aa r X i v : . [ ec on . T H ] F e b Robust double auction mechanisms*
Kiho YoonDepartment of Economics, Korea University145 Anam-ro, Seongbuk-gu, Seoul, Korea 02841 [email protected]://econ.korea.ac.kr/~kiho
Abstract
We study the robust double auction mechanisms, that is, the double auctionmechanisms that satisfy dominant strategy incentive compatibility, ex-post in-dividual rationality, ex-post budget balance and feasibility. We first establishthat the price in any deterministic robust mechanism does not depend on thevaluations of the trading players. We next establish that, with the non-bossinessassumption, the price in any deterministic robust mechanism does not dependon players’ valuations at all, whether trading or non-trading, i.e., the price isposted in advance. Our main result is a characterization result that, with thenon-bossiness assumption along with other assumptions on the properties of themechanism, the posted price mechanism with an exogenous rationing rule is theonly deterministic robust double auction mechanism. We also show that, evenwithout the non-bossiness assumption, it is quite difficult to find a reasonablerobust double auction mechanism other than the posted price mechanism withrationing.Keywords: double auction, robust mechanism design, dominant strategy, budgetbalance, ex-post individual rationalityJEL Classification: C72; D47; D82 * This work was supported by a Korea University Grant (K1911191). Introduction
Double auctions are widely used in many two-sided market situations such as stockexchanges as well as commodity markets. In a typical double auction, (i) there are manysellers and many buyers who have private information about their true valuations for thegood, (ii) they simultaneously submit their respective offers and bids to the mechanism(or the center, the mediator, the clearing house, the social planner, etc.), and (iii) themechanism determines the volume and the terms of trade according to a pre-determinedrule. Double auctions approximate many other trading settings in the real world. Theyalso have great theoretical value in the study of price formation under private information.In this paper, we study the double auction mechanisms that satisfy dominant strategyincentive compatibility, ex-post individual rationality, ex-post budget balance and feasibil-ity. We call this family of double auction mechanisms as robust double auction mechanisms.A special instance of the robust double auction mechanisms is the robust bilateral tradingmechanism for the environments in which there is one seller who initially owns one indi-visible good and there is one buyer who is interested in obtaining the good. Hagerty andRogerson (1987) and ˇCopiˇc and Ponsat´ı(2016) have shown that the only robust bilateraltrading mechanism is the posted price mechanism, i.e., the mechanism in which a price isposted in advance and the seller and the buyer either trade at this price or do not trade atall. The purpose of this paper is to obtain a similar characterization for the general robustdouble auction mechanisms.In the next section, we first establish that the price in any deterministic robust mech-anism does not depend on the valuations of the trading players. This result in particularimplies that a deterministic robust bilateral trading mechanism is a deterministic postedprice mechanism. In contrast, we show that a robust double auction mechanism may notbe a posted price mechanism when there are more than two players. We next establish Hence, Hagerty and Rogerson’s conjecture (in page 97 of their paper) that the posted price mechanismmay be the only robust mechanism in ‘ n -person trading problems’ is incorrect. Section 4concludes.There is a large literature on double auction mechanisms: Representative works in-clude Wilson (1985), Rustichini et al. (1994), Satterthwaite and Williams (2002), Crippsand Swinkels (2006) and Satterthwaite et al. (2014). Most of the papers in this literatureexamine double auction mechanisms under the Bayesian incentive compatibility postulate.In contrast, the current paper focuses on robust double auction mechanisms. Robustnessis desirable since the equilibrium behavior does not depend on the fine details of the envi-ronments, such as players’ beliefs about each other. We note that the only other doubleauction mechanisms that satisfy dominant strategy incentive compatibility is McAfee’s(1992) mechanism and Yoon’s (2001) modified Vickrey double auction mechanism, butthese mechanisms do not satisfy ex-post budget balance. Value-respecting mechanisms are those mechanisms that select the traders according to the valua-tions. See Section 3 for details. See also Kojima and Yamashita (2017) and the references therein for double auctions with interde-pendent values. There is a set S = { , . . . , m } of sellers and a set B = { , . . . , n } of buyers for a good.Each seller owns one indivisible unit of the good to sell, and each buyer wants to buyat most one unit of the good. Hence, there are m units of the good available. Seller i ’sprivately known valuation for the good is denoted by s i , and buyer j ’s privately knownvaluation for the good is denoted by b j . We assume that s i ∈ [0 ,
1] for all i ∈ S as well as b j ∈ [0 ,
1] for all j ∈ B . Let s = ( s , . . . , s m ) and b = ( b , . . . , b n ). In addition, let v = ( s, b )and V = [0 , m + n . Hence, V is the set of possible valuation profiles of the sellers andthe buyers. We use the usual notation such as v = ( s i , s − i , b ) = ( s, b j , b − j ). We also use v = ( v i , v − i ) for i ∈ S and v = ( v j , v − j ) for j ∈ B . That is, ( v i , v − i ) = ( s i , s − i , b ) for i ∈ S where v i = s i and v − i = ( s − i , b ), and ( v j , v − j ) = ( s, b j , b − j ) for j ∈ B where v j = b j and v − j = ( s, b − j ). See Bergemann and Morris (2013) for an excellent introduction. p i : V → [0 ,
1] be the (probabilistic) allocation rule for seller i ∈ S , and let q j : V → [0 ,
1] be the (probabilistic) allocation rule for buyer j ∈ B . Both p i ( v ) and q j ( v )denote the probability of trade. In addition, let x i : V → IR be the transfer rule for seller i ∈ S such that x i ( v ) is the receipt of the money seller i gets, and let y j : V → IR bethe transfer rule for buyer j ∈ B such that y j ( v ) is the payment of the money buyer j makes. Note that x i ( v ) may be positive or negative: when it is negative, seller i pays thatamount. Likewise, y j ( v ) may be positive or negative: when it is negative, buyer j receivesthat amount. The payoff of seller i ∈ S and buyer j ∈ B is denoted respectively by u i ( v ) = x i ( v ) − p i ( v ) s i and u j ( v ) = q j ( v ) b j − y j ( v ) . Let p = ( p , . . . , p m ) , q = ( q , . . . , q n ) , x = ( x , . . . , x m ), and y = ( y , . . . , y n ). Themechanism ( p, q, x, y ) is said to be dominant strategy incentive compatible if u i ( s, b ) ≥ x i ( s ′ i , s − i , b ) − p i ( s ′ i , s − i , b ) s i ∀ i ∈ S , ∀ s i , ∀ s ′ i , ∀ s − i , ∀ b ; u j ( s, b ) ≥ q j ( s, b ′ j , b − j ) b j − y j ( s, b ′ j , b − j ) ∀ j ∈ B , ∀ b j , ∀ b ′ j , ∀ b − j , ∀ s ( IC )and ex-post individually rational if u i ( v ) ≥ ∀ i ∈ S and u j ( v ) ≥ ∀ j ∈ B , ∀ v ∈ V. ( IR )The mechanism ( p, q, x, y ) is said to be feasible if X i ∈S p i ( v ) = X j ∈B q j ( v ) ∀ v ∈ V ( P )and ex-post budget balancing if X i ∈S x i ( v ) = X j ∈B y j ( v ) ∀ v ∈ V. ( BB )We will call the mechanisms that satisfy ( IC ), ( IR ), ( P ), and ( BB ) as robust doubleauction mechanisms. We first have the following standard lemma. Note well that we may have P i ∈S p i ( v ) > P j ∈B q j ( v ) if the mechanism withholds/destroys someunits. We exclude this possibility. emma 1. The mechanism ( p, q, x, y ) is dominant strategy incentive compatible (IC) ifand only if(i) ∀ i ∈ S , ∀ v − i : p i ( v ) is weakly decreasing in v i .(ii) ∀ j ∈ B , ∀ v − j : q j ( v ) is weakly increasing in v j .(iii) ∀ i ∈ S , ∀ v : u i ( v ) = R v i p i ( w, v − i ) dw + u i (1 , v − i ) .(iv) ∀ j ∈ B , ∀ v : u j ( v ) = R v j q j ( w, v − j ) dw + u j (0 , v − j ) . Proof: Omitted since it is standard. See Myerson (1981), Myerson and Satterthwaite(1983), etc.
Q.E.D.
We will restrict our attention to the deterministic allocation rules that take only thevalue zero or one. Thus, p i : V → { , } for all i ∈ S and q j : V → { , } for all j ∈ B . Let S ( v ) = { i ∈ S| p i ( v ) = 1 } denote the set of trading sellers at v , and let B ( v ) = { j ∈ B| q j ( v ) = 1 } denote the set of trading buyers at v .A possible property that the mechanism may satisfy is that all sellers face one identicalprice and all buyers face another identical price. Assumption 1. (Common price) ∀ v ∈ V, ∀{ i, i ′ } ⊆ S ( v ) , ∀{ j, j ′ } ⊆ B ( v ): x i ( v ) = x i ′ ( v ) and y j ( v ) = y j ′ ( v ) . Hence, there are two prices, the seller price (ask price) of π s ( v ) and the buyer price(bid price) of π b ( v ). Of course, these two prices may be the same. Note that π s ( v ) and π b ( v ) may vary with v . This assumption is reasonable since all units of the good areidentical and the players supply or demand at most one unit. In fact, this assumptionseems to be a prerequisite for any centralized trading institution. Example 4 below showsthat we may pair one seller and one buyer and let the trade occurs only between them.Then, each seller-buyer pair is a separated market and different prices across the pairs mayemerge. Although we do not explicitly consider decentralized trading or resale possibilitiesin the current mechanism design approach, if the buyers face different prices, say, then itis conceivable that a buyer of the more expensive unit could find a new trade opportunity6ith a seller to the advantage of both.Another possible property that the mechanism may satisfy is that the payoff of aplayer with the worst possible valuation is equal to zero. Assumption 2. (Zero payoff for the worst type) ∀ i ∈ S , ∀ v − i : u i (1 , v − i ) = 0 and ∀ j ∈ B , ∀ v − j : u j (0 , v − j ) = 0.Note that the condition u j (0 , v − j ) = 0 for the buyer j ∈ B holds if and only if y j (0 , v − j ) = 0 holds. The latter condition is often assumed in the literature. Observe that( IR ) implies y j (0 , v − j ) ≤ y j (0 , v − j ) ≥
0, leadingus to y j (0 , v − j ) = 0 and hence u j (0 , v − j ) = 0. The seller case is essentially symmetric.Observe that ( IR ) implies x i (1 , v − i ) ≥ p i (1 , v − i ) whereas the no subsidy condition implies x i (1 , v − i ) ≤ p i (1 , v − i ), leading us to u i (1 , v − i ) = 0.We now characterize some necessary conditions that a robust double auction mecha-nism must satisfy. We use the notation | A | to denote the cardinality of an arbitrary set A ,and the notation v − T to denote the valuation profile of the players not in the set T ⊆ S ∪B . Proposition 1.
Let the mechanism ( p, q, x, y ) be a deterministic robust double auctionmechanism, that is, dominant strategy incentive compatible ( IC ) , ex-post individually ra-tional ( IR ) , feasible ( P ) , and ex-post budget balancing ( BB ) . Under Assumptions 1 and 2,the mechanism ( p, q, x, y ) must be the following form: for all v ∈ V and T ( v ) = S ( v ) ∪ B ( v ) ,(i) | S ( v ) | = | B ( v ) | ;(ii) x i ( v ) = y j ( v ) = π ( v − T ( v ) ) for all i ∈ S ( v ) and for all j ∈ B ( v ) ;(iii) x i ( v ) = y j ( v ) = 0 for all i / ∈ S ( v ) and for all j / ∈ B ( v ) ;(iv) v i ≤ π ( v − T ( v ) ) for all i ∈ S ( v ) and v j ≥ π ( v − T ( v ) ) for all j ∈ B ( v ) . Remark:
Property (i) says that the number of trading sellers must be equal to the numberof trading buyers. Property (ii) says that all trading sellers and trading buyers face thesame price π ( v − T ( v ) ) that is independent of their valuations. Note in particular that there The no subsidy condition is (i) ∀ j ∈ B , ∀ v : y j ( v ) ≥
0, and (ii) ∀ i ∈ S , ∀ v : x i ( v ) ≤ p i ( v ).
7s only one price, i.e., the seller price is identical to the buyer price. Property (iii) says thatthe sellers and the buyers who do not trade neither receive nor pay any money. Property(iv) says that every trading seller’s valuation is less than or equal to the price and everytrading buyer’s valuation is greater than or equal to the price.Proof: It is obvious that ( P ) implies property (i). Next, since p i ( v i , v − i ) ∈ { , } , property(i) of Lemma 1 implies: for a given v − i , there exists z i ( v − i ) ∈ [0 ,
1] such that p i ( v i , v − i ) = (cid:26) v i < z i ( v − i );0 if v i > z i ( v − i ).Likewise, property (ii) of Lemma 1 implies: for a given v − j , there exists z j ( v − j ) ∈ [0 , q j ( v j , v − j ) = (cid:26) v j < z j ( v − j );1 if v j > z j ( v − j ).That is, z i ( v − i ) = sup { v i ∈ [0 , | p i ( v i , v − i ) = 1 } for i ∈ S ; z j ( v − j ) = inf { v j ∈ [0 , | q j ( v j , v − j ) = 1 } for j ∈ B . Then, by property (iii) of Lemma 1 and Assumption 2, u i ( v ) = (cid:26) z i ( v − i ) − v i when p i ( v i , v − i ) = 1;0 when p i ( v i , v − i ) = 0.Likewise, by property (iv) of Lemma 1 and Assumption 2, u j ( v ) = (cid:26) v j − z j ( v − j ) when q j ( v j , v − j ) = 1;0 when q j ( v j , v − j ) = 0.Hence, we have:(i) For a seller i ∈ S : x i ( v ) = n z i ( v − i ) when p i ( v ) = 1;0 otherwise.(ii) For a buyer j ∈ B : y j ( v ) = n z j ( v − j ) when q j ( v ) = 1;0 otherwise.8e then have X i ∈ S ( v ) x i ( v ) = X j ∈ B ( v ) y j ( v ) ∀ v ∈ V by ( BB ) and the fact that x i ( v ) = 0 for all i / ∈ S ( v ) and y j ( v ) = 0 for all j / ∈ B ( v ).Since P i ∈ S ( v ) x i ( v ) = | S ( v ) | π s ( v ) and P j ∈ B ( v ) y j ( v ) = | B ( v ) | π b ( v ) by Assumption 1 and | S ( v ) | = | B ( v ) | by ( P ), we have π s ( v ) = π b ( v ). Let us denote this price by π ( v ).If T ( v ) = ∅ , then π ( v ) = z k ( v − k ) for all k ∈ T ( v ). Hence, π ( v ) does not depend on v k for any k ∈ T ( v ). So, we can write this price as π ( v − T ( v ) ) for all v ∈ V and T ( v ) ⊆ S ∪ B .This proves that properties (ii) and (iii) hold. It is now clear that ( IR ) implies property(iv). Q.E.D.
Observe that, for bilateral trading with one seller and one buyer, i.e., S = B = { } ,we always have T ( v ) = S ∪ B unless T ( v ) = ∅ . Hence, π ( v ) does not depend on either s or b whenever T ( v ) = ∅ . This mechanism can be implemented by a posted pricemechanism: A constant price π is posted in advance and the trade occurs when andonly when s < π < b . Since it is straightforward to show that Assumptions 1 and2 are satisfied for bilateral trading environments, Proposition 1 implies one of Hagertyand Rogerson’s (1987) results that a deterministic robust double auction mechanism forbilateral trading environments is a deterministic posted price mechanism.
Corollary 1.
A deterministic double auction mechanism for bilateral trading environ-ments is robust if and only if it is a deterministic posted price mechanism.
Proof: Observe that Assumption 1 is trivially satisfied for bilateral trading environments.Observe next that p ( s , b ) = q ( s , b ) by ( P ) and x ( s , b ) = y ( s , b ) by ( BB ).Hence, the total surplus, i.e., the sum of players’ payoffs, is p ( s , b )( b − s ). Thisbecomes (i) p (1 , b )( b − ≤ s = 1, and (ii) − p ( s , s ≤ b = 0.Since the total surplus is nonpositive when either s = 1 or b = 0 whereas each player’s We assume that players decide not to trade when they are indifferent between trade and no trade.We follow this convention throughout the paper. IR ), the only possibility is that each player gets a zero payoff.Thus, Assumption 2 is always satisfied for bilateral trading environments. Proposition1 then implies that a deterministic mechanism in these environments is robust only if itis a deterministic posted price mechanism. The converse also holds since a deterministicposted price mechanism is clearly robust. Q.E.D.
In contrast, a robust double auction mechanism may not be a posted price mechanismwhen there are more than two players, as the following example demonstrates.
Example 1.
Suppose there are one seller and two buyers, and consider a mechanism inwhich (i) players first submit their valuations (which may or may not be true valuations),(ii) the price is set to the lower of the buyers’ reported valuations, and (iii) a trade occursbetween the seller and the buyer of the higher valuation when and only when the seller’svaluation is less than the price. When s = 0 . b = 0 .
7, and b ∈ (0 . , . π = b depends on buyer 2’s valuation.Observe in particular that the mechanism in this example is dominant strategy in-centive compatible ( IC ) mainly due to the fact that the traders cannot affect the termsof trade. On the other hand, the non-trading buyer affects the terms of trade withoutaltering her own allocation or payment. We now introduce another property often used inthe literature to obtain a sharper result. The mechanism ( p, q, x, y ) is said to be non-bossy if(i) For all i ∈ S and for all v i , v ′ i , and v − i : if p i ( v i , v − i ) = p i ( v ′ i , v − i ) and x i ( v i , v − i ) = x i ( v ′ i , v − i ), then p ( v i , v − i ) = p ( v ′ i , v − i ), q ( v i , v − i ) = q ( v ′ i , v − i ), x ( v i , v − i ) = x ( v ′ i , v − i )and y ( v i , v − i ) = y ( v ′ i , v − i ).(ii) For all j ∈ B and for all v j , v ′ j , and v − j : if q j ( v j , v − j ) = q j ( v ′ j , v − j ) and y j ( v j , v − j ) = y j ( v ′ j , v − j ), then p ( v j , v − j ) = p ( v ′ j , v − j ), q ( v j , v − j ) = q ( v ′ j , v − j ), x ( v j , v − j ) = x ( v ′ j ,v − j ) and y ( v j , v − j ) = y ( v ′ j , v − j ). 10n words, a mechanism is non-bossy if no player can change others’ allocations or paymentswithout changing his/her allocation or payment. Assumption 3. (Non-bossiness) The mechanism ( p, q, x, y ) is non-bossy.The non-bossiness assumption is an extra property imposed on the mechanism ontop of the properties required by Proposition 1 under which the posted price mechanismresults. We note that this assumption is also imposed (along with additional assumptions)in the works of Barber`a and Jackson (1995) and Miyagawa (2001) cited in the introduction.We have:
Proposition 2.
Let the mechanism ( p, q, x, y ) be a deterministic robust double auctionmechanism, that is, dominant strategy incentive compatible ( IC ) , ex-post individually ra-tional ( IR ) , feasible ( P ) , and ex-post budget balancing ( BB ) . Under Assumptions 1, 2,and 3, the mechanism ( p, q, x, y ) must be the following form: there exists a constant price π such that, for all v ∈ V and T ( v ) = S ( v ) ∪ B ( v ) ,(i) | S ( v ) | = | B ( v ) | ;(ii) x i ( v ) = y j ( v ) = π for all i ∈ S ( v ) and for all j ∈ B ( v ) ;(iii) x i ( v ) = y j ( v ) = 0 for all i / ∈ S ( v ) and for all j / ∈ B ( v ) ;(iv) v i ≤ π for all i ∈ S ( v ) and v j ≥ π for all j ∈ B ( v ) . Proof: We know by Proposition 1(ii) that the price does not depend on the valuationprofile of the traders, i.e., π ( v ) = π ( v − T ( v ) ) for all v ∈ V . We now show that it does notdepend on the valuation profile of the non-traders, either. Let there be a non-trading seller i ∈ S c ( v i , v − i ) ∩ S c ( v ′ i , v − i ). This means p i ( v i , v − i ) = 0 = p i ( v ′ i , v − i ), and x i ( v i , v − i ) =0 = x i ( v ′ i , v − i ) by Proposition 1(iii). We have π ( v i , v − T ( v ) \{ i } ) = π ( v i , v − i ) = π ( v ′ i , v − i ) = π ( v ′ i , v − T ( v ) \{ i } ) where the second inequality holds by Assumption 3. The argument for anon-trading buyer is similar. Q.E.D. This property is first introduced by Satterthwaite and Sonnenschein (1981). See Thomson (2016)for a thorough discussion of non-bossiness. π ) and the demand (the number ofbuyers with valuations greater than π ).A rationing rule can take many different forms: It may depend on the names ofthe players, on the reported valuations of the players, on nothing at all (i.e., it may becompletely random), etc. We give a classification of rationing rules as follows: A rationingrule is said to be endogenous if it is based on the reported valuations of the players, andit is said to be exogenous if it is not endogenous.We show in the appendix that an endogenous rationing rule violates either dominantstrategy incentive compatibility or non-bossiness. To understand this intuitively, supposethat, when the supply is greater than the demand, the sellers are selected in an increas-ing order of valuations, that is, the seller with the lowest valuation is selected first, theseller with the second lowest valuation is selected second, and so on until the demand isfulfilled. Then, a seller with s i < π has an incentive to bid lower. The buyer’s incentiveto misreport when the demand is greater than the supply can be symmetrically described.This reasoning applies as well to any other endogenous rationing rule as long as a player’sbid affects his/her own allocation. Next, if the rationing rule is such that a player’s bidaffects another player’s allocation while not affecting his/her allocation, then it is bossy.Hence, if the double auction mechanism is dominant strategy incentive compatible ( IC )and non-bossy, then its rationing rule must be exogenous.We next observe that any deterministic exogenous rationing rule can be described asfollows. Let σ : S → S be a permutation of sellers such that seller σ (1) is ordered first, Note well that a rationing rule may be deterministic or probabilistic. An example of the latter is therandom rationing in which each player is selected with equal probability. Since a probabilistic rationingrule induces a probabilistic mechanism, we restrict our attention to deterministic exogenous rationing σ (2) is ordered second, and so on. Likewise, let β : B → B be a permutation ofbuyers such that buyer β (1) is ordered first, buyer β (2) is ordered second, and so on. Wenow define a deterministic robust double auction mechanism with a given pair ( σ, β ) ofpermutations. Definition 2. ( The deterministic posted price mechanism with an exogenous rationingrule ) A constant price π is posted in advance, and a rationing rule ( σ : S → S , β : B → B ) isgiven. Players submit their respective valuations, which may or may not be true valuations.Let ˆ S ( v ) = { i ∈ S| v i < π } and ˆ B ( v ) = { j ∈ B| v j > π } . If | ˆ S ( v ) | = | ˆ B ( v ) | , then all tradersin the set ˆ S ( v ) and ˆ B ( v ) trade at price π . Otherwise, rationing is needed: if | ˆ S ( v ) | > | ˆ B ( v ) | ,then | ˆ B ( v ) | sellers in ˆ S ( v ) are selected in the order of σ to trade with the buyers in ˆ B ( v );if | ˆ S ( v ) | < | ˆ B ( v ) | , then | ˆ S ( v ) | buyers in ˆ B ( v ) are selected in the order of β to trade withthe sellers in ˆ S ( v ). Remark:
An alternative form of this mechanism is that each player only indicates his/herwillingness to trade at π , i.e., each player responds with ‘yes’ or ‘no’ to trade at the priceof π . Hence, each seller responds with ‘yes’ if and only if his valuation is less than π , andeach buyer responds with ‘yes’ if and only if her valuation is greater than π .The following example illustrates how a deterministic posted price mechanism withan exogenous rationing rule works. Example 2.
The posted price is π = 0 .
6. There are 4 sellers and 3 buyers. Let σ (1) =3 , σ (2) = 4 , σ (3) = 2 and σ (4) = 1 and let β (1) = 1 , β (2) = 2 and β (3) = 3. When s =0 . , s = 0 . , s = 0 . s = 0 . b = 0 . , b = 0 . b = 0 .
4, we have ˆ S ( v ) = { , , } and ˆ B ( v ) = { , } . Since | ˆ S ( v ) | > | ˆ B ( v ) | , the seller side has to be rationed in the order of σ . Hence, sellers 2 and 3 are to trade with buyers 1 and 2.We have: rules. roposition 3. The deterministic posted price mechanism with an exogenous rationingrule is a robust double auction mechanism that satisfies Assumptions 1, 2, and 3.
Proof: Let us first show that ( IC ) holds. Consider a seller i ∈ S , and let s i be his truevaluation. We divide the cases.Case 1 s i < π : He does not have an incentive to bid lower since it does not change theprobability that he trades. He does not have an incentive to bid higher either since (i) itdoes not change the probability that he trades (if his bid is less than π ), or (ii) it rendersthe probability of trade to zero (if his bid is greater than or equal to π ) from which he getsa payoff of zero. On the other hand, he gets a nonnegative payoff from bidding truthfully.Case 2 s i ≥ π : If he bids s ′ i greater than or equal to π , no change occurs. If s ′ i < π , thenhe has a nonnegative probability of trade from which he gets a nonpositive payoff. On theother hand, he gets a payoff of zero from bidding truthfully.The buyer case can be similarly shown. It is now straightforward to see that the determin-istic posted price mechanism with an exogenous rationing rule satisfies ( IR ), ( P ), ( BB )and Assumptions 1, 2, and 3. Q.E.D.
Summarizing the discussion, we obtain the following characterization result.
Theorem 1.
The mechanism ( p, q, x, y ) is a deterministic robust double auction mecha-nism that satisfies Assumptions 1, 2, and 3 if and only if it is a deterministic posted pricemechanism with an exogenous rationing rule. Hence, the deterministic posted price mechanism with an exogenous rationing rule isthe only robust double auction mechanism that satisfies Assumptions 1, 2, and 3. We havealready seen in Example 1 that there exists a robust double auction mechanism other thanthe posted price mechanism when Assumption 3 is not satisfied and there are only threeplayers. This example can be extended as follows.
Example 3.
Suppose there are two seller and two buyers, and consider a mechanism inwhich (i) players first submit their valuations (which may or may not be true valuations),14ii) the price is set to the lower of the buyers’ reported valuations, and (iii) a trade occursbetween seller 1 and the buyer of the higher valuation when and only when seller 1’svaluation is less than the price.Observe that seller 2 in this example is a dummy player who is never involved inthe trade. Observe also that this example can be easily extended to the case of arbitrarynumber of sellers and buyers by adding more dummy players. Hence, there exists a robustdouble auction mechanism other than the posted price mechanism for any number ofplayers when Assumption 3 is not satisfied. On the other hand, both Assumption 1 andAssumption 2 are essential for Proposition 1 as the following examples demonstrate.
Example 4.
Suppose there are two sellers and two buyers, and consider a mechanism inwhich (i) two distinct constant prices π s = π b and π s = π b are posted in advance, and(ii) a trade occurs between seller 1 and buyer 1 when and only when s < π s = π b < b and a trade occurs between seller 2 and buyer 2 when and only when s < π s = π b < b .Hence, seller 1 and buyer 1 face the posted price π s = π b and trade only between them,and likewise for seller 2 and buyer 2. This mechanism is a deterministic robust doubleauction mechanism that violates Assumption 1.In this example, there are two posted price mechanisms separately applied to twopairs of traders, the pair of seller 1 and buyer 1 and the pair of seller 2 and buyer 2. Example 5.
Suppose there are one seller and two buyers, and consider a mechanism inwhich (i) two prices, the seller price π s and the buyer price π b , are posted in advancewith π s < π b , (ii) a trade occurs between seller 1 and buyer 1 when and only when s < π s < π b < b , (iii) when a trade occurs, seller 1 receives π s , buyer 1 pays π b , andbuyer 2 receives π b − π s , that is, x ( v ) = π s , y ( v ) = π b , and y ( v ) = π s − π b , and(iv) when there is no trade, every player’s monetary transfer is equal to zero, that is, x ( v ) = y ( v ) = y ( v ) = 0. This mechanism is a deterministic robust double auctionmechanism without one price for all traders. The reason Proposition 1 does not hold in15his example is that Assumption 2 is not satisfied. Note that Assumption 2 is satisfied in Example 4 whereas Assumption 1 is (trivially)satisfied in Example 5.
Are there robust double auction mechanisms that satisfy Assumptions 1 and 2 butnot Assumption 3? That is, are there robust double auction mechanisms other than theposted-price mechanisms with rationing? We already know by Examples 1 and 3 thatthere are such mechanisms. In this section, we further pursue this question to obtain amore elaborated characterization.Let us consider the following variant of Example 1. Suppose there are one seller andtwo buyers, and consider a mechanism in which (i) the price is set to buyer 2’s reportedvaluation b , and (ii) a trade occurs between seller 1 and buyer 1 when and only when s < b < b . In this example, the only possible trade is between seller 1 and buyer 1,and buyer 2 contributes to the determination of the price. It is easy to see that this is arobust double auction mechanism. However, this mechanism has the undesirable featurethat buyer 2 never has an opportunity to trade even when her valuation is high. Note alsothat the mechanism in Example 3 shares this undesirable feature.This leads us to consider those mechanisms that respect all players’ valuations, whichwe will call the value-respecting mechanisms. If the selection of traders depends on play-ers’ valuations, then it must be the case that the sellers are selected in an increasing ofvaluations and the buyers are selected in a decreasing order of valuations. That is, sellerswith lower valuations and buyers with higher valuations are given priorities. Otherwise, aseller (buyer) who does not trade but has a lower (higher) valuation than a seller (buyer)who trades may have an incentive to bid higher (lower) to match the latter’s valuation, We note that we can construct a similar mechanism for the case when there are more than one sellerand more than one buyer. Note well that the selection of traders is different from the rationing rule. T ( v ) = S ( v ) ∪ B ( v ). The following mechanism is a natural mechanism with thoseproperties. Definition 3. ( The linear price mechanism with an exogenous rationing rule ) Playersfirst submit their respective valuations in this mechanism, which may or may not betrue valuations. Let us order sellers’ submitted valuations in an increasing order andbuyers’ submitted valuations in a decreasing order. Thus, we have the order statistics s (1) ≤ s (2) ≤ · · · ≤ s ( m ) of sellers’ valuations and the order statistics b (1) ≥ b (2) ≥ · · · ≥ b ( n ) of buyers’ valuations. Ties in the valuations can be broken in any predetermined way. Forconvenience, we follow the convention that s (0) = b ( n +1) = 0 and s ( m +1) = b (0) = 1. Let κ be the number such that s ( κ ) ≤ b ( κ ) and s ( κ +1) > b ( κ +1) . That is, κ = max { k | s ( k ) ≤ b ( k ) } .Let us call κ as the tentative volume of trade. When κ < min { m, n } , the posted price π isset to π = c s ( κ +1) + c s ( κ +2) + · · · + c m − κ s ( m ) + d b ( κ +1) + d b ( κ +2) + · · · + d n − κ b ( n ) for given nonnegative numbers c , . . . , c m , d , . . . , d n . We assume these coefficients aresuch that 0 ≤ π ≤ , π between zero and one. Letˆ S ( v ) = { i ∈ S| s i ≤ s ( κ ) and s i < π } and ˆ B ( v ) = { j ∈ B| b j ≥ b ( κ ) and b j > π } . If | ˆ S ( v ) | = | ˆ B ( v ) | , then all traders in the set ˆ S ( v ) and ˆ B ( v ) trade at price π . Otherwise,rationing is needed: if | ˆ S ( v ) | > | ˆ B ( v ) | , then | ˆ B ( v ) | sellers in ˆ S ( v ) are selected in the orderof σ : S → S to trade with the buyers in ˆ B ( v ); if | ˆ S ( v ) | < | ˆ B ( v ) | , then | ˆ S ( v ) | buyers inˆ B ( v ) are selected in the order of β : B → B to trade with the sellers in ˆ S ( v ). Any pre-specified rule for the case when κ = min { m, n } is fine for our purpose, so we do not specifyit. s i ≤ s ( κ ) ( b j ≥ b ( κ ) , respec-tively) depending on the level of the price π actually realized.(iv) The actual volume of trade is less than or equal to the tentative volume of trade κ .(v) The players are obliged to follow the trading decision after the submission of valua-tions.The following example illustrates how a linear price mechanism with an exogenousrationing rule works. Example 6.
The price π is set to b ( κ +1) . That is, d = 1 and c = · · · = c m = d = · · · = d n = 0. There are 4 sellers and 4 buyers. Let both σ : S → S and β : B → B be identitymappings. When s = 0 . , s = 0 . , s = 0 . s = 0 . b = 0 . , b = 0 . , b = 0 . b = 0 .
4, we have κ = 3, π = 0 .
4, ˆ S ( v ) = { , } and ˆ B ( v ) = { , , } . Since | ˆ S ( v ) | < | ˆ B ( v ) | ,the buyer side has to be rationed. Hence, both sellers 1 and 2 and buyers 1 and 2 trade.Note that both seller 3 and buyer 3 do not trade in spite of κ = 3: Seller 3 does not tradesince the price π is lower than his valuation whereas buyer 3 does not trade since she isordered after buyers 1 and 2 in the rationing rule β .We now show that, when there are at least two sellers as well as at least two buyers,this mechanism does not satisfy ( IC ). We start with the following lemma. Lemma 2.
Let min { m, n } ≥ . If a linear price mechanism with an exogenous rationingrule satisfies ( IC ) , then c = c = · · · = c m = 0 . Proof: We first show that c = 0. Suppose to the effect of contradiction that c > s ( κ ) < b ( κ +1) < s ( κ +1) < c s ( κ +2) and b (1) is18ufficiently high. Then, seller i with s i = s ( κ +1) has an incentive to bid arbitrarily closeto 0 when σ (1) = i . In that case, the new tentative volume of trade becomes κ + 1 and thenew price becomes π ′ ≥ c s ( κ +2) , which is higher than s ( κ +1) . Since there exists at leastone buyer whose valuation b (1) is higher than π ′ , seller i gets a positive payoff instead ofa payoff of zero from reporting truthfully.We next show that c l = 0 for all l = 2 , . . . , m . Suppose to the effect of contradictionthat c l >
0. Let the realized valuations be such that b ( κ +1) < s ( κ ) < s ( κ +1) < π and b (1) is sufficiently high. For instance, let s ( κ +1) be small enough to satisfy s ( κ +1) < c l s ( κ + l ) .Then, seller i with s i = s ( κ +1) has an incentive to bid arbitrarily close to zero when σ (1) = i since (i) it would increase the probability of trade from zero to one as well as (ii) it wouldnot change the price since the new price π ′ = c s ( κ +2) + · · · + c m − κ s ( m ) + d b ( κ +1) + d b ( κ +2) + · · · + d n − κ b ( n ) is equal to the original price π , from which he gets a positive payoff instead of a payoff ofzero from reporting truthfully. Q.E.D.
Hence, we have π = d b ( κ +1) + · · · + d n − κ b ( n ) . We now show that this mechanismdoes not satisfy ( IC ). Let the realized valuations be such that d b ( κ +2) + · · · + d n − κ − b ( n ) < b ( κ +1) < s ( κ +1) < b ( κ ) and s (1) is sufficiently low. Then, buyer j with b j = b ( κ +1) has an incentive to bid arbitrarilyclose to 1 when β (1) = j . In that case, the new tentative volume of trade becomes κ + 1and the new price becomes π ′ = d b ( κ +2) + · · · + d n − κ − b ( n ) , which is lower than b ( κ +1) .Since there exists at least one seller whose valuation s (1) is lower than π ′ , buyer j getsa positive payoff instead of a payoff of zero from reporting truthfully. Summarizing thediscussion, we have: It helps to think of the case when κ = 1. Observe that the tentative volume of trade does not change. Observe also that, although s ( κ ) nowbecomes the ( κ + 1)-th lowest valuation of the seller, the price remains the same since c = 0. roposition 4. The linear price mechanism with an exogenous rationing rule is not arobust double auction mechanism when there are more than one seller and more than onebuyer.
Hence, even without Assumption 3 of non-bossiness, it seems quite difficult to finda value-respecting robust double auction mechanism other than the deterministic postedprice mechanism with an exogenous rationing rule. We want to note that, since we restrictour attention to deterministic allocation rules in this paper, we only considered determin-istic rationing rules. It is straightforward to see that Proposition 4 can be extended (withminor modification) to any linear price mechanism with an exogenous rationing rule, beit deterministic or probabilistic, in particular, the linear price mechanism with randomrationing in which each player is selected with equal probability. We have shown that the price in any deterministic robust double auction mechanismdoes not depend on the valuations of the trading sellers or the trading buyers. We have alsoshown that, with the non-bossiness assumption, the price in any deterministic robust dou-ble auction mechanism does not depend on the players’ valuations at all, whether tradingor non-trading, i.e., the price is posted in advance. Hence, the only significant differenceamong deterministic robust double auction mechanisms that also satisfy Assumptions 1, 2,and 3 lies in the rationing rule to resolve the discrepancy between the supply (the numberof sellers with valuations less than the posted price) and the demand (the number of buyerswith valuations greater than the posted price).We have established a characterization result that the deterministic posted price mech-anism with an exogenous rationing rule is the only deterministic robust double auctionmechanism that satisfies Assumptions 1, 2, and 3. We have also introduced the class oflinear price mechanisms with an exogenous rationing rule to demonstrate that, even with- Random rationing is the most prominent rationing rule both in reality and in the literature.
Appendix
In this appendix, we show that an endogenous rationing rule violates either dominantstrategy incentive compatibility or non-bossiness. We present a slightly general setup. Letthere be n players and m identical objects. The objects may be tangible objects such ashouses, or intangible objects such as positions, entitlements, etc. Assume that m < n since the analysis is trivial otherwise. Each player desires at most one unit of the object.Let v i be player i ’s private valuation for the object, and let V i ⊆ (0 , ∞ ) be the set ofplayer i ’s possible valuations. Let v = ( v , . . . , v n ) be the profile of players’ valuations,and let V = Π ni =1 V i be the set of possible valuation profiles. An allocation is a vector( a , · · · , a n ) with a i ∈ [0 ,
1] for all i = 1 , · · · , n and P ni =1 a i = m . Hence, a i represents theprobability that player i gets the object. An allocation is deterministic when a i ∈ { , } for all i = 1 , . . . , n . Let A be the set of all possible allocations.A rationing rule (or an allocation rule) is a function f : V → A . Let f i ( v ) correspondsto player i ’s allocation. A rationing rule is an exogenous rationing rule if f ( v ) = f ( v ′ ) forall v, v ′ ∈ V , i.e., f does not depend on v at all. We have: Proposition A.
If a rationing rule is both dominant strategy incentive compatible ( IC ) and non-bossy, then it is an exogenous rationing rule. Proof: We first show that f ( v i , v − i ) = f ( v ′ i , v − i ) for all i, v i , v ′ i , and v − i . Suppose not,that is, suppose there is i and v i , v ′ i and v − i with f ( v i , v − i ) = f ( v ′ i , v − i ). If f ( v i , v − i ) and f ( v ′ i , v − i ) are such that f i ( v i , v − i ) = f i ( v ′ i , v − i ), then it does not satisfy ( IC ). Or else, if21 i ( v i , v − i ) = f i ( v ′ i , v − i ) but f j ( v i , v − i ) = f j ( v ′ i , v − i ) for some j = i , then it does not satisfynon-bossiness. Next, since f ( v , . . . , v n ) = f ( v ′ , v , . . . , v n ) = f ( v ′ , v ′ , v , . . . , v n ) = · · · = f ( v ′ , . . . , v ′ n ), we are done. Q.E.D.
This proposition implies that an endogenous rationing rule in the text violates eitherdominant strategy incentive compatibility or non-bossiness.
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