadditive valuations). Michal Feldman, Brendan Lucier, and Vasilis Syrgkanis
Sequential auctions with additive bidders and hard budget constraints havebeen studied in only very limited settings in the economics literature and haverecently begun to attract the attention of the computer science community [14].Our result shows that if one allows for arbitrary additive valuations, then suchan auction process can lead to very high inefficiency.All of the negative results described above rely heavily on the fact that itemscan be sold in an arbitrary order. This leads naturally to the following design question: does there always exists an order on the items that results in betteroutcomes at a subgame perfect equilibrium? This can be interpreted as a mech-anism design problem, in which the auctioneer wishes to choose the order inwhich items are sold in order to mitigate the social impact of strategic bidding.We conjecture that a concrete class of item orders (that we propose) always con-tains a good order that leads to the VCG outcome at equilibrium, for the class of single-valued unit-demand valuations. We leave the resolution of this conjectureas an open problem.
Sequential auctions have been long studied in the economics literature. Weber[23] and Milgrom and Weber [17] analyzed first- and second-price sequential auc-tions with identical items and unit-demand bidders in an incomplete-informationsetting and showed that the unique symmetric equilibrium is efficient and theprices have an upward drift. The behavior of prices in sequential studies wassubsequently studied in [1,16]. Boutilier el al. [7] studies first-price auctions ina setting with uncertainty, and devised a dynamic-programming algorithm forfinding the optimal strategies (assuming stationary distribution of others’ bids).The setting of multi-unit demand has also been studied under the complete-information model. Several papers studied the two-bidder case, where there isa unique subgame perfect equilibrium that survives the iterated elimination ofweakly dominated strategies (IEWDS) [11,20]. Bae et al. [3,2] studied the case ofsequential second-price auctions of identical items with two bidders with concavevaluations and showed that the unique outcome that survives IEWDS achievesa social welfare at least 1 − e − of the optimum. Here we consider more thantwo bidders and heterogeneous items.Recently, Paes Leme et al. [19] analyzed sequential first- and second-price auc-tions for heterogeneous items and multi-unit demand valuations in the complete-information setting. For sequential first-price auctions they showed that whenbidders are unit-demand, every subgame perfect equilibrium achieves at least1/2 of the optimal welfare, while for submodular bidders the inefficiency cangrow with the number of items, even with a constant number of bidders. Thepositive results were later extended to the incomplete-information setting in [21]and to no-regret outcomes and budget-constrained bidders in [22]. In this workwe close the gap between positive and negative results and show that inefficiencycan grow linearly with the minimum of the number of items and bidders evenwhen bidders are either additive or unit-demand. imits of Efficiency in Sequential Auctions 5 This work can be seen as part of the recent interest line of research onsimple auctions. The closest literature to our work is the that of simultane-ous item-bidding auctions [5,8,4,13,10,22], which is the simultaneous counter-part of sequential auction. In contrast to sequential auctions, in simultaneousitem auctions constant efficiency guarantees have been established for generalcomplement-free valuations, even under incomplete-information settings or out-comes that emerge from learning behavior. We refer to [18] for a recent surveyon the efficiency of simultaneous and sequential item-auctions.
We consider settings with n bidders and m items, where every bidder i ∈ [ n ] hasa valuation function v i : 2 [ m ] → R + , associating a non-negative real value withevery subset of items. We denote the set of bidders by [ n ] and the set of itemsby [ m ]. The valuation function is assumed to be monotone (i.e., v i ( T ) ≤ v i ( S )for every T ⊆ S ). An allocation is a vector x = ( x , . . . , x n ), where x i denotesthe set of items allocated to bidder i , and such that x i ∩ x j = ∅ for every i = j . Sequential item auctions.
The auction proceeds in steps, where a single item issold in every step using a first-price auction. In every step t = 1 , . . . , m , everybidder i offers a bid b i ( t ), and the item is allocated to the agent with the highestbid for a payment that equals his bid. Each bid in each step can be a functionof the history of the game, which is assumed to be visible to all bidders. Moreformally, a strategy of bidder i is a function that, for every step t , associates abid as a function of the sequence of the bidding profiles in all periods 1 , . . . , t − utility of an agent is defined, as standard, to be his value for the items hewon minus the total payment he made throughout the auction (i.e., quasi-linearutility). We will also assume that the bid space is discretized in small negligible δ -increments, and for ease of presentation we will use b + to denote the bid b + δ .This setting is captured by the framework of extensive-form games (see, e.g.,[19]), where the natural solution concept is that of a subgame-perfect equilibrium (SPE). In an SPE, the bidding strategy profiles of the players constitute a Nashequilibrium in every subgame. That is, at every step t and for every possiblepartial bidding profile b (1) , b (2) , . . . , b ( t −
1) up to (but not including) step t ,the strategy profile in the subgame that begins in step t constitutes a Nashequilibrium in the induced (i.e., remaining) game. Elimination of Weakly Dominated Strategies.
We wish to further restrict ourattention to “natural” equilibria, that exclude (for example) dominated over-bidding strategies. We therefore consider a natural and well-studied refinementof the set of subgame perfect equilibria: those that survive iterated elimina-tion of weakly dominated strategies (IEWDS). A strategy s is weakly domi-nated by a strategy s ′ if, for every profile of other players’ strategies s − i , wehave u i ( s, s − i ) ≤ u i ( s ′ , s − i ), and moreover there exists some s − i such that u i ( s, s − i ) < u i ( s ′ , s − i ). Roughly speaking, under IEWDS, each player removes Michal Feldman, Brendan Lucier, and Vasilis Syrgkanis from her strategy space the set of all weakly dominated strategies. This removalmay cause new strategies to become weakly dominated for a player, which arethen removed from her strategy space, and so on until no weakly dominatedstrategies remain. We defer a formal definition of IEWDS to Appendix A.We will focus on subgame perfect equilibria of sequential first-price itemauctions that survive IEWDS. It is shown in [19] that there always exists suchan equilibrium. We note one necessary property of an equilibrium satisfyingIEWDS: in every subgame beginning at a time t = m (i.e., when the last itemis being sold), for every possible bidding history up to that round, each playerwill bid no more than his marginal value for the final item. In other words, noplayer can credibly threaten to overbid on the last item for sale. Price of anarchy.
The price of anarchy (PoA) measures the inefficiency thatcan arise in strategic settings. The PoA for subgame perfect equilibria is definedas the worst (i.e., largest) possible ratio between the welfare obtained in theoptimal allocation and the welfare obtained in any subgame perfect equilibriumof the game. We note that all of our lower bounds on the price of anarchy willinvolve “natural” equilibria that survive IEWDS.
To develop some intuition regarding the strategic considerations that might takeplace in sequential auctions, we give a simple example in which one bidder hasvalue for many items (i.e., wholesale buyer) and another bidder has value foronly one item (i.e., retail buyer).In particular, consider a sequence of two auctions for two identical itemsand two buyers, A and B . Buyer A is a “wholesale” buyer, having an additivevaluation with a value of 9 for each of the two items. Buyer B is a “retail” buyer,who wants only one item (unit-demand) and has a value of 5 for either of thetwo. The items are sold sequentially using a first-price auction for each item.Consider the situation from the perspective of the additive buyer A . Thinkingstrategically and farsightedly, he reasons that if he wins the first auction, thenin the second auction he will have to compete with buyer B and will thereforehave to pay 5 dollars to win the second item. If, however, he lets buyer B winthe first item, then buyer B will have no value for the second item and hence theonly undominated strategy for buyer B will be to bid 0 in the second auction,and hence buyer A will win the second item for free. What must buyer A pay inorder to win the first item? Buyer B knows that if the first item goes to buyer A , then buyer B will certainly lose the second item as well; therefore buyer B is willing to pay up to 5 for the first item. Therefore, in order to win the firstitem, buyer A will have to bid at least 5 in the first auction.Thus bidder A needs to choose between the following two options: he caneither win both auctions and pay a price of 5 for each one of them, or let bidder B win the first auction and win only the second auction but pay nothing. Observethat the first option gives bidder A a utility of 8 (= 2 · (9 − imits of Efficiency in Sequential Auctions 7 option gives him a utility of 9 (= 1 · (9 − A will chooseto forego the first item in order to improve his situation in the second one.Interestingly, this outcome is socially suboptimal, since the efficient outcome isfor bidder A to win both items — although bidder A has much more value forthe first item than bidder B , the first item is allocated to B in equilibrium.One can also take this example to the extreme where, e.g., bidder A ’s valueis set to 10 − ǫ for each item. In this case the unique subgame perfect equilib-rium that survives elimination of dominated strategies is a 4 / O ( m ) fraction of the optimalsocial welfare. We now present our main result by providing an instance of a sequential firstprice auction with unit-demand and additive bidders, where the social welfareat a subgame-perfect equilibrium that survives IEWDS achieves social welfarethat is only an O (min { n, m } )-fraction of the optimal welfare. Therefore, ourexample shows that inefficiency can arise at equilibrium in a robust manner. Theorem 1.
The price of anarchy of the sequential first-price item auctionswith additive and unit-demand bidders is Ω (min { n, m } ) . Moreover, this resultpersists even if we consider only equilibria that survive IEWDS. Informal Description.
Before we delve into the details of the proof ofTheorem 1, we give a high-level idea of the type of strategic manipulations thatlead to inefficiency and compare them with the simultaneous auction counterpartof our sequential auction.Consider an auction instance where two additive bidders have identical valuesfor most of the items for sale, but their valuations differ only on the last few itemsthat are sold. Specifically, assume that there are two items Z and Z , auctionedlast, such that only player 1 has value for Z and only player 2 has value for Z .We will refer to these items as the non-competitive items and to all other itemsas the competitive items . The additive bidders know that it is hopeless to tryto achieve any positive utility from the competitive items on which they haveidentical interests. The only utility they can ever derive is from the last, non-competitive items on which they don’t compete with each other. If these werethe only two players in the auction, then we would obtain the optimal outcome:the two bidders would simply compete on each of the competitive items, withone of them acquiring each competitive item at zero utility. The equilibrium that we describe is, in some sense, the unique natural equilibrium: ifwe were to ask players to submit bids sequentially within each auction, rather thansimultaneously, then there would be a unique equilibrium (solvable by backwardinduction), which is the equilibrium that we describe. In fact, optimality is always achieved when all bidders are additive, in general. Michal Feldman, Brendan Lucier, and Vasilis Syrgkanis
We now imagine adding unit-demand bidders to the auction in order to per-turb the optimality. Specifically, suppose there is a unit-demand bidder that hasvalue for the two non-competitive items , with the value for item Z i being slightlyless than player i ’s value for Z i , i ∈ { , } . This endangers the additive bidders’hopes of getting non-negligible utility, since competition from the unit-demandplayer may drive up the prices of Z and Z . The only hope that the additivebidders have is that the unit-demand bidder will have his demand satisfied priorto these final two auctions, in which case the unit-demand bidder would notbother to bid on them. Hence, the two additive bidders would do anything intheir power to guide the auction to such an outcome, even if that means sac-rificing all the competitive items! This is exactly the effect that we achieve inour construction. Specifically, we create an instance where this competing unit-demand bidder has his demand satisfied prior to the auctions for Z and Z ifand only if a very specific outcome occurs: the additive bidders don’t bid at allon all the competitive items, but rather other small-valued bidders acquire thecompetitive items instead. These small-valued bidders contribute almost nothingto the welfare, and therefore all of the welfare from the competitive items is lost.It is useful to compare this example with what would happen if the auc-tions were run simultaneously, rather than sequentially. This uncovers the crucialproperty of sequential auctions that leads to inefficiency: the ability to respondto deviations . If all auctions happened simultaneously, then the behavior of theadditive bidders that we described above could not possibly be an equilibrium:one additive bidder, knowing that his additive competitor bids 0 on all the com-petitive items, would simply deviate to outbid him on the competitive items andget a huge utility. However, because the items are sold sequentially, this devia-tion cannot be undertaken without consequence: the moment one of the additivebidders deviates to bidding on the competitive items, in all subsequent auctionsthe competitor will respond by bidding on subsequent competitive items, leadingto zero utility for the remainder of the auctions. Moreover, this response neednot be punitive, but is rather the only rational response once the auction has leftthe equilibrium path (since the additive bidders know that there is no way toobtain positive utility in subsequent auctions). Thus, in a sequential auction, anadditive player can only extract utility from at most one competitive item, whichis not sufficient to counterbalance the resulting utility-loss due to the increasedcompetition on the last non-competitive item. The Lower Bound.
We now proceed with a formal proof of Theorem 1.Consider an instance with 2 additive players, k unit-demand players and k + 3items. Denote with { a, b } the two additive players and with { p , . . . , p k } the k unit-demand players. Also denote the items with { I , . . . , I k , Y, Z , Z } . Thevaluations of the additive players are represented by the following table of v ij ,where ǫ > I k . . . I Y Z Z a ǫ . . . ǫ b . . . imits of Efficiency in Sequential Auctions 9 In addition the unit-demand valuations for the k players are given by thetable of v ij that follows (an empty entry corresponds to a 0 valuation), thoughnow a valuation of a player when getting a set S is max j ∈ S v ij : I k I k − I k − . . . I I Y Z Z p . . . − ǫ − ǫ − ǫp . . . δ p . . . δ δ . . . p k − δ k − δ k − . . .p k δ k δ k . . . The constants δ , . . . , δ k are chosen to satisfy the following condition: δ k > δ k − > . . . > δ > δ > ǫ (1)Note that, by taking ǫ to be arbitrarily small, we can take each δ i to be arbitrarilysmall as well.In the optimal allocation, player a gets all the items I , . . . , I k and Z , player b gets Z and player p gets Y . The resulting social welfare is k (1 + ǫ ) + 30. Weassume that the auctions take place in the order depicted in the valuation tables: { I k , . . . , I , Y, Z , Z } . We will show that there is a subgame perfect equilibriumfor this auction instance such that the unit-demand players win all the items I , . . . , I k . Specifically, player p i wins item I i , player a wins Z , player b wins Z ,and player p wins Y , resulting in a social welfare of 30 − ǫ + P ki =1 δ i . Taking δ sufficiently small, this welfare is at most 31. This will establish that the price ofanarchy for this instance is at least k (1+ ǫ )+3031 = O ( k ), establishing Theorem 1.Furthermore, we will show that this subgame perfect equilibrium is natural , inthe sense that it survives iterated deletion of weakly dominated strategies.The intuition is the following: after the first k auctions have been sold, player p has to decide if he will target (and win) item Y , or if he will instead targetitems Z and/or Z . If he targets item Y , he competes with player p andafterwards lets players a and b win items Z , Z for free. This decision of player p depends on whether player p has won item I , which in turn depends on theoutcomes of the first k − p can win item I onlyif player p has won item I . In turn, p can win I only if p has won item I and so on. Hence, it will turn out that in order for p to want to target item Y ,it must be that each item I i is sold to bidder p i . Thus, if either player a or b acquires any of the items I , . . . , I k , they will be guaranteed to obtain low utilityon items Z and Z . This will lead them to bidding truthfully on all subsequent I i auctions, leading to a severe drop in utility gained from future auctions.In the remainder of this section, we provide a more formal analysis of theequilibrium in this auction instance. We begin by examining what happens inthe last three auctions of Y, Z and Z , conditional on the outcomes of the first k auctions. We first examine the outcome of auctions Y, Z , Z conditional onthe outcome of auction I : – Case 1: p has won I Player p has marginal value of 10 − δ for item Y . Hence, he is willing tobid at most 10 − δ on item Y .Player p knows that if he loses Y then in the subgame perfect equilibriumin that subgame he will bid 10 − ǫ on Z and Z and lose. Thus he expectsno utility from the future if he loses Y . Thus he is willing to pay at most10 − ǫ for item Y .Since by assumption (1) δ > ǫ , player p will win Y at a price of 10 − δ .Then players a, b will win Z and Z for free. Thus the utilities in this casefrom this subgame are: u ( a ) = 10, u ( b ) = 10, u ( p ) = δ − ǫ , u ( p ) = 0. – Case 2: p has lost I Player p has marginal value of 10 for item Y . Hence, he is willing to bid atmost 10 on item Y .Player p performs the exact same thinking as in the previous case andthereby is willing to bid at most 10 − ǫ for item Y .Thus in this case p will win item Y at a price of 10 − ǫ . Then, as predicted, p will bid 10 − ǫ on Z and Z and lose. Thus the utilities of the players inthis case are: u ( a ) = ǫ , u ( b ) = ǫ , u ( p ) = 0, u ( p ) = ǫ .Now we focus on the auction of item I . As was explained in Paes Leme etal. [19] this auction will be an auction with externalities where each player has adifferent utility for each different winner outcome. This utilities can be conciselyexpressed in a table of v ij ’s where v ij is the value of player i when player j wins. The only players that potentially have any incentive to bid on item I are a, b, p , p , p . The following table summarizes their values for each possiblewinner outcome of auction I as was calculated in the previous case-analysis (wepoint that in the diagonal we also add the actual value that a player acquiresfrom item I to his future utility conditional on winning I ) .[ v ij ] = a b p p p a ǫ ǫ ǫ ǫb ǫ ǫ ǫ ǫp δ − ǫ p ǫ ǫ ǫ δ ǫp δ · hasn’t won I For example, player a obtains utility 10 if player p wins item I . We seefrom the table that, at this auction, everyone except p achieves their maximumvalue when p wins the auction. Player p has value for winning the auction onlyif he hasn’t won I . In addition, since δ > δ , if p hasn’t won I then he candefinitely outbid p on I and therefore p has no chance of winning the auctionof I . As we now show, this implies that there is a unique equilibrium of theauction conditioning on whether or not p has won I : – Case 1: If p has won I then he has no value for I . There exists an equi-librium in undominated strategies where and all players a, b, p , p will bid0, while p bids 0 + . In fact this is in some sense the most natural equilib-rium since it yields the highest utility for a and b . In this case the utility imits of Efficiency in Sequential Auctions 11 of the players from auctions I and onward will be: u ( a ) = 10, u ( b ) = 10, u ( p ) = δ − ǫ , u ( p ) = δ , u ( p ) = 0. – Case 2: If p has lost I , then he has value of δ > δ for I . Hence, p hasno chance of winning item I . Thus, the unique equilibrium that surviveselimination of weakly dominated strategies in this case is for player a to bid1 + , for player b to bid 1, for player p to bid 0, for player p to bid δ − ǫ andfor player p to bid δ . In this case the utility of the players from auctions I and on will be: u ( a ) = 2 ǫ , u ( b ) = ǫ , u ( p ) = 0, u ( p ) = ǫ , u ( p ) = 0.Using similar reasoning we deduce that player p i can win I i only if p i − haswon I i − . If at any point some p i does not win I i then players a and b knowthat from that point onward no p j can win auction I j , and therefore they willget only utility ǫ from Z , Z . Thus there will be no reason for players a and b to allow unit-demand players to continue to win items, and thus the onlyequilibrium strategies from that point on will be for a to bid 1 + on each of I i and b to bid 1. This will lead to player a to get utility O ( ǫ ) from each auctionfor items I i − , . . . , I , and player b to get no utility from these auctions. Thus,at any point in the auction, it is an equilibrium for players a and b to allowthe unit demand player p i to win auction I i conditional on the fact that theyhave allowed all previous unit-demand bidders to win. In particular, in the firstauction, it is an equilibrium for players a and b to allow player p k to win. Weconclude that the strategy profile we described is a subgame perfect equilibriumfor this auction instance. This completes the proof of Theorem 1Finally, as discussed throughout our analysis, the equilibrium described abovesurvives IEWDS. The reason is that, for every item k and bidder i , the proposedequilibrium strategy for bidder i does not require that he bid more than his valuefor item k less his utility in the continuation game subject to not winning item k . As discussed in Paes Leme et al. [19], this property guarantees that no playeris playing a weakly dominated strategy. We now provide some reinterpretations and extensions of our lower bound fromthe previous section, to show that linear inefficiency can occur under severalimportant classes of valuations.
Gross Substitutes.
Since the class of gross substitutes valuations includes alladditive and unit-demand valuations, the example from the previous sectionimmediately implies a linear price of anarchy for gross substitutes valuations.
Budget-Additive.
A valuation is budget additive if it can be written in the form v ( S ) = max n B, P j ∈ S v j o . As it turns out, in the example in the previoussection all valuations are budget additive. The additive players can be thoughtof as having infinite budget. Each of the unit-demand players p i for i ∈ [2 , k ] canbe thought as budget-additive with a budget of δ i and value δ i for items I i and I i +1 and 0 for everything else. Player p has budget of 10 and additive value of δ for I , 10 for Y and 0 for everything else. Player p has budget 10 − ǫ andadditive value of 10 − ǫ for each of Y, Z , Z and 0 for everything else. Thereforethe analysis in the previous section holds even for budget-additive valuations. Additive valuations with budget constraints on payments.
We show that thesame analysis can be applied to a setting in which each player i has an additivevaluation as well as a hard budget constraint B i on his payment. That is, hisutility is quasi-linear as long as his payment is below B i , but becomes minusinfinity if he pays more than B i . Formally, if a player i receives a set S and paystotal price p then his utility u i ( S, p ) is v i ( S ) − p if p ≤ B i , or −∞ otherwise.We will adapt the example from the previous section to the setting of budgetconstraints in a manner similar to the case of budget-additive valuations. Specif-ically, we set the budgets of the players as in the budget-additive case describedabove, but we treat them as payment budgets rather than a cap on valuations.We need to be slightly careful in our analysis under this adaptation, sinceit doesn’t only matter whether a player won or lost an item, but also at whichprice. Specifically, the equilibrium will alter slightly. The additive bidders, apartfrom letting bidder p i win I i , will also have to make him pay enough so that hehas no remaining budget with which to win the subsequent item I i +1 .For player p , we know that his budget is indeed almost exhausted at auction Y whenever he wins, since player p has a substantial value. Thus for auction Y no change in the equilibrium analysis takes place. However, when examiningauction I , if we consider the same equilibrium as in the previous section, thenplayer p i pays nothing and thus still has all his budget to bid on Y and win it.It is in the interest of the additive bidders to ensure that p not only wins, butalso pays at least ǫ , so that he doesn’t have enough budget to win item Y .Player p knows that if he loses the auction for item I then he can use hisbudget to get utility of ǫ from winning Y . If he wins I for a price of t ≥ ǫ then he gets no utility from the future and instead gets a utility of δ − t fromwinning I . Assuming that δ > ǫ , player p is willing to pay more than ǫ towin auction I . Thus, if we assume δ > ǫ , the additive players can bid enoughon item I that player p will win it at some price above ǫ , which will then resultin p winning Y and the additive bidders getting utility 10 from Z and Z . Asimilar analysis holds for the auction of each item I i , for i ∈ [2 , k ]: the additiveplayers need to make sure that each bidder p i wins I i , and also pays enoughso that he doesn’t have enough budget to tilt player p i +1 on getting his nextitem rather than I i +1 . However, observe that if player p i loses auction I i , thensubsequently the additive players will switch to winning all the remaining items,since there is no hope to make the unit-demand bidders win their items; so itis in the interest of each player I i to accept any price up to δ i and thereforethe additive players can completely exhaust his budget. With this change in theequilibrium strategies, our analysis in the previous section carries over, and weconclude that the price of anarchy in this instance is Ω ( k ). imits of Efficiency in Sequential Auctions 13 Our lower bound establishes that if items are sold sequentially, then arbitrarilyinefficient outcomes can result at equilibrium even when all agents have grosssubstitutes valuations. The constructions depend on the items being sold in anarbitrary order. A natural question arises: does there always exist an order overthe items such that the resulting outcome is efficient, or approximately efficient?In this section we discuss this problem in the context of unit-demand bidders.Recall that, for unit-demand bidders, selling items in an arbitrary order alwaysresults in an outcome that achieves at least half of the optimal social welfare.Additionally, it is known by [19] that if any order is allowed then the uniquesubgame-perfect equilibrium that survives IEWDS can be inefficient, achievingonly a 3 / / v i for getting one item from some interest set S i . We conjecture that, for the caseof single-valued unit-demand bidders, if the auctioneer can choose the order inwhich the objects are sold, then it is possible to recover the optimal welfare at allnatural equilibria. Indeed, we make a stronger conjecture: there exists an orderin which the VCG outcome (allocation and payments) occurs at equilibrium. Conjecture 1.
For every instance of single-valued unit-demand bidders, thereexists an order over the items such that the corresponding sequential auctionadmits a subgame perfect equilibrium that survives IEWDS and that replicatesthe VCG outcome.Observe that such a result cannot hold for both additive and unit-demand bid-ders as is portrayed by our simple example in Section 3, where all items areidentical and hence, under any ordering, the unique subgame-perfect equilibriumthat survivies IEWDS is inefficient. Our conjecture also stems from the fact thatfor the case of single-valued unit-demand bidders the optimization problem isa matroid optimization problem. It is known by [19] that a form of sequentialcut auction for matroids always leads to a VCG outcome. The difference is thatsequential item-auctions do not correspond to auctions across cuts of the ma-troid. However, it is feasible that under some ordering the same behavior as ina sequential cut auction will be implemented.As progress toward this conjecture, we will present a subset of item orderings,the augmenting path orderings , which we believe always contains an ordering thatsatisfies Conjecture 1. For instance, we show in Appendix B that the 3 / Consider a profile of single-valued unit-demand valuations. Let x denote theVCG allocation (i.e., x i is the item allocated to bidder i ). We also write x ( − i ) todenote the VCG allocation when bidder i is excluded. For each i , the allocations x and x ( − i ) define a directed bipartite graph between players and objects, wherethere is an edge between player k and item j if x ( − i ) k = j but x k = j , and thereis an edge from item j to player k if x ( − i ) k = j but x k = j . It is known that, foreach player i , this graph is always a directed path from player i to some otherplayer k ; this is the augmenting path for player i and player k is the price setter of player i , i.e. the VCG price of player i is v k . With no loss of generality weassume that every player has a price setter k .Given a welfare-optimal matching π , that matches each player i to an item π ( i ), consider the following forest construction. Consider all price setters in de-creasing value order. For each price setter k , we will create a tree and add it tothe forest, as follows. Consider all the items that are in the interest set of k , S k ,that are not yet in the forest. Add each such item to the tree as a child of player k . Next, from each such item j , consider its optimally matched player π − ( j )and add this player to the tree as a child of j . For each player i that was added,consider all items that are in the interest set of i , S i , that are not yet in theforest, and add each of these items to the tree as a child of i . We continue thisprocess, which is essentially a breadth-first traversal of the set of items, untilthere is no new item to be added.The above process creates a forest that contains a node for each item, foreach player that is allocated an item in the optimal allocation, and for eachprice setter. Additionally, each player belongs to the tree rooted at his pricesetter and his unique path in the tree to the price setter is an augmenting pathin the initial bipartite graph. The reasoning is as follows: each tree contains allpossible alternating paths ending at the price-setter, except alternating pathsthat contain items and players who have been included in the tree of a pricesetter with larger value. Since a player’s price setter is the largest unallocatedplayer with which he is connected, through an alternating path, the claim follows.We will refer to the above forest as the augmenting path graph G . Givenan augmenting path graph G , a post-order item traversal of G is a depth-first,post-order traversal of the nodes of G , restricted to the nodes corresponding toitems and rooted at price setters. Note that this is an ordering over the itemsin the auction. We also assume that trees are traversed in decreasing order ofprice-setters. Also note that this order is not necessarily unique, as it does notspecify the order in which the children of a given node should be traversed. Definition 2
The set of augmenting path orderings of the items is the set oforderings corresponding to post-order item traversals of G . Our (refined) conjecture is that, for every instance of single-valued unit-demand bidders, there exists an augmenting path ordering such that the corre-sponding sequential auction admits a subgame perfect equilibrium that replicatesthe VCG outcome. As an example, we show in Appendix B that this conjectureholds for the 3 / all augmenting path orderings lead to efficient outcomesat equilibrium: there are examples in which multiple augmenting path orderingsexist, and some orderings lead to inefficient outcomes at equilibrium. imits of Efficiency in Sequential Auctions 15 References
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A Iterated Elimination of Weakly Dominated Strategies
When considering subgame perfect equilibria of sequential item auctions, wewish to restrict our attention to “natural” equilibria, that exclude (for example)dominated overbidding strategies. We therefore consider a natural and well-studied refinement of the set of subgame perfect equilibria: those that surviveiterated elimination of weakly dominated strategies (IEWDS). A strategy s is weakly dominated by a strategy s ′ if, for every profile of other players’ strategies s − i , we have u i ( s, s − i ) ≤ u i ( s ′ , s − i ), and moreover there exists some s − i suchthat u i ( s, s − i ) < u i ( s ′ , s − i ). We can now define what it means for a strategyprofile to survive iterated elimination of weakly dominated strategies. Definition 1.
Given an n -player game defined by strategy sets S , . . . , S n andutilities u i : S × . . . × S n → R we define a valid procedure for eliminating weaklydominated strategies as a sequence { S ti } such that for each t there is an i suchthat S tj = S t − j for j = i , S ti ⊆ S t − t , and for all s i ∈ S t − i \ S ti there is some s ′ i ∈ S ti such that u i ( s ′ i , s − i ) ≥ u i ( s i , s − i ) for all s − i ∈ Q j = i S tj and the inequalityis strict for at least one s − i . We say that a strategy profile s survives iteratedelimination of weakly dominated strategies (IEWDS) if, for any valid procedure { S ti } , s i ∈ ∩ t S ti . B Augmenting Path Orderings: An Example
In [19], it was shown that there exist single-valued unit-demand auctions in whichinefficient outcomes can occur when items are sold sequentially in an arbitraryorder. In this section we motivate that augmenting path ordering by showingthat, for this example, the efficient outcome occurs when the items are soldaccording to their augmenting path order.We begin by recalling the example. There are three items, { A, B, C } , and 4players { a, b, c, d } . We fix an arbitrarily small constant ǫ >
0. Recall that thevaluation of each player is specified by a real value v and a set S of items ofinterest; the player then has value v for any item in S and value 0 for any otheritem. The valuations in our example are given by: – v a = ǫ and S a = { A } , – v b = 1 and S b = { A, C } , – v c = 1 and S c = { B, C } , and – v d = 1 − ǫ and S d = { B } .The welfare-optimal allocation is ( x a , x b , x c , x d ) = ( ∅ , { A } , { C } , { B } ), for a so-cial welfare of 3 − ǫ . The VCG prices are ( p a , p b , p c , p d ) = (0 , ǫ, ǫ, ǫ ). Note that,in the terminology of Section 6, player a is the price-setter for each of the otherplayers. In [19] it is shown that if the items are auctioned in the order ( A, B, C ),then the unique subgame perfect equilibrium that survives IEWDS leads to aninefficient outcome.What are the augmenting path orderings in this example? In this example,the augmenting path graph is a line, given by nodes ( a, A, b, C, c, B, d ) in that imits of Efficiency in Sequential Auctions 17 sequence. There is therefore a unique augmenting path ordering over the items:the order (
B, C, A ).We can now solve for the subgame perfect equilibrium of the auction whenitems are sold in this order. We do so by analyzing the item auctions in reverseorder. When item A is sold, the outcome depends on whether or not player b won item C : if so, player a will win item A for a price of 0, yielding u a = ǫ ; if not,then player b will win item A for a price of ǫ , yielding u b = 1 − ǫ and u a = 0. Thisallows us to determine the outcome of the auction for item C : because player b knows that she can win item A for a price of ǫ , she is willing to bid no morethan ǫ on item C . Thus, if player c did not previously win item B , then player c can win item C with a bid of ǫ + , yielding u c = 1 − ǫ . This ultimately allows usto determine the outcome of the first auction, the auction for item B . Becauseplayer c knows that he can win item B for a price of ǫ , she is willing to bid nomore than ǫ on item B . Since player d obtains positive utility only if she winsitem B , she is willing to bid as much as v d = 1 − ǫ on item B . We thereforehave that player d will choose to win item B with a bid of ǫ + , obtaining utility u d = 1 − ǫ . Applying our analysis of the subsequent auctions, we conclude thatbidder c will win item C for a price of ǫ , and then bidder b will win A for a priceof ǫ .Note that this subgame perfect equilibrium, which is the unique equilibriumin undominated strategies, precisely implements the VCG outcome. Moreover,our analysis extends easily to other values of v b , v c , and v d , as long as they areall greater than ǫ . C Not all Augmenting Path Orderings lead to Efficiency
We now show that if the augmenting path graph is not a line, then some aug-menting path orderings do not result in an efficient outcome, even if valuationsare unit demand single-valued.The example is as follows. There are 3 items, say { A, B, C } . There are 4players. Player 1 wants all items and has value 1. Player 2 wants only item B and has value 2. Player 3 wants item B or C and has value 3. Player 4 wantitem A or C and has value 4.In this example, the VCG outcome is ( x , x , x , x ) = ( ∅ , B, C, A ), and theVCG prices are ( p , p , p , p ) = (0 , , , i being the child of item x i . For this graph, every order over the items isan augmenting path order.Suppose the items are sold in the order ( A, B, C ). In the VCG outcome,player 4 obtains utility v − p = 3. In the sequential play corresponding to theVCG outcome, players 1 and 4 both bid their values on item A . Consider thefollowing deviation by player 4. When item A is sold, he bids 0, causing player1 to win item A . Item B will sell next; players 2 and 3 will bid on it. Considerwhat would happen if player 2 wins item B : in this case, players 3 and 4 both bidtheir values on item C , and hence player 4 wins C and player 3 ends with utility