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Bid rotation and collusion in repeated auctions

Journal of Economic Theory112(2003)79–105

Masaki Aoyagi1

ISER,Osaka University,6-1,Mihogaoka,Ibaraki,Osaka567-0047,Japan

Received24January2001;?nal version received8July2002

Abstract

This paper studies bidder collusion with communication in repeated auctions when no side transfer is possible.It presents a simple dynamic bid rotation scheme which coordinates bids based on communication history and enables intertemporal transfer of bidders’payoffs.The paper derives a suf?cient condition for such a dynamic scheme to be an equilibrium and characterizes the equilibrium payoffs in a general environment with af?liated signals and private or interdependent values.With IPV,it is shown that this dynamic scheme yields a strictly higher payoff to the bidders than any static collusion scheme which coordinates bids based only on the current reported signals.

JEL classi?cation:C72;D82

Keywords:Collusion;Auction;Repeated games

1.Introduction

It is well recognized that bidder collusion is a serious problem in many auctions: Collusion is documented in auctions for used machinery,timbers,frequency spectrums,Treasury securities,the procurement of construction work,etc.[3,13,17]. Despite its signi?cance as an empirical phenomenon,relatively little is understood about the theory of collusion in auctions,which is distinguished from the standard collusion theory by the presence of asymmetric information across bidders about their valuations of the object.

Most of the existing analysis of collusion in auctions is conducted in the one-shot framework.One important contribution in this case is made by McAfee and E-mail address:aoyagi@iser.osaka-u.ac.jp.

1ISER,Osaka University.

doi:10.1016/S0022-0531(03)00071-1

McMillan [14],who analyze bidder collusion with communication in ?rst-price auctions under the independent private values (IPV)assumption.Their key ?ndings include the identi?cation of the most ef?cient collusion schemes with and without side transfer.In particular,they show that full collusion is possible with side transfer,but that the scope of bidder collusion is severely limited without it.

If collusion is a product of frequent interaction,however,a more appropriate framework for analysis is that of repeated games,where the same set of bidders participate in a series of auctions held sequentially over time.2The purpose of this paper is to show that in in?nitely repeated auctions,collusion is possible through intertemporal payoff transfer even if there is no side payment of money.In other words,bidders in repeated auctions can collude through the adjustment of continuation payoffs in a way that partially compensates for the lack of monetary transfer.We derive a suf?cient condition for such a collusion scheme to be an equilibrium and characterize the equilibrium payoffs in a general environment with af?liated signals and private or interdependent values.Speci?cally,the collusion scheme considered in this paper builds on communication between the bidders,which we think is an integral part of many collusion practices.3The analysis shows how communication coordinates bidding in a dynamic environment.

We consider a model of in?nitely repeated auctions with two symmetric bidders.In every period,a single indivisible object is sold through the same auction format,and the bidders’private signals are drawn from the same distribution.

The bidders collude by coordinating their bids in each auction with the help of a communication device referred to here as a center.In each period,the center receives reports from the bidders about their private signals and then instructs them on what bid to submit in the stage auction.This stage mechanism,which chooses instructions as a function of reports,is called an instruction rule in this paper.collusion scheme represents the center’s choice of contingent on history.A collusion scheme is an equilibrium if truth-telling is incentive compatible and obedience to the instructions is rational for each bidder.The paper’s analysis focuses on a class of (grim-trigger)collusion schemes called bid rotation schemes.In these schemes,play begins with the collusion phase where no more than one bidder is instructed to bid in each stage auction,and any deviation from the center’s instruction triggers the punishment phase where the one-shot Nash equilibrium of the stage auction is played.

The bid rotation scheme constructed in this paper is as follows:Play during the collusion phase rotates among three (sub-)phases S ;A 1and A 2:In the original symmetric phase S ;the center uses the ef?cient instruction rule which instructs the bidder with the higher valuation (based on the reports)to bid the reserve price R if and only if his valuation exceeds R :It instructs the other bidder to stay out.In phase A i ei ?1;2T;the center uses an asymmetric instruction rule which favors bidder j a i in the sense that j ’s ex ante stage payoff is higher than that of bidder i :It can be seen 2

See also [10],who emphasize the need of a repeated model of collusion in auctions.

3The Japanese word for collusion in procurement auctions is ‘‘dang %o,’’which literally means ‘‘to discuss.’’M.Aoyagi /Journal of Economic Theory 112(2003)79–105

M.Aoyagi/Journal of Economic Theory112(2003)79–10581 that the ef?cient instruction rule used in phase S is not incentive compatible by itself since the bidders would overstate their signals in the hope of winning the object at the reserve price.The incentive for truth-telling in phase S is provided through the adjustment in continuation payoffs as follows:When bidder i’s reported signal is higher than that of j;a transition to phase A i takes place with positive probability so that bidder i’s continuation payoff would be lower.The instruction rule used in phase A i is chosen to be incentive compatible so that no further adjustment in continuation payoffs is necessary.After a?xed number of play in phase A i;the game returns to phase S:

The above bid rotation scheme is dynamic in the sense that the instruction rule used during the collusion phase is chosen as a function of past reports.In contrast, the collusion scheme proposed by McAfee and McMillan[14]is a static bid rotation scheme,which uses the same instruction rule throughout the collusion phase independent of the history.4It should be noted that in a static scheme,incentive compatibility of the instruction rule must hold period by period since no adjustment in continuation payoffs is possible.With this constraint relaxed,a dynamic bid rotation scheme can be strictly more ef?cient.Speci?cally,this paper shows that when the stage auction is?rst-price and when we have IPV,a dynamic bid rotation scheme as described above is an equilibrium for suf?ciently patient bidders and yields a strictly higher payoff than the optimal static scheme.

As mentioned above,most existing models of collusion in auctions are one-shot. Robinson[18]and von Ungern-Sternberg[20]are among the?rst to point out the vulnerability of the English and second-price sealed-bid auctions to bidder collusion.5Graham and Marshall[8]and Graham et al.[9]present particular side transfer schemes for second-price and English auctions with pre-auction commu-nication.Subsequently,Mailath and Zemsky[12]and McAfee and McMillan[14] identify the optimal schemes under the IPV assumption.The former examine collusion with side transfer in second-price auctions,while the latter look at that with and without side transfer in?rst-price auctions.Both papers conclude that ef?cient collusion is possible with side transfer.6More recently,Athey et al.[2],Johnson and Robert[11],and Skrzypacz and Hopenhayn[19]analyze collusion in repeated IPV auctions without side transfer.7Among them,Skrzypacz and Hopenhayn[19]prove the existence of a collusion scheme without communication that performs strictly better than the static scheme of McAfee and McMillan[14]when the stage auction is ?rst-price and the reserve price equals zero.While the intuition behind their results is closely related to ours,their formal logic is specialized to the particular auction format as well as the IPV assumption.In contrast,this paper presents a simple collusion scheme which is robust with respect to these speci?cations,and characterizes the collusive payoff explicitly.

4Note that in both‘‘static’’and‘‘dynamic’’schemes,any deviation from the instruction is met by reversion to the one-shot Nash equilibrium.

5Brusco and Lopomo[4]and Engelbrecht-Wiggans and Kahn[5]analyze collusion in(one-shot)multi-object ascending price auctions.

6An earlier version of this paper contains a generalization of this result.

7The present paper was developed independently of them.

It should be noted that the collusion scheme considered in this paper does not extract full surplus from the auctioneer since the instruction rule used in the asymmetric phase A i is not ef?cient.In other words,the scheme is not ?rst-best ef?cient from the point of view of the bidders.The question of ?rst-best ef?ciency in collusion without side transfer under asymmetric information is indeed very dif?cult.Ef?ciency results are available only in IPV models with ?nite signals:Fudenberg et al.

[7]show that the IPV model with ?nite signals and communication has the ‘‘product structure,’’which guarantees the existence of a near ef?cient equilibrium for suf?ciently low discounting.8In contrast,the present paper uses the continuous signal formulation,which is most common in the auction literature.Since conclusions based on ?nite signals do not necessarily extend to continuous signals in many mechanism design problems,we think it important to gain insight into the problem for this standard framework.9

The severest restriction of the present model is the assumption that the auctioneer uses the same auction format every period.While this may be an adequate description of some actual practices,it would be extremely important to analyze the alternative scenario where the auctioneer has the ability to react to bidder collusion.In the one-shot framework,Mailath and Zemsky [12]and McAfee and McMillan

[14]both discuss the choice of the reserve price as the auctioneer’s response to collusion.In repeated auctions,the corresponding treatment is to include the auctioneer as a player of the repeated game who chooses the reserve price as a function of history.The analysis of such a model would be very much involved and is left as a topic of future research.

Although this paper will focus on a two-bidder model for simplicity,its qualitative conclusions would go through with three or more bidders as long as attention is restricted to collusion by the grand coalition.When there are more than two bidders,however,a new set of questions will arise concerning the feasibility and pro?tability of collusion by a subcoalition of bidders.

We assume that the bidders’private signals are independent across periods.While there are many interesting situations that involve serially correlated signals,the following are some dif?culties associated with the analysis of such a model.First,with independent signals,bid rotation improves allocative ef?ciency by making the highest valuation bidder win at the reserve price (at least in the symmetric phase).When signals are correlated over time,on the other hand,the relationship between bid rotation and allocative ef?ciency is more subtle.In the extreme case of perfect correlation where the signals stay the same throughout the game,for example,rotating bids would only lower allocative ef?ciency.10Another simpli?cation 8

Based on this result,Athey and Bagwell [1]present an explicit characterization of the ?rst-best scheme in the IPV model with binary signals.

9The ?niteness of actions is critical for the type of argument used in [7].With communication,a mapping from the set of signals to the set of reports is part of a player’s action.This suggests that using a ?nite message space in communication is not suf?cient.

10Bid rotation could still improve the bidders’expected payoffs from the one-shot Nash equilibrium if the higher valuation bidder makes his opponent win from time to time in return for less aggressive bidding in other periods.M.Aoyagi /Journal of Economic Theory 112(2003)79–105

M.Aoyagi/Journal of Economic Theory112(2003)79–10583 possible with independent signals is that rules of transition between distinct phases of the collusion scheme can be taken as a function of current reports only and independent of past reports.With serially correlated signals,this would no longer be the case and the determination of transition probabilities would be much more dif?cult.Finally,the introduction of serial correlation also induces some fundamental asymmetries between the bidders during the course of play.11

The organization of the paper is as follows:Section2formulates a model of repeated auctions with the center as a mediation device.The dynamic bid rotation scheme is described in Section3.The main theorem in this section gives a suf?cient condition for this scheme to be an equilibrium and describes the equation that characterizes its payoff.In Section4,its performance is compared with that of static schemes.Section5analyzes a collusion scheme based on implicit communication through winning bids.

2.Model

There are two symmetric,risk-neutral bidders1and2and a center which coordinates their bidding in in?nitely repeated auctions.A single indivisible object is sold every period through a?xed auction format.In each period,bidder i receives a private signal s i A?0;1 about the value of the object.The probability distribution of the signal pro?le s?es1;s2Tis the same in every period and represented by the density function f whose support is the unit square?0;1 2:The signals are independent across periods.The conditional density of s i given s j is denoted f ieáj s jT; and the corresponding distribution function is denoted F ieáj s jTei a jT:With slight abuse of notation,we also use f i(resp.F i)to denote the marginal density(resp. distribution)of s iei?1;2T:Throughout,we assume that the signals s?es1;s2Tare af?liated.Af?liation includes independent signals as a special case,and is equivalent to the monotone likelihood ratio property in the current framework with only two signals s1and s2:The present analysis will use the following properties of af?liation

(e.g.,[15]):

f ies i j s jT

iseweaklyTincreasing in s j for any s iei a jT;e1T

F ies i j s jT

and if H:?0;1 2-R is increasing,then12

E?He?sTj s1p?s1p s10;s2p?s2p s20 is increasing in s i and s i0ei?1;2T:e2TGiven the signal pro?le s?es1;s2T;the expected value of the object to bidder i is denoted v iesT:We adopt the convention that the?rst argument of v i is s i(own signal) 11Asymmetry is problematic since it implies that incentive compatibility of a collusion scheme is now characterized by a system of(linear)differential equations rather than one(cf.(A.7)in the appendix).Such a system typically admits only a numerical solution and does not yield any characterization of the equilibrium payoff.

12Throughout,the expectation E is with respect to random variables written with tildes.

and the second is s j (the other bidder’s signal).The valuation function v i is the same for every period and v i e0;0Tis normalized to zero.Symmetry implies f es T?f es 0Tand v i es T?v j es 0Tfor every s ;s 0A ?0;1 2such that es 0i ;s 0j T?es j ;s i T:

We say that the values are private if v i es T?s i for every s and i ?1;2,and interdependent if for i ?1;2and j a i ;(i)v i :?0;1 2-R tis continuously differenti-

able,(ii)@

v i @s i es T40and @v i @s j es T40for every s ;and (iii)v i es TX v j es Tif s i X s j :Note that under the symmetry assumption,(iii)is implied by the standard single-crossing condition:@

v i i es T4@v

j i es Tfor every s A ?0;1 2such that v i es T?v j es T:

A stage auction is any transaction mechanism which determines the allocation of the good and monetary transfer according to a pair of sealed bids submitted by the bidders.The restriction to a seal-bid mechanism is for simplicity.Participation in the stage auction is voluntary so that the set of each bidder’s generalized bids is expressed as

B ?f N g ,R t;where N represents ‘‘no participation.’’The rule of the auction is summarized by measurable mappings c i and p i ei ?1;2Ton the set B 2of bid pro?les b ?eb 1;b 2T:c i eb Tis the probability that bidder i is awarded the good,and p i eb Tis his expected payment to the auctioneer.The functions c i and p i ei ?1;2Tare symmetric (c i eb T?c j eb 0Tand p i eb T?p j eb 0Tif eb 0i ;b 0j T?eb j ;b i T),and satisfy the following conditions.Assumption 1.(i)A bidder makes no payment when he chooses not to participate:p i eb T?0if b i ?N :

(ii)There exists a non-random reserve price R A ?0;v i e1;1TTsuch that a bidder may win the object only if he submits a bid at or above R :c i eb T?0if b i A f N g ,?0;R T:(iii)If only one bidder participates and submits bid R ;then he wins the object at price R :c i eb T?1and p i eb T?R if b i ?R and b j ?N :

(iv)There exists a symmetric Nash equilibrium in the (Bayesian)game in which each bidder’s strategy is a mapping z i :?0;1 -B and payoff function is

E ?c i ez i e?s

i T;z j e?s j TTv i e?s Tàp i ez i e?s i T;z j e?s j TT :Assumption 1holds for most standard auctions including the ?rst-and second-price auctions.Let g 0be the (ex ante)symmetric Nash equilibrium payoff to each bidder in the stage auction as described in Assumption 1(iv).Also,let g ?be the expected payoff to each bidder under truthful information sharing and ef?cient allocation with bidder i winning the object at price R if and only if s i 4s j and v i es T4R :

g ??E ?1f ?s i 4?s j ;v i e?s T4R g f v i e?s

TàR g :Clearly,2g ?gives the ?rst-best joint collusive payoff,and the bidders see a potential gain from collusion if g 0o g ?:This is the case to be studied in what follows.

Collusion in the repeated auction takes the following form:At the beginning of each period,the two bidders report their private signals s i to the center.Upon

receiving the report pro?le ?s

?e?s 1;?s 2TA ?0;1 2;the center chooses instruction to each bidder i on what (generalized)bid to submit in the stage auction.M.Aoyagi /Journal of Economic Theory 112(2003)79–105

In general,the bidders may report a false signal,and/or disobey the instruction. Bidder i’s reporting rule l i:?0;1 -?0;1 chooses report?s i as a function of signal s i; and his bidding rule m i:?0;1 2?B-B chooses bid b i in the stage action as a function of his signal,report and instruction.The reporting rule is honest if it always reports the true signal,and the bidding rule is obedient if it always obeys the instruction.

Denote by l?

i and m?

bidder i’s honest reporting rule and obedient action rule,

respectively.

For simplicity,we assume that the(generalized)bids in the stage auction are observable to every party including the center.13The observability of bids implies that a bidder’s deviation can be classi?ed into two types:A bidder commits an observable deviation when he chooses a bid different from the instruction given to him,and commits an unobservable deviation when he reports a false signal.

The center is formally a communication device as formulated by Forges[6]and Myerson[16].Its choice of instructions to the bidders given their reports is captured by an instruction rule q?eq1;q2T:?0;1 2-B2:q ie?sTis the instruction to bidder i when the report pro?le is?s:Let g iel;m;qTdenote bidder i’s stage payoff resulting

from any pro?leel;m;qTof reporting and bidding rulesel;mT?el1;m1;l2;m2Tand

instruction rule q:The instruction rule q is one-shot incentive compatible(one-shot IC)if neither bidder has an incentive to misreport his signal: g iel?;m?;qTX g iel i;l?j;m?;qTfor any reporting rule l i;i?1;2;and j a i:Note that one-shot incentive compatibility refers only to the incentive in reporting and presumes bidders’obedience to the given instructions.In particular,the instruction rule that instructs bidders to play the one-shot Nash equilibrium of the stage auction is one-shot IC.Lemma1in the next section identi?es some other instruction rules with this property.

Bidder i’s communication history in period t in the repeated auction game is the sequence of his reports and instructions in periods1;y;tà1:On the other hand, bidder i’s private history in period t is the sequence of his private signals s i in periods 1;y;tà1:Furthermore,the public history in period t is a sequence of instruction rules used by the center in periods1;y;t and(generalized)bid pro?les in the stage auctions in periods1;y;tà1:

Bidder i’s(pure)strategy s i in the repeated auction chooses the pairel i;m iTof

reporting and bidding rules in each period t as a function of his communication and private histories in t;and the public history in t:Let s?i be bidder i’s honest and

obedient strategy which plays the pairel?

i ;m?

Tof the honest reporting rule and

obedient bidding rule for all histories.

The collusion scheme t describes the center’s choice of an instruction rule in every period as a function of communication and public histories.At the beginning of each period,it publicly informs the bidders which instruction rule is used in that period.

13The conclusions in Sections3–5hold if only the identity of the winner is publicly announced by the auctioneer.When we take the interpretation that the center is a simple communication device which does not have any monitoring function,we can replicate the same results by letting the bidders report the outcome of each stage auction to the center.

M.Aoyagi/Journal of Economic Theory112(2003)79–10585

Our analysis will focus on the following class of ‘‘grim-trigger’’collusion schemes with two phases:The game starts in the collusion phase,and reverts to the punishment phase forever if and only if there is an observable deviation by either bidder in the sense described above.In the punishment phase,the bidders are instructed to play the one-shot Nash equilibrium of the stage auction speci?ed in Assumption 1(iv).A collusion scheme t in this class is static if it chooses the same instruction rule in every period during the collusion phase independent of the history,and is dynamic otherwise.Furthermore,the collusion scheme t employs bid rotation if no more than one bidder is instructed to participate in each stage auction during the collusion phase.

Let d o 1be the bidders’common discount factor,and P i es ;t ;d Tbe bidder i ’s average discounted payoff (normalized by e1àd T)in the repeated game under the strategy pro?le es ;t T:The collusion scheme t is an equilibrium if the pair s ??es ?1;s ?2Tof honest and obedient strategies constitutes a perfect public equilibrium of the repeated game:s ?i is optimal against es ?j ;t Tafter any public history of the game.14It follows from the de?nition that if t is an equilibrium static collusion scheme,then its instruction rule in the collusion phase is one-shot IC.

3.A dynamic bid rotation scheme

Let q ?be the ef?cient instruction rule that instructs bidder i to (i)bid R if his report ?s

i is higher than j ’s report ?s j ;and if his valuation v i e?s T(computed from the report pro?le ?s

)exceeds R ;and (ii)stay out otherwise:q ?i e?s T?R if ?s i 4?s j and v i e?s T4R ;N otherwise :

(Clearly,the (ex ante)payoff g i el ?;m ?;q ?Tassociated with q ?equals the ?rst-best level g ?although q ?is not one-shot IC.Consider next the asymmetric instruction rule q i that ‘‘favors’’bidder j over bidder

i as follows:Bidder j is instructed to (i)bid R if his valuation v j e?s

Texceeds R ;and (ii)stay out otherwise.On the other hand,bidder i is instructed to (i)bid R if his

valuation v i e?s

Texceeds R and if bidder j ’s valuation v j would not exceed R even when i ’s signal were 1,and (ii)stay out otherwise:

q i i e?s T?R if v i e?s T4R X v j e?s j ;1T;N otherwise ;(q i j e?s T?R if v j e?s T4R ;N otherwise :

(We refer to bidder j as the primary bidder under q i and bidder i as the secondary

bidder.Let %g ?g j el ?;m ?;q i Tand %g ?g i el ?;m ?;q i Tbe the expected stage payoffs to the

primary and secondary bidders,respectively,under q i :By de?nition,%g 4%g :14See [7].

M.Aoyagi /Journal of Economic Theory 112(2003)79–105

Lemma 1.q i is one-shot IC .

Proof.See the appendix.

Since 2g ?is the ?rst-best joint collusive payoff,it can be readily veri?ed that

2g ?4%g t%g ;e3Twhere the strict inequality is the consequence of the full support of the density

function f and R o v i e1;1T:Let t d be a dynamic bid rotation scheme such that:(a)The collusion phase consists of three subphases S ;A 1and A 2:S is the original symmetric phase where the ef?cient instruction rule q ?is used,while A i is the asymmetric adjustment phase where the instruction rule q i is used.

(b)Play begins in the symmetric phase S :After each period in phase S ;transition

to the asymmetric phase A i ei ?1;2Ttakes place with probability o i e?s

T;which is a function of the reported signals in the current period alone and given by o i e?s T?x e?s i Tif ?s i 4?s j ;0otherwise ;

(for some increasing function x :?0;1 -?0;1 :Play stays in phase S with probability

1ào 1e?s

Tào 2e?s T:(c)Each asymmetric phase A i lasts exactly for m periods and then play returns to S :

It should be noted that the transition probability x e?s

i Tdepends only on the higher of the two reports.There is a clear connection between the above collusion scheme and the one with side transfer in [14].In [14],each bidder is discouraged from overstating his signal by the transfer payment that is required from the bidder with the higher report.On the other hand,the deterrent in the above scheme is the possibility of a lower continuation payoff for such a bidder.This is a natural modi?cation of the side transfer scheme in view of the substitutability of continuation payoffs for monetary transfer in repeated games.

Write v i es i ;s i T??v

i es i Tand recall ?v i e0T?0by normalization.Let r A ?0;1 be the (unique)signal such that ?v

i er T?R :De?ne z i eb T?f i eb j b TF i eb j b T

:It can be seen that given any transition probability function x a bidder’s payoff u d ?P i es ?;t d ;d Tfrom the dynamic bid rotation scheme t d solves the following recursive equation of u :

u ?e1àd Tg ?td E ?1f ?s i 4?s j g f x e?s i T%u te1àx e?s i TTu g t1f ?s i o ?s j g f x e?s j T%u te1àx e?s j TTu g ;e4T

where %u ?e1àd m T%g td m u and %u ?e1àd m T%g td m u :On the other hand,x is obtained as a solution to the incentive compatibility condition for truth-telling in

phase S :As seen in the appendix,the incentive condition reduces to a linear

M.Aoyagi /Journal of Economic Theory 112(2003)79–10587

differential equation which has the payoff u d as a parameter.By substituting its solution into (4)and simplifying,we obtain the following equation of u ?u d :

j eu T u àg ?t

2u à%g à%

g u à%g y eu T?0;e5T

where y :e%g ;N T-R ttis de?ned by y eu T?Z 1r Z s i 0Z s i r f ?v i eb TàR g z i eb Te à%g à%g u à%g R s i b z i eg Td g d b f es Tds j ds i :Note that the continuity of y implies that of j :e%g ;N T-R :The following is our main theorem.

Theorem 1.Assume that the values are either private or interdependent.If j eu T?0has a solution u d strictly greater than g 0;then for a suf?ciently large discount factor d ;the dynamic bid rotation scheme t d is an equilibrium for some x eáT(transition probability function )and m (duration of phase A i ),and yields payoff u d :Proof.See the appendix.

It should be noted that (5)and its solution u d are independent of d although low discounting is required for t d to be an equilibrium.By construction,note that the bidders’overall payoff u d in the dynamic bid rotation scheme is a convex combination of their (ex ante)stage payoff g ?in phase S ;and the average of their stage payoffs in phases A 1and A 2:(Note that A 1and A 2are equally likely ex ante.)While j eu T?0cannot be solved analytically in general,it is possible to bound its solution from below as follows.Let

C ?Z 1

f ?v

i es i TàR g F j es i j s i Tf i es i Tds i :Then it is shown in the appendix that there exists a solution u d to (5)which satis?es

u d 4L

%g à%g g à%g t2C g ?t2C g à%g t2C %g t%g 2:e6TNote that we have L 4e%g t%g T=2since g ?4e%g t%g T=2by (3).In particular,(6)shows that the weight on the ef?cient

payoff g ?in L is an increasing function of the gap between the two asymmetric payoffs %g and %g :With %g t%g ?xed,therefore,the lower bound on the ef?ciency of the dynamic scheme t d is increased when we take the payoffs in the two asymmetric phases farther apart.15The following corollary is an immediate consequence of the above observation.15

Conversely,this also suggests the following:While we may specify any other (static or dynamic)path of play in phase A i ;the performance of t d cannot be increased by raising just the ef?ciency (e.g.,%g t%g )of that phase.It is important to maintain the gap %g à%g :M.Aoyagi /Journal of Economic Theory 112(2003)79–105

Corollary 1.Assume that the values are either private or interdependent.If g 0p L ;then for a suf?ciently large discount factor d ;the dynamic bid rotation scheme t d is an equilibrium for some x eáTand m :

Example 1.Suppose that the stage auction is second-price sealed-bid with the reserve price R equal to zero.In this case,it is well known that the bidding function

in the one-shot Nash equilibrium is given by z 0es i T??v

i es i T:Suppose further that the valuation function v i has the linear form v i es T?cs i te1àc Ts j ;where c A ?1=2;1 :As seen,the values are private if c ?1and interdependent otherwise.The one-shot Nash equilibrium payoff equals g 0?Z f s :s i 4s j g f v i es Tàz 0es j Tg f es Tds ?c Z f s :s i 4s j g

es i às j Tf es Tds ;

whereas %g ?0and %g ?Z ?0;1 2

f cs i te1àc Ts j

g f es Tds :From these,we can verify that 12e%g t%g Tàg 0?12Z ?0;1 2

f s j àc j s j às i j

g f es Tds :Hence,the suf?cient condition in Corollary 1holds if E ??s j X cE ?j ?s j à?s i j :Wit

h f ?xed,therefore,if this inequality holds under private values ec ?1T;then it also holds under interdependent values ec o 1T:Furthermore,since E ??s j 412E ?j ?s

i à?s

j j for any f as can be readily veri?ed,there exists %c 41=2such that if c p %c ;then there exists an equilibrium dynamic bid rotation scheme.

The last observation in the above example has the following simple generalization.Suppose that the two bidders have almost common values so that their valuation of the object is similar for every signal pro?le.It then follows that any allocation of the object between them leads to almost the same level of ef?ciency.In particular,the allocation in phase A i described above is almost (?rst-best)ef?cient,and so is the dynamic bid rotation scheme t d :Formally,given any e 40;we say that the values are e -common if sup s A ?0;1 2j v 1es Tàv 2es Tj o e :

Corollary 2.Let e 40be given ,and assume that the values are interdependent and e -common.If the dynamic bid rotation scheme t d is an equilibrium ,then it is e -ef?cient in the sense that its payoff u d satis?es u d X g ?àe :

Proof.See the appendix.

In particular,when the stage auction is either ?rst-or second-price sealed-bid,the one-shot Nash equilibrium payoff g 0of an e -common value model is bounded away from g ?in the limit as e -0:For e X 0small,therefore,u d 4g 0holds by Corollary 2

M.Aoyagi /Journal of Economic Theory 112(2003)79–10589

so that by Corollary 1,the dynamic bid rotation scheme t d is in fact an (e -ef?cient)equilibrium for d close to one.16

4.Dynamic vs.static collusion schemes

This section compares the performance of the dynamic bid rotation scheme described in Section 3with that of static collusion schemes.The characterization of the optimal static collusion scheme is available only in the independent private values (IPV)environment,and our primary focus is on this case.A brief discussion of a more general environment is given at the end of this section.With IPV,we show that the dynamic scheme yields a strictly higher payoff to a bidder than the optimal static scheme as identi?ed by McAfee and McMillan [14].

As a ?rst step,Theorem 2below states that the dynamic bid rotation scheme is in fact an equilibrium for d close to one when the stage auction is either ?rst-price or second-price.Recall that the reserve price R is allowed to be any number in the interval ?0;v i e1;1TT:

Theorem 2.Assume independent private values (IPV).Suppose that the stage auction is either ?rst-price or second-price sealed-bid.Then for a suf?ciently large discount factor d ;the dynamic bid rotation scheme t d is an equilibrium for some x eáTand m ;and its payoff u d is strictly higher than both e%g t%

g T=2and g 0:Proof.See the appendix.

McAfee and McMillan [14]identify the highest payoff achieved by a collusion scheme with no side transfer for one-shot ?rst-price sealed-bid auctions in the IPV environment.Their theorem applies directly to the current repeated game framework and yields the characterization of the most ef?cient static collusion scheme.Let

h i es i T?1àF i es i T

f i es i T

be the inverse hazard rate of s i :The following theorem is stated without a proof as it is a straightforward application of Theorem 1of McAfee and McMillan [14].Theorem 3.Assume independent private values (IPV ).Suppose that the stage auction is ?rst-price sealed-bid .For a suf?ciently large discount factor d ;the most ef?cient equilibrium static collusion scheme without side transfer is a grim-trigger scheme.The corresponding payoff u s is given as follows:

(i)If h i eáTis (weakly )increasing ,then u s equals the one-shot equilibrium payoff g 0:(ii)If h i eáTis (weakly )decreasing ,then u s equals e%g t%

g T=2:16

With e -common values,the static bid rotation scheme that randomly allocates the object with probability 1/2to each bidder during the collusion phase tends to be an equilibrium as well.As seen in Corollary 4,however,such a scheme is dominated by a dynamic scheme.M.Aoyagi /Journal of Economic Theory 112(2003)79–105

Note that the instruction rule used in a static collusion scheme must be one-shot IC since no incentive for truthful reporting can be provided through the adjustment in continuation payoffs.McAfee and McMillan [14]show that the ef?cient static scheme t s when h i eáTis decreasing (case (ii)above)is described as follows:The (mixed)instruction rule ?q :?0;1 2-D B 2used in the collusion phase is such that

e?q i e?s T;?q j e?s TT?eR ;N T

if ?s i 4R X ?s j ;12eN ;R Tt12eR ;N Tif ?s i ;?s j 4R ;eN ;N Tif ?s

i ;?s j p R :8><>:In other words,the representative bidder is chosen at random with probability one-half when both valuations exceed R :Clearly,such an instruction rule is one-shot IC.Following any deviation from the instruction,play reverts to the punishment phase where the bidders are instructed to play the one-shot Nash equilibrium z 0:Comparison of Theorems 2and 3immediately reveals that when h i eáTis monotone,the dynamic bid rotation scheme t d outperforms any static bid rotation scheme.The following corollary summarizes this observation.

Corollary 3.Assume independent private values (IPV ).Suppose that the stage auction is ?rst-price sealed-bid ,and that h i eáTis monotone.Then for a suf?ciently large discount factor d ;the dynamic bid rotation scheme t d is an equilibrium for some x eáTand m ;and yields a strictly higher payoff than any equilibrium static bid rotation scheme .The intuition behind Corollary 3for a decreasing h i (case (ii))is as follows:The instruction rule ?q described above equals a convex combination of q i and q j de?ned

in the previous section:?q ?12q i t12q j :It follows that the bidder’s payoff in the

optimal static scheme exactly equals the average of his payoffs in phases A 1and A 2of the dynamic scheme t d :Since the allocation in phase S is ef?cient,the bidder’s overall payoff in t d is strictly higher.

Example 2.Suppose that the stage-auction is ?rst-price,and that s i has the uniform distribution over ?0;1 :Assume that d is suf?ciently large.17Since h i 0es i To 0for every s i ;the best static scheme yields u s ?e%g t%g T=2by Theorem 3.Table 1below presents the values of u d ;u s and the one-shot equilibrium payoff g 0as fractions of the ?rst-best ef?ciency level g ?for various values of R :It can be seen that t d extracts at least close to 90%of the surplus.

In a general environment without IPV,little is known about the optimal one-shot scheme,and hence no clear-cut comparison of dynamic and static collusion schemes is possible.Even without IPV,however,any static scheme that uses a mixed instruction rule as in case (ii)above is dominated by a dynamic scheme that belongs

to the class described in Section 3.Formally,let ?q

1and ?q 2be any pair of (possibly 17When R ?0;for example,t d is an equilibrium if d X 0:95:

M.Aoyagi /Journal of Economic Theory 112(2003)79–10591

mixed)asymmetric one-shot IC instruction rules such that ?q 11es T??q 22es 0Tand ?q 12es T?

?q 21es 0Tif es 10;s 20T?es 2;s 1T;and

g i e?q i ;l ?;m ?To g j e?q i ;l ?;m ?T;i ?1;2;j a i :

In other words,bidder 2is favored over bidder 1under q 1and their roles are exactly reversed under q 2:Suppose that #t s is a symmetric static bid rotation scheme which

uses the mixed instruction rule ?q ?12?q 1t12?q 2in the collusion phase.

Corollary 4.Assume that the values are either private or interdependent.Suppose that #t s is a symmetric static bid rotation scheme as described above which yields payoff ?u s o g ?to each bidder.If #t s is an equilibrium ,then there exists a dynamic bid rotation scheme #t d of the class described in Section 3which is an equilibrium for some d and

yields a strictly higher payoff than ?u

s :Proof.Denote %g 0?g i e?q i ;l ?;m ?Tand %g 0?g j e?q i ;l ?;m ?Tei ?1;2;j a i T:Let #t d be a dynamic bid rotation scheme as described in Section 3with the exception that it uses

the instruction rule ?q

i in phase A i ei ?1;2T:It then follows from (6)that its payoff ?u d 4e%g 0t%g 0T=2??u s :Since ?u s X g 0by assumption,we also have ?u d 4g 0so that #t d is an equilibrium for some d by Corollary 1.&

5.Implicit communication through winning bids

In the previous sections,bidder communication is explicit in the sense that reporting of private signals is done separately from bidding in the stage auction.This section shows that even if there is no explicit communication,collusion may be sustained by implicit communication through bids submitted to the stage auction when the identity of the winner as well as his bid is publicly announced by the auctioneer.In other words,communication studied in this section is signaling of private information with bids.It is ex post in the sense that the messages are exchanged only after the stage auction is concluded,and will be used to determine

Table 1

g ?u d g ?u s g ?g 0g ?0.0

0.33330.8860.7500.5000.1

0.28350.8990.7860.5710.2

0.23470.9120.8180.6360.3

0.18780.9260.8480.6960.4

0.14400.9380.8750.7500.5

0.10420.9490.9000.7990.6

0.06930.9610.9230.8470.70.04050.9810.9440.889

M.Aoyagi /Journal of Economic Theory 112(2003)79–10592

continuation play only.18For simplicity,we assume that the stage auction is ?rst-price sealed-bid,and that a public randomization device is available.Let w i es i T?E ?v i e?s Tj ?s i ?s i ;?s j o s i be the expected value of the object to bidder i with signal s i conditional on the knowledge that j ’s signal is lower than s i :The function w i is strictly increasing over e0;1 since for s i 04s i 40;w i es i T?Z s i 0v i es i ;s j Tf es j j s i Tds j

o Z s i 0

v i es i 0;s j Tf es j j s i Tds j

?E ?v i es i 0;?s j Tj ?s i ?s i ;?s j o s i

p E ?v i es i 0;?s j Tj ?s i ?s i 0;?s j o s i 0 ?w i es i 0T;

where the ?rst inequality follows from the strict monotonicity of v i ;and the second from (2).We make the regularity assumption that the derivative w i 0is continuous.Suppose that the reserve price R satis?es R o w i e1T;and let r A ?0;1 be the signal such that w i er T?R :For e 40small,consider the bidding function z e :?0;1 -B such that z e es i T?R te R s i r f w i ea TàR g z i ea Te àR s i a z i eb Td b d a if s i X r ;N otherwise :8<:e7TWhen s i X r ;partial integration of the right-hand side of (7)yields z e es i T?e1àe TR te w i es i Tàe Z s i r

w 0i ea Te àR s i a z i eb Td b d a :

It follows that z e es i To e1àe TR te w i es i Tfor s i 4r :When the bidders bid according to z e ;hence,the winning bid is always within e f w i e1TàR g of R :Furthermore,z e is differentiable over ?r ;1 and satis?es

z 0e es i T?z i es i Tf e w i es i Tte1àe TR àz e es i Tg :e8TSince (8)implies z 0e es i T40for s i 4r from the above observation,the private signal of the winner (if any)is fully revealed.It can also be seen that when both bidders use z e ;i ’s (interim )expected payoff from winning the stage auction with signal s i 4r and bid b i ?z e es i Tis strictly positive since it equals w i es i Tàz e es i T40:

Let %v i es i T?E ?v i e?s

Tj ?s i ?s i be the expected valuation conditional only on one’s own signal.The collusion scheme with implicit communication through bids is described as follows:

(a)The collusion phase consists of the symmetric phase S ;and the asymmetric phases A 1and A 2:In phase S ;bidder i with signal s i bids b i ?z e es i T:In phase A i ;bidder j is favored over bidder i and bids are determined as follows:b i ?R if %v i es i TX R ;N otherwise (and b j ?R te if %v j es j TX R te ;N otherwise :

(18

This model is a special case of the general formulation in Section 3where the instruction to bidder i is independent of j ’s report.M.Aoyagi /Journal of Economic Theory 112(2003)79–10593

(b)Play begins in phase S ;and transition to phase A i takes place with probability x eb i T(based on the public randomization)if bidder i wins with bid b i A ?R ;z e e1T :(c)Each asymmetric phase A i lasts for m periods and then play returns to S :In this scheme,deviations are observable if either bidder wins with a bid above z e e1Tin phase S ;or if the primary (resp.secondary)bidder wins with a bid different from R te (resp.R )in phase A 1or A 2:As before,reversion to the punishment phase takes place following any such deviation.On the other hand,

deviations are unobservable if,for example,bidder i bids b i ?z e e?s i Tin phase S when his true type is s i a ?s i :Of course,neither type of deviation should be pro?table in equilibrium.

It can be seen that the absence of pre-auction information sharing entails the following inef?ciencies:First,when the values are interdependent,the winner’s valuation conditional on both bidders’signals could be lower than his bid (or even the reserve price).Second,any participating bidders in phase S must bid strictly above the reserve price R (albeit by a small amount)in order to achieve an ef?cient allocation.Third,the primary bidder in phase A 1or A 2must win with a bid strictly above R in order to avoid a tie.

For any e X 0;let

g ?ee T?E ?1f ?s i 4max f ?s j ;r gg f v i e?s

Tàz e e?s i Tg ;%g ee T?E ?1f %v i e?s

i TX R te g f v i e?s TàR àe g ;%

g ee T?E ?1f %v i e?s i TX R ;%v j e?s j To R te g f v i e?s TàR g :When e 40;g ?ee Trepresents each bidder’s (ex ante)expected stage payoff in phase S ;and %g ee Tand %g ee Trepresent the (ex ante)expected stage payoffs of the primary and secondary bidders,respectively,in the asymmetric phases.We assume that %g and %g are both differentiable as functions of e :

Consider the following equation of u parameterized by e o 1:

j eu ;e T u àg ?ee Tt

2u à%g ee Tà%g ee Tu à%g ee Ty eu ;e T?0;e9Twhere

y eu ;e T?Z

1r Z s i 0Z s i r f ?v i eb Tàe1àe TR àe w i eb Tg z i eb Te à%g ee Tà%g ee Tu à%g ee TR s i b z i eg Td g d b f es Tds j ds i :

As in Section 3,(9)characterizes the equilibrium payoff of the above dynamic bid-rotation scheme with implicit communication.Suppose now that j eu ;0T?0(with e ?0)has a solution u e0Tstrictly greater than the one-shot equilibrium payoff g 0:By the same argument as in Section 3,we have u e0T4f %g e0Tt%g e0Tg =24%g e0T;and hence @j @u eu e0T;0T40as can be readily veri?ed.It then follows from the implicit function theorem that if e 40is small,j eu ;e T?0has a solution u ee Tclose to u e0T:Theorem 4describes the conditions for u ee Tto be an equilibrium payoff of the above collusion scheme.

M.Aoyagi /Journal of Economic Theory 112(2003)79–105

Theorem 4.Assume that the values are either private or interdependent ,and that the stage auction is ?rst-price sealed-bid with the reserve price R o w i e1T:If the equation j eu ;0T?0has a solution u e0Tstrictly greater than g 0;then the following holds for a suf?ciently large discount factor d :Given any k 40;there exists a dynamic bid rotation scheme with implicit communication as described above that is an equilibrium and yields payoff u 4u e0Tàk :

Proof.See the appendix.

It should be noted that the discount factor d required is independent of the level of approximation k :When the values are private or when the reserve price equals zero,there is no inef?ciency associated with bidding based only on one’s own signal.In this case,therefore,the ef?ciency loss caused by the absence of information sharing can be made negligible when we take a small e 40:

Corollary 5.Suppose that the stage auction is ?rst-price sealed-bid,and that one of the following holds :(i)the values are private and the reserve price R o w i e1T;(ii)the values are interdependent and the reserve price R ?0:If the dynamic bid rotation scheme t d with explicit communication in Section 3is an equilibrium and yields payoff u d (for some d ),then the following holds for a suf?ciently large discount factor :Given any k 40;there exists a dynamic bid rotation scheme with implicit communication as described above that is an equilibrium and yields payoff u 4u d àk :

Proof.When the values are private or when R ?0;we have g ?e0T?g ?;%g e0T?%g ;

and %g e0T?%g ;where g ?;%g ;and %g are as de?ned in Section 3.It follows that (9)with e ?0is equivalent to (5),and hence that u d ?u e0T:The conclusion then follows

from Theorem 5.&

Acknowledgments

I am grateful to Uday Rajan,Phil Reny,and an associate editor and a referee of this journal for helpful comments.The material presented in Section 5is based on the referee’s suggestion.Financial support from the Faculty of Arts and Sciences of the University of Pittsburgh is gratefully acknowledged.

Appendix

The following notation is used for the discussion of the interdependent values case in the Appendix.For each s i A ?0;1 ;let k i es i Tbe the opponent’s signal s j such that v i es i ;s j T?R if there exists any such s j A ?0;1 :Let k i es i T?1if v i es i ;s j To R for every s j A ?0;1 ;and k i es i T?0if v i es i ;s j T4R for every s j A ?0;1 :Note by de?nition that r ?k i er T:It follows from the continuity and strict monotonicity of v i that k i is well M.Aoyagi /Journal of Economic Theory 112(2003)79–10595

de?ned.Also,we have by de?nition

v i es i ;k i es i TTX R

if k i es i T?0;?R if k i es i TA e0;1T;

p R if k i es i T?1:

8><>:Furthermore,since v i is continuously differentiable,k i es i Tis continuously differenti-able itself at any s i such that k i es i TA e0;1Tby the implicit function theorem.Fig.1depicts k i for a generic (interdependent)valuation function v i :

Proof of Lemma 1.The conclusion is straightforward if the values are private.We will show below that q i is one-shot IC for bidder i in the interdependent values case.

A similar argument proves the same for bidder j :Let G i i es i ;?s

i Tbe bidder i ’s interim stage payoff under q i when he has signal s i and reports ?s

i :Let k i be as de?ned above,and x A ?0;1 denote the minimum of s j such that v j es j ;1TX R (Fig.1).Assume v i es i ;1TX R since the conclusion is trivial otherwise.

Note ?rst that if ?s

i is such that k i e?s i TX x ;then G i i es i ;?s i T?0for any s i since v i e?s

i ;s j TX R implies s j X k i e?s i TX x so that v j es j ;1TX R :Suppose now that k i e?s i To x :In this case,we have

G i i es i ;?s i T?Z x

k i e?s i Tf v i es TàR g f j es j j s i Tds j :

Since v i es i ;s j TX R for any s j X k i es i Tand v i es i ;s j Tp R for any s j p k i es i T;it follows from

the above equation that G i i es i ;?s

i Tis maximized when ?s i ?s i :&

Fig.1.Ordering of v 1;v 2and R in [0,1]2.

M.Aoyagi /Journal of Economic Theory 112(2003)79–105

Proof of Theorem 1.Let u 4g 0be a solution to (5).Take any m A N such that

m 41u à%g Z 1r f ?v i ea TàR g z i ea Td a :We claim that there exists %d o 1such that for any d 4%d ;e1àd Tf ?v i e1TàR g td g 0o e1àd m T%g td m u ;eA :1Tand

1àd d e1àd m Teu à%g TZ 1r f ?v i ea TàR g z i ea Td a o 1:eA :2TThis is clear for (A.1)since u 4g 0by assumption.For m as speci?ed,(A.2)also holds

since

lim d -1à1àd d e1àd m T?lim d -1à1d e1td t?td T?1m

:Recall that x ea Tdenotes the probability of transition to phase A i when i ’s report a is higher than that of bidder j :Take any d 4%d and let x be given by x ea T?1àd d e1àd m Teu à%g TR a r f ?v i eb TàR g z i eb Te à%g à%g u à%g R a b z i eg Td g d b if a 4r ;0otherwise :8><>:eA :3TNote that x ea TA ?0;1 by our choice of m and d so that it is indeed a probability.In what follows,we ?x u ;m ;d 4%d ;x as above and prove the theorem in three steps:Step 1shows that each bidder’s payoff is u when they play the honest and obedient

strategy s ?i under t d :Steps 2and 3then prove the non-pro?tability of observable and unobservable deviations,respectively.By the principle of optimality in dynamic programming,the latter two steps can be accomplished by checking the pro?tability of one-step deviations.For simplicity,we write %u ?e1àd m T%g td m u and %u ?e1àd m T%g td m u for the expected payoffs of the primary and secondary bidders,respectively,at the beginning of the asymmetric phase A 1or A 2:

Step 1:P i es ?;t d ;d T?u :Using symmetry,we can rewrite (4)as

u ?g ?àd e1àd m T1àd e2u à%g à%g TZ 1r Z s i 0

x es i Tf es Tds j ds i :eA :4TSubstitution of (A.3)into (A.4)shows that the above recursive equation is equivalent to (5).Since u solves (5)by assumption,the desired conclusion follows.

Step 2:No observable deviation (in bidding )is pro?table .When bidder i with any signal or report (whether truthful or not)disobeys the instruction,the maximal

instantaneous gain from the deviation is bounded above by ?v

i e1TàR and the continuation payoff equals g 0:On the other hand,the lowest payoff along the path

of play equals %u ?e1àd m T%g td m u (by Step 1)when phase A i is just beginning.Hence,(A.1)implies that no observable deviation is pro?table.

M.Aoyagi /Journal of Economic Theory 112(2003)79–10597

Step 3:No unobservable deviation (in reporting )is pro?table .Since a bidder is made strictly worse off by disobeying the instruction,a pro?table deviation is possible only when he misreports his signal and then obeys the instruction.Since the instructions rules q 1and q 2are one-shot IC by Lemma 1,bidder i has no incentive to misreport a signal in phases A i and A j :Likewise,he has no incentive for misreporting in the punishment phase.It remains to check the incentive for misreporting in phase S :

Let p i es i ;?s

i Tdenote bidder i ’s interim (intertemporal)expected payoff in any period in phase S when he has signal s i and reports ?s

i while bidder j reports his signal truthfully.

The discussion below assumes interdependent values.Derivation in the case of private values is similar and hence omitted.We can express p i using the continuation

payoffs u ;%u and %u as follows:For ?s i 4r ;p i es i ;?s i T?e1àd TZ ?s i k i e?s i Tf v i es TàR g f j es j j s i Tds j

td Z ?s i 0

?x e?s i T%u tf 1àx e?s i Tg u f j es j j s i Tds j tZ 1

?s i ?x es j T%u tf 1àx es j Tg u f j es j j s i Tds j !;eA :5T

and for ?s

i p r ;p i es i ;?s i T?d Z ?s

i 0?x e?s i T%u tf 1àx e?s i Tg u f j es j j s i Tds j tZ 1

?s i ?x es j T%u tf 1àx es j Tg u f j es j j s i Tds j !:eA :6T

Upon simpli?cation,we see that (A.5)is equivalent to

p i es i ;?s

i T?e1àd T

Z ?s i k i e?s i Tf v i es TàR g f j es j j s i Tds j

td u àeu à%u Tx e?s i TF j e?s i j s i Tte%g à%u TZ

1?s i x es j Tf j es j j s i Tds j !:

Since k i is differentiable almost everywhere,we can differentiate the above with

respect to ?s

i to obtain @p i @?s

i es i ;?s i T?e1àd T?f v i es i ;?s

i TàR g f j e?s i j s i Tàk 0i e?s i Tf v i es i ;k i e?s i TTàR g f j ek i e?s i Tj s i T àd ?eu à%u Tf x 0e?s i TF j e?s i j s i Ttx e?s i Tf j e?s i j s i Tg te%u àu Tx e?s i Tf j e?s i j s i T ?e1àd T?f v i es i ;?s

i TàR g f j e?s i j s i Tàk 0i e?s i Tf v i es i ;k i e?s i TTàR g f j ek i e?s i Tj s i T àd ?eu à%u Tx 0e?s i TF j e?s i j s i Tte%u à%u Tx e?s i Tf j e?s i j s i T :M.Aoyagi /Journal of Economic Theory 112(2003)79–105