Coordination in Auctions with Entry

journal of economic theory 82, 425450 (1998) article no. ET952452

Coordination in Auctions with Entry* Colin M. Campbell Department of Economics, The Ohio State University, Columbus, Ohio 43210-1172 campbell.371osu.edu Received December 14, 1995; revised June 8, 1998

We examine self-enforcing bidder coordination in auctions in which bidding is costly. Conditions on the distribution of valuations are identified that guarantee multiple equilibria. Under slightly stronger conditions, within the set of ``sunspot'' payoffs only those that are convex combinations of payoffs of extreme asymmetric equilibria are interim efficient for the bidders. If preplay communication is possible, every no-communication equilibrium payoff is interim Pareto-dominated for the bidders by some cheap-talk equilibrium payoff. When the number of auctions is large, communication equilibria exist that yield ex-post efficient payoffs for the bidders almost surely. Journal of Economic Literature Classification Numbers: D44, D82.  1998 Academic Press

1. INTRODUCTION Coordination among competing bidders has been a topic of some interest in the large body of work on auctions. The general approach of the theoretical strain of this literature has been to consider a fixed auction rule (or rules) that is common andor familiar, and to ask how competing bidders may coordinate their actions in such an auction to mutually beneficial ends. Examples include McAfee and McMillan [14] and Pesendorfer [16] for sealed-bid first-price auctions, Mailath and Zemsky [10] for sealed-bid second-price auctions, Avery [1] for open ``English'' auctions, and Graham and Marshall [8] for both second-price and English auctions. An important contribution of these papers is the demonstration that ``standard'' predictions about bidder behavior under a given auction rule, e.g., that * This paper is derived from Chapter Two of the author's Ph.D. dissertation. Thanks to Steven Matthews, William Rogerson, Robert Marshall, Jan Vleugels, and participants at the 1996 North American Meetings of the Econometric Society for many helpful suggestions and comments, as well as to an associate editor and two anonymous referees for insights into how to improve on previous versions of the paper.

425 0022-053198 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

426

COLIN M. CAMPBELL

symmetric bidders will use symmetric strategies that entail no coordination, are not the only predictions that have a strong theoretical basis. This paper attempts to follow in this tradition by examining second-price auctions with the features that submitting a bid entails some fixed cost, and that any potential bidder has the option of not submitting a bid and thus avoiding this cost. Auctions that feature avoidable pre-bid fixed costs are often called ``auctions with entry,'' because in contrast to the standard auction environments that are studied, the number of potential bidders who in fact submit a bid is endogenous. Within the literature on auctions with entry there are two rather different environments that are modeled. In one, the fixed cost represents a cost of acquiring private information that aids in making an informed bid. Thus, at the time that the entry decision is made, the potential bidders are identical with respect to the information that can be acquired via entry. This is the environment studied in, for instance, French and McCormick [7] and McAfee and McMillan [12]. In the other environment, the potential bidders have as much information as they ever will at the time of the entry decision, so that the cost is one simply of submitting a bid. The literature examining this environment is rather small, with Samuelson [17] and Daniel and Hirshleifer [5] being rare examples. We focus here on the latter class of auctions with entry, for two reasons. First, while ``information acquisition costs'' is an appealing specification with an abundance of real-world examples, other fixed costs of bidding can have many sources and are perhaps underrepresented in the literature. One category of these sources includes costs that must be incurred in the physical submission of the bid, e.g., the time and travel expenses of appearing at an auction, or the costs of putting a bid into the proper format. Such costs can be nontrivial: Mills [15] notes that bidding costs incurred by a typical bidder in a government procurement auction often run into the millions of dollars. Another category includes costs that must be incurred for a would-be participant to be recognized as an active bidder. A seller of a good at auction might require all bidders to have access to a minimum amount of bidding funds, compelling some bidders to borrow; or a government selling a state-owned industry for privatization may impose capital requirements on bidders. In all cases, the presence of the fixed costs can be expected to have strategic and behavioral consequences for the bidders. The second reason for examining auctions with costly bidding is that they are open to forms of bidder coordination that are, in our opinion, quite compelling. The cited literature has taken two broadly-defined approaches to the modeling of bidder coordination. One approach is to expand the auction game to include a pre- (and possibly post-) auction phase in which the bidders engage in some coordination activity. The

COORDINATION IN AUCTIONS

427

auction itself is then only one stage in the larger game played by the bidders. In Graham and Marshall [8], McAfee and McMillan [14], Mailath and Zemsky [10] and Pesendorfer [16], the pre-auction phase entails information sharing by the bidders, and all but Pesendorfer [16] feature side-payments (and in some cases a reallocation of the object) after the auction. The other approach, found in Avery [1] as well as in papers by Bikchandani [2] and Bikchandani and Riley [3], is to show that an auction rule may induce a game with multiple equilibria, and to provide some basis (e.g., Pareto- dominance) for believing that bidders might coordinate to one equilibrium rather than another. In this paper we will adopt both approaches. Section Two of the paper describes our auction environment, in which a seller uses a second-price auction for which bidding is costly to sell an object to one of two bidders. When bidding is costly, each bidder has a strict preference for not submitting a losing bid; this will be the source of the possibility of coordination. In Section Three we consider equilibria of the auction. While a symmetric equilibrium of such an auction always exists when the bidders are symmetric, we identify a condition on the distribution of bidder types ensuring that asymmetric equilibria also exist. We then identify an additional condition on the distribution ensuring that within the set of ``sunspot'' equilibria, in which players coordinate to an equilibrium via some public correlation device, every type of bidder prefers that they coordinate to the most asymmetric equilibria. The conditions that are invoked are strong, but, we attempt to argue, encompass a nontrivial class of distributions. In Section Four we examine endogenous coordination by bidders by allowing them to engage in cheap talk before bidding decisions are made. One criticism of the existing literature on pre-auction coordination is that it does not address the issue of how bidders can enforce their agreements to coordinate. The schemes of Graham and Marshall [8], McAfee and McMillan [14], Mailath and Zemsky [10] and Pesendorfer [16] all require some bidders to engage in behavior that is not strictly self-interested, and therefore demand a means of commitment such as binding contracts or repeated interaction. Coordination that results just from allowing cheap talk has the advantage of being fully self-enforcing, since cheap-talk equilibria are simply Bayesian equilibria of the larger game that includes cheap talk. Farrell and Gibbons [6] and Matthews and Postlewaite [11] study the possibilities for coordination via cheap-talk in double auctions, in which a single buyer and a single seller bargain over an exchange price, but such coordination is substantively different than coordination between bidders competing on the same side of a market. Naturally, for information exchange to affect entry decisions the bidders must have private information before these decisions are made, so information

428

COLIN M. CAMPBELL

exchange is of interest only in the ``bidding-cost'' environment, and not in the ``information acquisition-cost'' environment. Our results show that when bidders are symmetric, under the same condition that guarantees that bidders prefer to randomize over the most asymmetric equilbria, for every sunspot equilibrium with no exchange of private information there is a cheap-talk equilibrium involving information exchange (with public correlation) that is better for all types of bidder. Finally, we identify a cheap-talk equilibrium that exists for an arbitrary number of objects to be auctioned simultaneously, and show that as this number grows large the probability that the lower-valuation bidder submits a bid on any given object goes to zero; this may be regarded as the ex-post efficient outcome for the auction of that object from the bidders' standpoint. Before proceeding, we emphasize two points. First, while we feel that the possibility of coordination by competing bidders through self-enforcing cheap talk is an interesting and novel result, this possibility arises solely because of the existence of the bidding costs. The coordination that takes place does not affect what bidders bid, but whether they bid. Indeed, in a second-price auction with no bidding costs, bidders bidding their true values is the only reasonable self-enforcing prediction of the game. Thus, we do not view the self-enforcement obtained here as an improvement on the aforementioned papers, which consider only the case in which bidding is costless, but rather as an indication that perhaps auctions characterized by costly bidding are particularly conducive to bidder coordination. Second, we focus only on a fixed auction rule and ignore all issues of optimal mechanism design by the seller of the object. This is in the tradition of the previous literature on bidder coordination, and is also the approach of the large literature on standard auction rules when bidders have correlated information (e.g., first and second-price common-value auctions). Our defense for doing so is much the same: auctions in which bidding is costly are real institutions, and their use suggests that they merit study as exogenous mechanisms in spite of theoretical suboptimality. In fact, as bidding costs may be interpreted as communication costs, we strongly suspect that the optimal mechanism for the seller in such an environment would be a sequential search mechanism as characterized in McAfee and McMillan [13], which would leave no room for buyer coordination since there are no simultaneous decisions made about incurring fixed costs. However, as in the case of common-value auctions, we do not feel that the existence of a superior, but rather different, mechanism nullifies all interest in behavior that results when the more common mechanism is taken as given.

COORDINATION IN AUCTIONS

429

2. THE AUCTION ENVIRONMENT Two risk-neutral bidders 1 value an object. The bidders have independent private valuations: bidder m # [1, 2] has valuation % m , where % m is a random variable drawn from a continuously differentiable distribution Fm( } ) defined on [0, % ] and satisfying F m(%)<1 for all %<%; and % 1 is independent of % 2 . The bidders compete for the object in a sealed-bid second-price auction, i.e., the bidder submitting the highest bid wins the object and pays to the seller the amount of the other bidder's bid if one was made, paying zero otherwise. Ties are resolved via uniform randomization. In order to submit a bid, bidder m must incur a cost of k m # (0, % ). It is assumed that F m(k m )>0 for both m. The bid decisions are made simultaneously. Only bidder m observes the realization of % m , and he makes this observation before he must decide whether to submit a bid. For some but not all of the results it will be assumed that the sellers are ex-ante homogeneous (shortened hereafter to just ``homogeneous''), i.e., that F1(%)=F 2(%) for all % # [0, % ], and that k 1 =k 2 . A bidder's pure strategy in this game is a function mapping his possible types % m , into actions b m(% m ). An action may either be a real number, representing a bid, or <, representing no bid. Throughout the paper, we will restrict our attention to undominated strategies. As a result, because the auctions are second-price, we need only consider bidder m 's pure strategies of bidding his true cost % m and not submitting a bid. The restriction to this class of strategies allows a very simple characterization of strategies that are candidates for best responses. Fix an arbitrary (mixed) strategy _ &m for the bidder who is not m. The payoffs to bidder m from not submitting a bid (0) and from submitting a losing bid (&k m ) are independent of his type. His payoff conditional on submitting a winning bid is strictly increasing in % m . Therefore, regardless of bidder m's beliefs about how bidder &m will play, if bidder m submits a bid as a best response when his type is %, he must also submit a bid as a best response whenever his type is %$>%. Because F m( } ) is assumed to be continuous, this means that all candidates for best-response strategies are cutoff strategies of the form % m # [0, % ]. The strategy % m , for bidder m is interpreted as, ``bid my type % m whenever % m >% m ; do not bid otherwise.'' In any equilibrium, % m , is the maximum type of bidder m that earns a nonpositive payoff by bidding. The set of cutoff strategy profiles for the players lie in [0, % ]_[0, % ], and because the F m( } ) are continuous, it is easily verified that the players' payoffs are continuous in the cutoffs, and that each bidder's best response 1 For the remainder of the paper, ``bidder'' refers to someone who can choose to submit a bid, rather than to someone who in fact does so.

430

COLIN M. CAMPBELL

to any cutoff strategy of the opponent is a unique cutoff strategy. Brouwer's fixed-point theorem therefore guarantees the existence of an equilibrium in cutoff strategies.

3. EQUILIBRIA In this section we examine equilibria of the auction. We will identify conditions under which multiple equilibria exist, and provide results on which of the equilibria the bidders may be expected to coordinate to given the multiplicity. We begin with the case of homogeneous bidders. In this case there is always exactly one symmetric equilibrium, i.e., one in which the bidders use an identical cutoff. Let F( } ) be the common distribution function, and k the common cost of bidding. Consider now the number %* satisfying F(%*)%*=k. It is an equilibrium for each bidder to use the cutoff %* if it exists; furthermore, since F(%) % is strictly increasing in % on [k, % ] and is less than k for %=k and greater than k for %=%, such a %* exists and is unique. We now turn to when other, asymmetric equilibria exist even when the bidders are homogeneous. Let 1 %>%$ , be an indicator function, defined in the standard way. Returning to the general environment, the expected payoff to bidder m from submitting a bid when his type is % m , and his opponent is using the cutoff % &m is F&m(% &m ) % m +1 %m >% &m

|

%m %&m

(% m &x) dF &m(x)&k m .

 &m may therefore be Bidder m's best-response cutoff %* m as a function of % defined by the condition F &m(% &m ) %*m +1 %*m >% &m

|

%* m %&m

(%*m &x) dF &m(x)k m ,

. < only if %* m =%  )=k m and is strictly decreasing, except This implicit function satisifies %* m(% possibly when it takes on the value %. Thus, its inverse %*m&1( } ) is well 1 , % 2 ) must theredefined on (k m , %* m(0)). Any interior cutoff equilibrium (% &1      fore satisfy, for instance, % 2 =% 2*(% 1 )=%*1 (% 1 ). The set of equilibria has certain properties that will be useful to invoke in the analysis that follows. First, since the best-response functions are downward-sloping in (% 1 , % 2 ) space, if (% A1 , % A2 ) and (% B1 , % B2 ) are distinct equilibria with % Am >% Bm , then it is necessarily the case that % A&m <% B&m .

COORDINATION IN AUCTIONS

431

Thus, there is a natural ordering of equilibria, e.g., by the value of % 1 . Second, for any distributions F m( } ), except possibly for knife-edge specifications of the bidding costs k m there will be a finite number of equilibria, and at any equilibrium (% 1* , % 2*) % 2*( } ) will cross from either above or below in  and that lim %  k1k +  1 &1(%)=%, (% 1 , % 2 ) space. Noting that % *(k 2 1 )% 1 %* there are two possibilities: either (k 1 , % ) is an equilibrium, or % 2*(k 1 )<%. In the latter case, the equilibrium (% 1 , % 2 ) characterized by the smallest % 1 among all equilibria must either be (%, k 2 ), or must be an interior equi 1* &1( } ) from below (see Figure 1). Thus, librium at which % *( 2 } ) cuts % any set of equilibria must satisfy at least one of the following properties: (1) there is a boundary equilibrium, in which one bidder never submits a

FIG. 1.

Equilibrium at which % 2*( } ) cuts % 1* &1( } ) from below.

432

COLIN M. CAMPBELL

bid, or (2) there is at least one interior equilibrium at which % *( 2 } ) cuts %*1 &1( } ) from below. Taken together, the two properties above imply that a sufficient condition for the existence of an asymmetric equilibrium in the homogeneous bidder case is that the symmetric equilibrium (%*, %*) satisfies &1$ (%*) (the differentiability of which is guaranteed by the % 2*$(%*)<% * 1 differentiability of the F m( } )). Under what exogenous conditions is this likely to be the case? Consider type %* of bidder 2, who is indifferent between bidding and not bidding in this equilibrium. In particular, this type earns a profit by bidding only when bidder 1's type is less than %*. Suppose that, conditional on having a type less than %*, bidder 1 is rather likely to have a type very close to %*. This is true if the hazard rate ( f (%*))(F(%*)) is large. Then if bidder 1 uses a slightly larger cutoff than %*, this results in a relatively large increase in expected profit for bidder 2, implying a relatively large decrease in bidder 2's optimal cutoff, which is the desired condition. Formally: Proposition 3.1. For homogeneous bidders, if ( f (%))(F(%))>1% for all % # (k, % ), an asymmetric equilibrium exists. Proof. It must be shown that the condition guarantees that &1$ (%*). The best response functions are identical in this % 2*$(%*)<% * 1 homogeneous case, so the derivatives of % 2*( } ) and %*1 &1( } ) are reciprocals when evaluated at %*. It therefore suffices to show that the derivative of % 2*( } ) is less than &1 when evaluated at %*. For % 1 >%*, % 2*(% 1 )=k(F(% 1 )). Differentiating and substituting the equilibrium condition % 2*(%*)=%* yields that the derivative when evaluated at &%* is %*( f (%*))(F(%*)). Since %* necessarily lies in (k, % ), the condition guarantees the desired result. K Note that the proof only requires that the hazard-rate condition hold at the symmetric equilibrium cutoff profile. However, the desire to impose a condition on the exogenous environment rather than on the endogenous equilibrium obliges us to require it of all possible values for the equilibrium. Since 1% is the hazard rate when F( } ) is uniform, the condition is equivalent to requiring that the distribution put greater probability on high values of % (conditional on %>k), in the sense of hazard-rate dominance, than does the uniform distribution. 2. We comment briefly here on a possible refinement of equilibria in this game, beyond the restriction to equilibria in cutoff strategies. Equilibria at 2

Campbell [4] shows the existence of asymmetric equilibria for the uniform distribution. The proof there also demonstrates that distributions violating the hypothesis of Proposition 3.1 for all % but still ``close'' to the uniform distribution must still have asymmetric equilibria, so the condition is sufficient but not necessary.

COORDINATION IN AUCTIONS

433

 *1 &1( } ) from below satisfy the following notion of which % *( 2 } ) cuts % stability: if one defines the best response dynamic (% t+1 , % t+1 )= 1 2 t t  ), % *(% )), then for initial pairs of cutoffs near such an equilibrium, (% *(% 1 2 2 1 the dynamic converges to that equilibrium. Equilibria at which % 2*( } ) cuts %*1 &1( } ) from above have the property that the dynamic never converges to the equilibrium unless the equilibrium itself is the initial condition. Further but more, except for the knife-edge case in which for some m % * m(k &m )=%  for all %>k &m any boundary equilibrium also satisfies this %* m(%)<% notion of stability. 3 Since, as was argued before, the equilibria characterized by the smallest and largest equilibrium cutoffs for bidder 1 (which are necessarily the equilibria characterized by the largest and smallest, respectively, cutoffs for bidder 2) are either boundary equilibria or are interior equilibria at which % 2*( } ) cuts %*1 &1( } ) from below, these extreme equilibria are necessarily stable in this sense. In the coordination equilibria that follow we will give these equilibria, referred to hereafter as the most asymmetric equilibria because of their places in the ordering, special attention; the guarantee of stability is one justification for doing so. Sunspot Equilibria In this subsection we explore how bidders fare by coordinating to one equilibrium rather than another when multiple equilibria exist. In one sense, the answer is not interesting. As discussed previously, when the equilibria are ordered by their values of bidder 1's cutoff, say from the smallest to the largest, the decreasing best-response functions imply that the same ordering ranks them in decreasing order by their values of bidder 2's cutoff. It is easy to see that for any two distinct cutoff strategies %<%$ of his opponent, a bidder's payoff given best-response behavior is higher when his opponent uses %$. This implies that for any two distinct equlibria A and B with % A1 <% B1 , it is necessarily the case that bidder 1 earns a higher payoff in equilibrium A than in equilibrium B, and that bidder 2 earns a higher payoff in equilibrium B than in equilibrium A. Thus, no two equilibria can be Pareto ranked for the bidders, and the notion of ``coordination'' has little meaning other than the mutual belief that a particular one of the equilibria will be played. To enrich the possibilities for mutually beneficial coordination by the bidders, we consider the set of ``sunspot'' equilibria of the game. The game is altered by allowing the players access to a publicly observable randomizing device. The players may use cutoff strategies that depend on the realization 3 This refinement can be formalized by extending the notion of persistent equilibrium (Kalai and Samet [9]) for finite games to this game. The extension is possible because the set of cutoff strategies is compact and convex, and payoffs are continuous in the cutoffs. The extension requires a weaker definition of a persistent retract to guarantee existence of a persistent equilibrium, but the modified refinement is identical to persistence for finite games.

434

COLIN M. CAMPBELL

of this device. Any equilibrium of this expanded game is such that for any realization of the device, each bidder makes the same conjecture about which equilibrium of the base auction game will be played. If the support of the realizations of the device is sufficiently rich (e.g., the set [0, 1]), then any probability distribution over the set of equilibria of the base auction game can be supported as a sunspot equilibrium of the game with the device. Such equilibria are a subset of the set of correlated equilibria of the auction game. However, they put less stringent requirements on the external device than do general correlated equilibria, whose devices may be required to ``talk'' privately to players, rather than simply generating a publicly observed random outcome. Our goal is to show that when sunspot coordination is possible, both bidders may prefer to coordinate to one of the two most asymmetric equilibria. We identify a condition under which this is true for the homogeneous bidder case. Before stating the result, we give some intuition for why bidders might prefer to select most asymmetric equilibria when there are multiple equilibria from which to choose. Consider first a bidder with a valuation so low that among all equilibria he only submits a bid in the most asymmetric equilibrium most favorable to him. Since he earns a payoff of zero in all other equilibria, he clearly prefers a sunspot equilibrium that puts positive probability on this most asymmetric equilibrium being played to one putting zero probability on it. Next, consider a bidder of type %, who submits a bid in all equilibria except possibly in the most asymmetric equilibrium that is least favorable to him. Note that when bidders are homogeneous, asymmetric equilibria come in symmetric pairs: if (%, %$) is an equilibrium, then (%$, % ) necessarily is also. The difference between the most asymmetric equilibrium most favorable to this bidder and his next favorite equilibrium is that in the former his opponent is less likely to enter a high bid. The difference between the most asymmetric equilibrium least favorable to him and his next least-favorite equilibrium is that in the latter his opponent is less likely to enter a low bid. Since the bidder gains more when his opponent refuses to enter high bids than when he refuses to enter low ones, then on average the bidder will prefer a mix of the most asymmetric equilibria than a mix of the next-most asymmetric equilibria, if the probability that his opponent will have a high valuation is sufficiently high. Thus, our formal result will require some condition guaranteeing that high valuations are sufficiently likely relative to low valuations. The condition that we will invoke is that the density f ( } ) is nondecreasing on [k, % ]. Proposition 3.2. For homogeneous bidders, let F( } ) satisfy f ( } ) nondecreasing on [k, % ]. Then any sunspot equilibrium that with positive probability selects an auction equilibrium other than a most asymmetric

COORDINATION IN AUCTIONS

435

equilibrium has a payoff that is not interim efficient for the bidders within the set of all sunspot equilibrium payoffs. Proof.

See appendix.

Remark. As the nondecreasing f ( } ) assumption may strike the reader as objectionably strong, we comment on it here. Examination of the proof reveals that the result holds for far weaker assumptions on the set of endogenous equilibria. For example, we could alternatively assume that f( } ) is nondecreasing only between the lower and higher cutoffs in the most asymmetric equilibrium (even this assumption is much stronger than is needed). However, as with Proposition 3.1, we desire a condition on the fundamentals of the environment. Thus, the condition is sufficient but not necessary. One reasonable objection is that the assumption rules out the limit case when % =, since any distribution with support [0, ) must exhibit a declining density for some range of large arguments. However, consider this limit case, and let % be the best response to a cutoff of k. Since the best-response cutoff to % is greater than k under the assumption that F(%)<1 for all finite %, all equilibrium cutoffs must lie in (k, % ). Thus, the condition f ( } ) nondecreasing on (k, % ) is sufficient, and the unbounded support case is permitted. 4 Furthermore, take a distribution satisfying f( } ) nondecreasing on [k, % ], and suppose that the most asymmetric equilibrium cutoffs (% l , % h ), with % l <% h , are interior. From previous  *1 &1(%) for all % # (k, % l ). Now arguments, this implies that %* 2 (%)<% construct a new distribution G( } ) such that G(%)=F(%) for all % # [k, % h ], but with G(%)F(%) for all % # [k, % ]. It is readily verified that (% l , % h ) remains an equilibrium when the distribution is G( } ), and that the new %*1 &1(%) function lies strictly above the old one for % # (k, % l ). Thus, (% l , % h ) also remain the most asymmetric equilibrium cutoffs. Since g( } ) coincides with f ( } ) on the interval, the conclusion of Proposition 3.2 must hold. But g( } ) need not be nondecreasing on [% h , % ]. Thus, given the family of distributions satisfying the stronger condition, a larger family can be readily generated for which the proposition still holds. Finally, Campbell [4] shows that an interim Pareto-improvement results for the uniform distribution, implying by continuity that such an improvement must result for at least some distributions characterized by densities that are decreasing on a portion of the relevant range of cutoffs. All told, as a sufficient but unnecessary condition we suggest that the assumption permits a restricted but still interesting subset of possible distributions. In addition to the result of Proposition 3.2, note that because every type of bidder m at least weakly prefers more probability to be placed on equilibria in which bidder &m uses a larger cutoff, the payoffs of all sunspot 4

We thank a referee for bringing this to our attention as a potential objection.

436

COLIN M. CAMPBELL

equilibria that only put probability on the most asymmetric equilibria are interim efficient within the set of all sunspot equilibrium payoffs. Also, it can be shown that f ( } ) nondecreasing on [k, % ] and ( f (k))(F(k))>1k imply the hypothesis of Proposition 3.1, so Proposition 3.2 holds in a case in which multiple equilibria are known to exist. Propositions 3.2 and 3.1 together suggest that randomization over most asymmetric equilibria can be an extremely effective way for bidders to coordinate their actions in this auction environment. The interim efficiency of such sunspot equilibria means that the equilibria are not subject to some types of bidder trying to convince the other bidder that it would be mutually beneficial to play a different sunspot equilibrium. And once the auction equilibrium is chosen, that equilibrium is necessarily stable. In addition, if there are multiple auctions, held either simultaneously or over time, such that any given bidder's valuations are independent and identically distributed over those auctions, then the per-auction sunspot payoffs can be achieved by playing different equilibria in different auctions rather than by randomizing. For instance, for sequential auctions, bidders might ``take turns'' using the aggressive strategy in the most asymmetric equilibrium profile, yielding a higher time-zero expected payoff than if, say, they always played the symmetric equilibrium. 5 And finally, since an equilibrium is played in each individual auction, the bidders' coordination is by definition self-enforcing.

4. CHEAP-TALK EQUILIBRIA In the previous section we considered only equilibria that entailed no sharing of private information by the bidders: ``coordination'' meant simply a selection from among several equilibria of the auction game. In this section we explore the opportunities for the bidders to improve their payoffs further by engaging in cheap talk after they have learned their private information, but before they make their bidding decisions. As in all Bayesian games in which players have private information, the introduction of cheap talk is liable to expand the set of payoffs for the players that can be supported in equilibrium, although a strict expansion need not occur. Rather than attempting to find the entire set of cheap-talk equilibria for the auction, we settle for identifying some that have nice welfare properties for the bidders. These equilibria fall into two classes, where the defining 5 Bikchandani and Riley [3] note a similar opportunity for bidders in sequential secondprice common-value auctions.

COORDINATION IN AUCTIONS

437

feature of a class is the rule that players use to talk to each other. For the first class, we are able to show that under the same condition required for Proposition 3.2 to hold, any sunspot equilibrium of the no-communication game is interim Pareto-dominated by some cheap-talk equilibrium in this class. Thus, cheap talk can always strictly improve on the coordination available without cheap talk. The second class of cheap-talk equilibria exist when there are multiple objects to be auctioned simultaneously, and when any given bidder's valuations for the objects are independent and identically distributed. In this case, we are able to obtain a very strong welfare result when the number of objects is large: for a given number of objects N there exists a cheap-talk equilibrium such that in the auction for object n, the ex-ante probability that the bidder with the lower valuation for n submits a bid, or that the bidder with the higher valuation for n does not submit a bid even though his valuation is strictly larger than his cost of bidding, is arbitrarily small if N is sufficiently large. Thus, in a proportion very close to 1 of auctions, the bidders will enjoy their highest possible ex-post joint payoff. The equilibria in this class generate such efficiency because in the limit they fully reveal the bidders' private information, allowing them to coordinate to the best possible joint outcome. We model cheap talk in a standard way. Each bidder has access to some space of messages he can send to the other bidder. For our purposes it is sufficient that the space of possible messages for each bidder be the [0, 1] continuum, and that the bidders simultaneously send exactly one message each, after they have learned their private information. Following the message exchange, the bidders play the auction game. A bidder m 's strategy now has three components: a talking strategy , m : [0, % ] Ä [0, 1], which determines what message he will send as a function of his type; a belief function ; m : [0, 1] Ä F [0, % ], where F [0, % ] is the set of all probability distributions on (the Borel sets of) [0, % ], which describes how he updates his beliefs about &m 's type as a function of the message &m sends; and a bidding function b m : [0, % ]_X Ä R _ <, where X is a set of publicly observable variables that includes the bidders' messages (and possibly public randomizations). A perfect Bayesian equilibrium of the game is any strategy profile such that (1) each bidder's belief function results from Bayesian updating of his prior according to his opponent's talking strategy whenever possible, (2) bidders play best responses at the bidding phase according to their beliefs and to the public information available, and (3) bidders play best responses in the talking phase given the belief functions and the bidding rules. 6 As in Section Three we are interested only in undominated strategies, and so can restrict attention to 6 Any perfect Bayesian equilibrium payoff of this game can be supported by a sequential equilibrium, so we use the weaker but simpler equilibrium concept.

438

COLIN M. CAMPBELL

bidding functions that are cutoff rules, where the cutoffs used can now be conditioned on messages sent in the talking phase. ``I 'm Out'' Equilibria When searching for equilibria of a cheap-talk game in which a player reveals some private information, one must ask which types of that player, or sets of types, can credibly separate themselves. In the auction game studied here, consider a bidder m whose valuation is less than k m . Regardless of m 's conjectures about the actions of his opponent &m, m will not submit a bid, since doing so would yield a negative payoff with certainty. Thus, m would lose nothing by revealing to &m that his valuation is less than k m . To determine whether separation of these types is possible in a cheap-talk equilibrium, it therefore only needs to be checked that types greater than k m would not wish to misrepresent themselves as having types less than k m given &m's response to the two messages ``less than k m '' and ``greater than k m ''. Note that if &m believes that m's valuation is less than k m , then in any equilibrium &m must believe that m will surely not submit a bid. The only best-response candidate for &m given such a belief is therefore to use a cutoff of k &m . However, this is the worst possible cutoff for &m to use from m's standpoint, within the class of cutoffs that are candidates for best responses. Thus, in any equilibrium no bidder with a type greater than k m would wish to make &m believe that his type was less than k m , and separation is possible. To formalize the kind of equilibrium we have in mind, partition the message space [0, 1] into two sets, say M0 and M1 . The cheap-talk equilibrium is as follows. Each bidder m uses a talking strategy such that he sends a message from M0 if and only if his valuation is less than or equal to k m , and sends a message from M1 otherwise. Each bidder m 's belief function maps any message in M0 into (F &m ( } ))(F &m (k &m )) on [0, k &m ] (i.e., the posterior associated with the information that &m 's type is less than or equal to k &m ), and maps any message in M1 into (F &m ( } )&F &m (k &m ))(1&F &m (k &m )) on (k &m , % ] (i.e., the posterior associated with the information that &m's type is greater than k &m ). In the auction, if bidder &m announced a message in M0 , then bidder m uses a cutoff of k m . If both bidders announce messages in M1 , then they play some equilibrium (possibly a sunspot equilibrium) consistent with their updated beliefs. We call such equilibria ``I'm Out'' equilibria, because sending a message in M0 carries the equilibrium meaning that you will certainly not submit a bid. Equilibria within this class differ in their payoffs only insofar as they differ in the auction equilibria played when both bidders have valuations greater than k m . To say something about these equilibria, we return to the

COORDINATION IN AUCTIONS

439

case of homogeneous bidders, for which several results were derived in the previous section. First, the hazard rate for the bidders' posterior beliefs is ( f (%))(F(%)&F(k)), which is greater than ( f (%))(F(%)) for all % # (k, % ]. Thus, if the condition ( f (%))(F(%))>1% for all % # (k, % ] holds, guaranteeing multiple equilibria in the no-communication game, then the condition holds for the posterior beliefs and multiple continuation equilibria exist. In addition, if f ( } ) is nondecreasing on [k, % ], then the posterior density ( f ( } ))(1&F(k)) is also nondecreasing on this domain, and so the result that only sunspot equilibria putting positive probability only on most asymmetric equilibria of the auction are efficient for the bidders also carries over for the set of sunspot equilibria in the ``I'm Out'' class. Our goal is to show that any no-communication sunspot equilibrium is Pareto dominated for the bidders by some sunspot equilibrium in the ``I'm Out'' class. 7 There are clear advantages of an ``I'm Out'' equilibrium to the bidders: in the event that one bidder has a valuation below his cost of bidding, the other bidder learns this. If the other bidder has a valuation such that he would not have submitted a bid in an equilibrium with no communication, but does profit by being the sole bidder, the equilibrium allows him to realize this profit. Since the bidder with a valuation below his bidding cost neither gains nor loses from the information exchange, the communication is Pareto-improving. What makes the total effect on the bidders' welfare ambiguous is what happens when both have valuations above their respective bidding costs. The following proposition establishes that under the same condition guaranteeing the result of Proposition 3.2, the efficiency gain is unambiguous: Proposition 4.1. For ex-ante homogeneous bidders, let F( } ) satisfy f ( } ) nondecreasing on [k, % ]. Then for any sunspot equilibrium S of the nocommunication game, there is a sunspot ``I 'm Out'' equilibrium S$ of the cheap-talk game such that every type of bidder earns a higher payoff in S$ than in S, some strictly higher. Proof.

See appendix.

The proof essentially just establishes that in the most asymmetric equilibrium of the auction following announcements by both bidders that their types are at least k, the passive cutoff is higher than is the passive cutoff of the most asymmetric equilibrium of the no-communication game. The intuition for why this is preferred is the same as the intuition for why Proposition 3.2 holds: for distributions that put sufficient probability on 7 Campbell [4] also shows that for any distribution F( } ) the symmetric equilibrium is interim Pareto-dominated for the bidders by an ``I'm Out'' equilibrium in which they use a symmetric cutoff after both announcing that they will potentially submit a bid.

440

COLIN M. CAMPBELL

high types, a bidder m gains by moving to an equilibrium in which &m doesn't enter as often when &m 's type is high. This gives us the hoped for result that regardless of the equilibrium being played by the bidders when they do not communicate, engaging in cheap talk can always raise the interim payoff for every type of bidder. Multiple-Auction Equilibria The final opportunity for bidder coordination we will explore exists when there are multiple objects to be auctioned simultaneously. The environment we consider is essentially just a replication of the one-object environment. Bidders' valuations over the objects are linear; i.e., there are no complementarities or substitutabilities across objects. Also, bidder m 's valuation for object n and his valuation for different object n$ are assumed to be independent and identically distributed, and his bidding cost k m is the same for each object. These conditions imply that any (non-sunspot) equilibrium of the N-object auction game is necessarily just an N-dimensional cross-product of (non-sunspot) equilibria of the single-auction game. Thus, a cheap-talk equilibrium in this environment will be of special interest only if the talking strategies somehow depend explicitly on the multiplicity of objects for sale. To motivate the kind of equilibrium we have in mind, consider the following informal argument. You and I are bidding for two objects, and my valuation for object 1 is higher than my valuation for object 2. Suppose I announce to you that I value object 1 more than 2, without saying anything else about my valuations. If you believe my announcement, in any equilibrium you must change your beliefs such that you think I am likely to have a high valuation for 1 (relative to what you thought before the announcement), and likely to have a low valuation for 2. This should in some sense make you want to bid less aggressively for 1 (use a lower cutoff) and more aggressively for 2 than you would have otherwise, because your optimal cutoff depends on your calculation of how likely I am to submit a high bid. But if I have the ability to make you bid less aggressively for one object and more aggressively for another simply by making such an announcement, I should prefer to make you bid less aggressively for the object I value more, since I enjoy a greater benefit from it when I win. Thus, if I make a consistent conjecture of how you will react to my announcement, my announcement that I value 1 more than 2 is credible. It is evident that if this argument is valid, it can be applied to an arbitrary number of objects. That is, if there are N objects being auctioned simultaneously, then an announcement by a bidder that ranks all N of them according to his valuations is liable to be credible, if appropriate

COORDINATION IN AUCTIONS

441

equilibria can be identified following the announcements. The class of equilibria we consider here entail talking strategies that do exactly this. In particular, partition the message space into N ! sets. Since there are N! possible orderings of the objects, it is possible to employ a talking strategy that maps rankings into elements of the partition in a one-to-one fashion. Doing so fully reveals a bidder's ranking of the objects by valuation (ties occur with probability zero because of the continuity of F m ( } )), but reveals no other private information. We refer hereafter to such talking strategies as ranking talking strategies. We have promised that the equilibria we consider for this case deliver ex-post efficiency. Three kinds of ex-post inefficient outcomes are possible in our auction. In one, both bidders bid on an object, which wastes bidding costs since only one bidder can win. In a second, no bidder bids on an object and one of them has a valuation higher than his bidding cost, but does not bid for fear that he will not win, or will have to pay too high a price if he does win. In a third only the bidder with the lower potential profit (valuation minus bidding cost) for an object submits a bid. In the absence of side payments it is possible that one would not want to classify this third outcome as inefficient, but showing that such events can be avoided as well strengthens our results. Suppose that each bidder had full information about the other's valuation for an object. Then there is an equilibrium that achieves ex-post efficiency in the above sense: the bidder with the lower potential profit does not bid, while the bidder with the higher potential profit bids if and only if that profit is nonnegative. Thus, talking strategies that fully reveal the bidders' private information could lead to ex-post efficient equilibria of the auction subgame. However, fully-revealing talking strategies cannot be part of any equilibrium of the whole cheap-talk game: if bidder &m believes bidder m 's announcement about his exact valuations for each object, then for every object for which his true valuation was at least k m bidder m would have an incentive to claim that his valuation was %, thus deterring &m from submitting bids for those objects. The key to our result is that the ranking talking strategies come arbitrarily close to revealing all of a bidder's private information as the number of objects grows unboundedly. This follows from a simple law-of-large-numbers argument. For example, if N=1001 and bidder m ranks object 1 501st, then it is highly likely that m's valuation for object 1 lies very close to the median of his distribution. Likewise, bidder m will be able to make an excellent low-variance prediction about bidder &m 's valuation for object 1, and this mutual knowledge will allow the bidders to coordinate to an equilibrium of the auction for object 1 that yields ex-post efficiency with a very high probability. To state the formal result, for an equilibrium E of the N-object auction let E n (% 1n , % 2n ) be the sum of the bidders' payoffs from the auction for

442

COLIN M. CAMPBELL

object n when bidder m has valuation % mn . For any type profile (% 1 , % 2 ), define ?*(% 1 , % 2 )#max[0, % 1 &k 1 , % 2 &k 2 ]. As pointed out above, for any E and (% 1n , % 2n ) it is the case that ?*(% 1n , % 2n )E n(% 1n , % 2n ). Let RN be the set of all equilibria of the N-object auction in which both bidders use ranking talking strategies. Proposition 4.2. For any =>0 there exists N such that NN implies that there exists E # RN such that the ex-ante probability that E n (% 1n , % 2n ){?*(% 1n , % 2n ) is less than = for all n. Proof.

See appendix.

The proof shows that there is a mapping from the bidders' ranking announcements to equilibria of the auction subgame that validates the intuition given in the example at the beginning of this subsection. Specifically, the higher a ranking you announce for an object, the more passive a bidding strategy your opponent will use in the auction for that object. Since you have more to gain by your opponent using a more passive bidding strategy the higher your valuation is, your ranking announcement is credible. The asymptotic efficiency result holds because as N becomes large, bidders are able to estimate each other's valuations with a high degree of accuracy by using the rankings. The equilibrium selection rule is such that in the auction for object n, the equilibrium most favorable to the bidder who has the highest potential profit from n on the basis of the rankings is chosen (which as has been established previously is a persistent equilibrium; see footnote 3); since in the limit the rankings accurately reveal which bidder should bid, as well as revealing his valuation, with high probability the equilibrium outcome is that he will bid and his opponent will not. 8 We end this section by emphasizing one point regarding the efficiency result. Many results in economics are some variation on the idea that when an environment is ``large'' (with respect to, e.g., the number of agents or commodities), efficiency can be attained. Here it is true that efficiency can only result for a large number of objects. However, it is not the number of objects per se that generates the efficiency, but rather the opportunity for bidders credibly to reveal their private information. In particular, if the bidders did not use talking strategies that compared their valuations across items, then any equilibrium of the N-object game would yield a per-auction total payoff to the bidders that was no greater than the largest total payoff available in an equilibrium of the single auction. No equilibrium of the 8 This result will also hold even if each bidder's cost of bidding is different for different objects, as long as these costs are private information and are independent and identically distributed across objects. In this case the bidders must rank the objects by % mn &k mn .

COORDINATION IN AUCTIONS

443

single auction can achieve ex-ante efficiency, so the combination of a large number of objects and cheap talk is necessary to our result.

5. CONCLUSION We have attempted to provide insight into the opportunities for bidder coordination in second-price auctions with bidding costs. Our approach was nonsystematic and noncommital: we explored several sunspot equilibria and cheap-talk equilibria of the environment, without attempting to characterize the entire set of such equilibria or selecting a unique equilibrium from those we did identify. Rather, we chose to describe some equilibria and suggest that they represent effective ways for bidders to coordinate, on the basis of welfare rankings and a stability condition. Unlike the existing theoretical work on bidder coordination, which considers the standard auction environment in which bidding is costless, bidder coordination in our environment does not affect how much bidders bid, but only whether they bid. While this departure perhaps puts our work on the fringe of the literature, it has bought us opportunities for coordination that are novel and, in our opinion, compelling, as they require only selfenforcing behavior on the part of the bidders. All of our ``coordination'' consists simply of the playing of equilibria in which publicly observable information, either exogenously generated or endogenously revealed by the bidders through cheap talk, leads the bidders to play particular equilibria of the bidding subgame. Ex-ante, this coordination can yield all types of bidder higher interim payoffs than if no coordination took place and bidders simply played one specific equilibrium of the bidding subgame. All results were derived for a fixed auction rule that is a theoretically suboptimal selling mechanism. Despite this, since auctions in which it is costly to bid do exist, we feel that treating the rule as exogenous is an interesting and valid approach to the study of bidder coordination. Indeed, we would be interested to learn about the opportunities for self-enforcing bidder coordination in other ``standard'' auction designs, such as the first-price sealed-bid auction, when bidding is costly.

APPENDIX Proof of Proposition 3.2 The proposition holds trivially if the symmetric equilibrium is unique. Otherwise, there exist two distinct most asymmetric equilibria. Let ;#(% min, % max ) be the most asymmetric equilibrium in which bidder 1 uses the smallest cutoff, and ;$#(% max, % min ) (by symmetry

444

COLIN M. CAMPBELL

of asymmetric equilibria under homogeneity) the most asymmetric equilibrium in which he uses the largest cutoff. For any equilibrium E of the auction game, let ? m (E; %) be bidder m 's interim payoff in E when his type is %, m # [1, 2]. Since the most asymmetric equilibria are a symmetric pair, ? m (;; %)=? &m ( ;$; %) for all %. Let a sunspot equilibrium S put probability _>0 on some auction equilibrium E$=(% $1 , % $2 ) that is not a most asymmetric equilibrium. Define the number * to satisfy *? 1 ( ;; % )+(1&*) ? 1 (;$; % )=? 1 (E$; % ). Since ; and ;$ are the most asymmetric equilibria, and because equilibrium payoffs are decreasing in one's best-response cutoff, it is necessarily the case that * # (0, 1). Define sunspot equilibrium S$ to be identical to S, except that it puts no probability on E$, instead putting an extra *_ probability on ; and an extra (1&*) _ probability on ;$. It will be shown that every type of each bidder earns a higher payoff in S$ than in S, some strictly so. For either bidder m, any type % m % $m earns a zero payoff in E$, and so is at least as well off in S$; all those types that bid in some most asymmetric equilibrium are strictly better off. Next, consider any type % m # [% $m , % max ]. The difference between such a type's payoff in S$ and in S is % _[* m [F(% max ) % m & k] & F(% $& m ) % m & 1 %m >% $&m  %m$&m (% m & x) dF(x) + k], where * m equals * for m=1 and 1&* for m=2. This difference is readily shown to be concave in % m . Thus, since it is positive for % m =% $m , it is nonnegative for all such % m if and only if it is nonnegative for % m =% max. Therefore, restrict attention to types % m # [% max, % ]. The difference between such % a type's payoff in S$ and in S is _[* m[F(% m ) % m & %mmax x dF(x)&k]+ %m (1&* m ) max[F(% m ) % m & % min x dF(x)&k, 0]&F(% m ) % m + %%m$ x dF(x) &m +k]. The maximum term is zero only when % max =%. If % max <%, the expression is constant in % m . In either case, the proof is completed by establishing that type % of each bidder prefers S$ to S. By construction of S$ type % of bidder 1 is indifferent, so the result will follow by showing that type % of bidder 2 earns an interim payoff at least as high in S$. We consider two possible cases. First, suppose that % max <%. Then the difference in payoffs over S$ and S for type % of bidder 2 is % max % $ _[(1&*)  %$ x dF(x)&*  %1min x dF(x)]. Substituting for * from its defining 1  max equation yields that this difference is identical to _[ %max[%  $1, % $2 ] x dF(x)&  $1, % 2$ ]  $1 % $2 .  min[% x dF(x)]. Without loss of generality assume that % % min It is clear that if F(% max )&F(% $2 )F(% $1 )&F(% min ), this difference is strictly positive. Thus, consider only the case in which F(% max )&F(% $2 )< F(% $1 )&F(% min ). Because % max <% by hypothesis, we have that type % max of bidder 2 earns a zero expected payoff from bidding in equilibrium ;, and that type % $2 of bidder 2 earns a zero expected payoff from bidding in equilibrium E$. Substituting these two conditions yields that the difference is max equal to _[2  %% $2 x dF(x)&F(% max ) % max +k+F(% $2 ) % $2 &k]. Substituting

COORDINATION IN AUCTIONS

445

the conditions that type % min in of bidder 1 earns a zero pay-off in equilibrium ; and that type % $1 of bidder 1 earns a zero payoff in equilibrium max E$ further yields that the payoff can be rewritten _[2  %% $2 x dF(x)& F(% max )% max +F(% $2 ) % $1 +F(% $2 ) % $2 &F(% max ) % min ]. Since f ( } ) is nondecreasing on [k, % ], this expression is greater than or equal to _[2(F(% max )&F(% $2 )) ((% max + % $2 )2) & F(% max) % max + F(% $2 ) % $1 + F(% $2 ) % $2 & F(% max) % min ]= _[F(% max )(% $2 &% min )&F(% $2 )(% max &% $1 )]. f ( } ) nondecreasing on [k, % ] implies that F( } ) is weakly convex on this domain, so the hypothesized condition F(% max )&F(% $2 )% max &% $1 , establishing that the above expression is positive. This completes the proof for the case % max <%. For the case in which % max =%, the difference in type % of bidder 2's payoffs in S$ and S, is _[(1&*)[% & %% max x dF(x)&k]&% + %% $1 x dF(x)+k]. Substituting for * yields _[ %% $2 x dF(x)&% + %% 1$ x dF(x)+k]. As (k, % ) is an equilibrium it is known that % & %k x dF(x)k. Thus, this difference is  min[% 1$ , %2$ ] greater than or equal to _[ %max[% x dF(x)]. Again let  1$ , % $2 ] x dF(x)& k % $1 % $2 without loss of generality. This difference is clearly positive for 1&F(% $2 )F(% $1 )&F(k). Thus, concentrate only on the case 1&F(% $2 )< F(% $1 )&F(k). The difference between the payoffs can be written _[2  %% 2$ x dF(x)+ %% 21$$ x dF(x)&% +k]; substituting the condition that type % $1 earns a zero payoff by bidding in equilibrium E$ yields _[2  %% 2$ x dF(x)+ F(% $2 ) % $2 &k&% +k]. From here the argument proceeds in identical fashion to the previous case, using the weak convexity of F( } ) on [k, % ] to show that this expression is positive given that 1&F(% $2 )
446

COLIN M. CAMPBELL

 max Consider then only the case % max <%. Let (% min IO , % IO ) be the most asymmetric equilibrium of the auction after both bidders have announced that they have valuations higher than k, in which bidder 1 uses the more aggressive strategy. It will first be shown that given the convexity of F( } ) on  max. [k, % ], it is necessarily the case that % max IO % max Consider the function g(%)=F(% )(%&% min )& %% min x dF(x) (%&k) max min max &k). It is clear that g(% )=0. % <% implies that F(%max ) % max & (% % max max min  % min x dF(x)=k. Also, F(% ) % =k. Substitution of these two conditions shows that g(% max )=0. The derivative of g( } ) is F(% max)&1(% max&k) [%f(%)(%&k)+ %% min x dF(x)]. If f ( } ) is nondecreasing on [k, % ], then the derivative of g( } ) is decreasing on [% min, % max ]. Thus, g( } ) is concave on this domain, and g(%)0 for all % # [% min, % max ].  Let % satisfy (F(% max )&F(k))(1&F(k)) % max & %%l x(dF(x)(1&F(k)))=k if a solution exists. % is the cutoff for bidder &m to which % max is a best response for bidder m, conditional on the information that &m's valuation is at least k. It is easily seen that % >% min, since (% min, % max ) is an equilibrium of the no-communication game when bidder 2 thinks that 1's valuation may be below k. Suppose it is also the case that % <% max. Then by the above g(% )0, or F(% max )(% &% min )& %% min x dF(x)(% &k)(% max &k)0. Substituting the first condition above for k in the equation defining % yields that ( %% min x dF(x))(% max &k)=F(k). Substituting this into the above inequality yields F(% max )(% &% min )&F(k)(% &k)0. Substituting the condition for % min to be a best response to % min yields [F(% max )&F(k)]% &(1&F(k))k0, or (F(% max)&F(k))%(1&F(k))k. This says that when bidder m uses a cutoff of % max and bidder &m has learned that bidder m 's valuation is at least k, bidder &m must use a cutoff no larger than % as a best response. This implies that for the continuation game following an announcement by both players that their types are at least k, [k, % ]_[% max, % ] is closed under best responses, so the most asymmetric equilibrium must entail a passive cutoff that is at least as large as % max. If % % max, or if there is no solution for %, then it is necessarily true that (F(% max)&F(k))(1&F(k)) % h
COORDINATION IN AUCTIONS

447

n is nonempty and closed. By the ordering of auction equilibria, it is possible  min to define (% max 1n (i 1n , i 2n ), % 2n (i 1n , i 2n )) as the equilibrium of the auction for object n in which bidder 1 uses the largest cutoff among all possible equilibrium cutoffs he might use, and bidder 2 uses the smallest equilibrium  max cutoff. (% min 1n (i 1n , i 2n ), % 2n (i 1n , i 2n )) is defined analogously. For any number * # [0, 1], define F m&1(*)#max[% | F m (%)=*]. This definition of the inverses of the distribution functions is necessary because we have not ruled out flat regions in the distributions. Define the following mapping from ranking announcements to equilibria of the auction subgame: if F 1&1(i 1n N)&k 1 F 2&1(i 2n N)&k 2 , then the bidders  max play equilibrium (% min 1n (i 1n , i 2n ), % 2n (i 1n , i 2n )) in the auction for object max n; otherwise, they play (% 2n (i 1n , i 2n ), % min 2n (i 1n , i 2n )) in the auction for  object n. Let (%* (i , i ), % * (i , i )) be the function representing this 1n 1n 2n 2n 1n 2n rule. We claim that this mapping satisfies the following monotonicity property: for any bidder m, any i mn and i$mn and any i &mn , if i$mn >i mn then * % * &mn (i mn , i &mn )% &mn (i $mn , i &mn ), where we have abused notation slightly and reversed the order of the rankings when m=2. In other words, for any fixed announcement by your opponent, announcing a higher ranking for object n induces your opponent to use a cutoff in the auction for n that is weakly larger. Proving this relies on the fact that &m's posterior about m 's valuation for object n following a ranking announcement of i$mn first-order stochastically dominates &m's posterior after an announcement of i mn . This is an easily established feature of order statistics and is not derived here. Given this, we consider only the case for m=1, as the case of m=2 involves an identical proof. First, suppose that F 1&1(i$1n N)&k 1 F 2&1(i 1n N)&k 2 . By the stochastic dominance property mentioned above, it is readily shown that * ] when the announcement pair is (i$1n , i 2n ), [k 1 , %* 1n (i 1n , i 2n )]_[% 2n (i 1n , i 2n ), % is closed under best responses in auction for object n, so that it must be the  2n (i 1n , i 2n ). By the equilibrium selection function, case that % max 2n (i $1n , i 2n )%* * this implies the desired result that %* 2n (i 1n , i 2n )% 2n (i$1n , i 2n ). Next, suppose that F 1&1(i$1n N)&k 1 %&mn  %mn (% mn &x) dF &m (x | i &mn )&k m , &mn

448

COLIN M. CAMPBELL

and the difference between this payoff when &m uses the cutoff % &mn and when &m uses the cutoff %$&mn <% mn is [F &m (% mn | i &mn )&F &m (%$&mn | i &mn )] % mn +1 %mn >%$&mn

|

min[%mn, % &mn ] %$&mn

(% mn &x) dF &m (x | i &mn ).

This difference is clearly nondecreasing in % mn . Now consider objects 1 and 2 and suppose without loss of generality that % m1 >% m2 for bidder m. Take as fixed any two rankings i m and i $m satisfying i m >i $m , and any two rankings i &m and i$&m . Let ? mn (% mn ; (i, i $), ( j, j $)) be bidder m 's expected payoff from the auction for object n # [1, 2] when he announces ranking i for object 1 and ranking i$ for object 2, and when bidder &m announces ranking j for object 1 and ranking j $ for object 2. By the argument of the preceeding paragraph, and the monotonicity result, it is necessarily true that ? m1 (% m1 ; (i m , i $m ), (i &m , i $&m ))&? m1 (% m1 ; (i $m , i m ), (i &m , i $&m )) &(? m2 (% m2 ; (i $m , i m ), (i $&m , i &m ))&? m2 (% m2 (i m , i $m ), (i $&m , i &m )))0 and ? m1 (% m1 ; (i m , i $m ), (i $&m , i &m ))&? m1 (% m1 ; (i $m , i m ), (i $&m , i &m )) &(? m2 (% m2 ; (i $m , i m ), (i &m , i $&m ))&? m2 (% m2 ; (i m , i $m ), (i &m , i $&m )))0. Adding these inequalities and rearranging yields ? m1 (% m1 ; (i m , i $m ), (i $&m , i &m ))+? m2 (% m2 ; (i m , i $m ), (i &m , i$&m )) +? m1 (% m1 ; (i m , i $m ), (i $&m , i &m ))+? m2 (% m2 ; (i m , i $m ), (i $&m , i &m )) ? m1 (% m1 ; (i $m , i m ), (i &m , i $&m )+? m2 (% m2 ; (i $m , i m ), (i &m , i $&m )) +? m1 (% m1 ; (i $m , i m ), (i $&m , i &m ))+? m2 (% m2 ; (i $m , i m ), (i $&m , i &m )). Noting that all rankings for bidder &m are equally likely ex-ante, this result says that conditional on bidder &m having rankings i &m and i $&m (in some order) for objects 1 and 2, bidder m earns a higher expected payoff in the auctions for objects 1 and 2 by announcing that he has ranking i m for object 1 and ranking i $m for m object 2 than he earns by announcing the opposite ranking. Since i &m and i $&m are arbitrary, bidder m must also have this preference unconditionally. Since 1 and 2 are arbitrary, it must be that for any two rankings for any two objects, each bidder must prefer to give the higher ranking to the object he values more. Thus, the announcement in which he truthfully reveals his ranking is a best-response.

COORDINATION IN AUCTIONS

449

This proves the existence of an equilibrium in which bidders use ranking talking strategies, for any N. In particular, one equilibrium was found for each N, and the equilibria across different N were qualitatively similar. We now show that the sequence of these equilibria has the desired asymptotic efficiency property. Since bidders' valuations are independent and identically distributed across the objects, we can consider only object 1 without loss of generality. No bidder m who values object 1 at less than k m will submit a bid for 1 in any equilibrium. Consider then states of the world in which % 11 , and % 21 satisfy % 11 &k 1 >max[% 21 &k 2 , 0]. Conditional on one of these states occuring, there exists a number :>0 such that Pr(% 11 &k 1 & max[% 21 &k 2 , 0]<:)<- = . We will further condition on states of the world in which % 11 &k 1 &max[% 21 &k 2 , 0]:, fixing an arbitrary % 11 and % 21 that satisfy this inequality. Let r N 1m be the true ranking of object 1 for bidder m in the N-object environment. Treating the ranking as a random variable whose distribution depends on N, by the law of large numbers we have that for both m and any #>0 and #$>0, Pr( |r N m1N&F m (% m1 )| #)<#$ for N sufficiently large. Valuations on flat spots of the bidders' distributions are never realized (regardless of the size of N), so without loss of generality it is also true that N for any $>0 and $$>0, Pr(|F &1 m (r m1N)&% m1 | $)<$$ for N sufficiently large. Choose $<:2, $$F 2 ((r 21 )N)&k 2 , and that F 11N)&k 1  :&$. Thus, consider only states of the world in which these inequalities hold. By the first inequality, the equilibrium most favorable for bidder 1 is selected by the mapping found in the construction of the cheap-talk equiN &1 N  librium. Also, we claim that [k 1 , F &1 1 (r 11N)&$]_[F 2 (r 21N)+$, % ] is closed under best responses in such states. To show this, it is sufficient to N establish that type F &1 1 (r 11N)&$ of bidder 1 bids as a best response when N &1 N bidder 2 uses a cutoff of F &1 2 (r 21N)+$, and that type F 2 (r 21N)+$ does N not bid as a best response when bidder 1 uses a cutoff of F &1 1 (r 11N)&$. &1 N &1 N Type F 1 (r 11N)&$ of bidder 1 earns a payoff of F 1 (r 11N)&$&k 1  :&2 $ if he submits a bid and bidder 2 does not, and loses no more than k 1 if he and bidder 2 both submit bids. Thus, if the probability * that bidN der 2's type is no larger than F &1 2 (r 21N)+$ satisfies *k 1 (1&*)(:&2 $), &1 N then type F 1 (r 11N)&$ must enter. The condition that $$<(:&2$) N (:&2 $+k 1 ) ensures this. Type F &1 2 (r 21N)+$ of bidder 2 earns a payoff &1 N of F 2 (r 21N)+$&k 2 % +$&k 2 if he bids and bidder 1 does not, and N &1 N loses at least k 2 +(F &1 1 (r 11N)&$)&F 2 (r 21N)+$):&2 $ if both bidders N bid. Thus, if the probability *$ that bidder 1's type is at least F &1 1 (r 11N)&$

450

COLIN M. CAMPBELL

N satisfies *$(% +$&k 2 )(1&*$)(:&2 $), then type F &1 2 (r 21N)+$ must not enter. The condition that $$<(:&2 $)(% &k 2 +:&$) guarantees this. Thus, the equilibrium played must have cutoffs that lie in N &1 N  [k 1 , F &1 1 (r 11N)&$]_[F 2 (r 21N)+$, % ]. But in all states of the world considered here, which occur with probability greater than 1&=, only bidder 1 submits a bid when the bidders play this equilibrium. This is the efficient outcome. The proof is completed by noting that an identical argument will generate an N 2 guaranteeing efficiency with a probability of at least 1&= in the states of the world when it is efficient for only bidder 2 to submit a bid. N can then be chosen to be the maximum of N 1 , and N 2 . K

REFERENCES 1. C. Avery, Strategic jump bidding in English auctions, Rev. Econ. Stud. 65 (1998), 185210. 2. S. Bikchandani, Reputation in repeated second-price auctions, J. Econ. Theory 46 (1988), 97119. 3. S. Bikchandani and J. G. Riley, Equilibria in open common value auctions, J. Econ. Theory 53 (1991), 101130. 4. C. M. Campbell, ``Cheap-Talk Coordination in Dissipative Auctions,'' Ohio State University Departmant of Economics Working Paper No. 96-21, 1996. 5. K. Daniel and D. Hirshleifer, ``A Theory of Costly Sequential Bidding,'' Working paper, University of Chicago, 1995. 6. J. Farrell and R. Gibbons, Cheap talk can matter in bargaining, J. Econ. Theory 48 (1989), 221237. 7. K. R. French and R. E. McCormick, Sealed bids, sunk costs, and the process of competition, J. Bus 57 (1984), 417442. 8. D. Graham and R. C. Marshall, Collusive bidder behavior at single object second price and English auctions, J. Polit. Econ 95 (1987), 12171239. 9. E. Kalai and D. Samet, Persistent equilibria, Int. J. Game Theory 13 (1984), 129144. 10. G. Mailath and P. Zemsky, Collusion in second price auctions with heterogeneous bidders, Games Econ. Behav. 3 (1991), 467486. 11. S. A. Matthews and A. Postlewaite, Pre-play communication in two-person sealed-bid double auctions, J. Econ. Theory 48 (1989), 238263. 12. R. P. McAfee and J. McMillan, Auctions with entry, Econ. Lett. 23 (1987), 343347. 13. R. P. McAfee and J. McMillan, Search mechanisms, J. Econ. Theory 44 (1988), 99123. 14. R. P. McAfee and J. McMillan, Bidding rings, Amer. Econ. Rev. 82 (1992), 579599. 15. M. Mills, Giving contractors a break, Cong. Quart. 51 (1993), 2947. 16. M. Pesendorfer, ``A Study of Collusion in First-Price Auctions Part I: Cartel Agreements,'' Northwestern University, December, 1995. 17. W. F. Samuelson, Competitive bidding with entry costs, Econ. Lett. 17 (1985), 5357.