|
Article Excerpt
I apply the notion of a self-confirming equilibrium (SCE) to study how feedback in first price auctions influences bidders' perceptions about their strategic environment, and consequently their bidding behavior. In a private values setting, revealing the two highest bids at the end of each auction is sufficient for bidders to have correct beliefs (justifying the assumption of Nash equilibrium). In contrast, in every symmetric SCE of a symmetric, affiliated, private values model, bidding strategies and revenue are (weakly) higher if only the highest bid is revealed. I also consider interdependent valuations and discuss implications for the empirical auction literature.
**********
"The analysis of past bids can provide a wealth of valuable information to the contractor in clarifying the relationships among many different competitive factors."--William R. Park, Construction Bidding for Profit, 1979
1. Introduction
* In practice, the information that bidders receive about auction outcomes is often a design choice of the auctioneer. In the private sector, it is common for the auctioneer to reveal only the winning bid (and many times not even this) at the end of each auction. (1) In contrast, in the public sector, the law usually mandates all bids to be revealed. In both cases, it may be of interest to know whether a different information policy would result in different outcomes.
I study how different information feedback policies affect competitive bidding in first price auctions. In particular, I focus on a specific role of feedback: the amount of information regarding outcomes of past, similar auctions may presumably affect bidders' beliefs about their strategic environment, including their beliefs about how competitors are bidding in these auctions as well as beliefs about their own valuation of the object or procurement cost (in a common value setting). This role of feedback is ignored in the literature when attention is restricted to the (Bayesian) Nash equilibrium (NE) of the first price auction, which requires bidders to have correct beliefs about the equilibrium strategies of other players and about the distribution of valuations, so that, for example, bidders correctly anticipate their probability of winning with any possible bid. In contrast, in this article players' beliefs about their probability of winning the auction are influenced by the amount of information that they observe from previous auctions.
The proposed setup is inspired by the empirical literature on auctions, where the stage game is a standard, single-object, first price sealed-bid auction and data are observed from play of several identical and independent auctions. (2) In this context, the empirical literature asks whether the primitives of the stage game (such as the distribution of valuations) can be identified from the data available to the researcher under the assumption that bidders play a Nash equilibrium, and then proceeds to estimate the model (sec Athey and Haile, 2006, forthcoming for a review of this literature). But just as identification depends on the data observed by a researcher, whatever players can learn about their strategic environment is also likely to depend on the data that are available to them. As suggested by the quote at the beginning of the article, bidders do seem to behave as empirical researchers and use information about past bids to learn about their strategic environment. (3)
I model the effect of different feedback policies on bidders' beliefs about opponents' strategies and the primitives of the economy by assuming that bidders play a self-confirming equilibrium (SCE) (Battigalli, 1987; Fudenberg and Levine, 1993a; Dekel et al., 2004). An SCE is often interpreted as the outcome of a learning process, in which players revise their beliefs using observations of previous play. (4) An SCE is similar to an NE in that it is a steady-state solution concept applied to a well-defined stage game where players choose strategies that are optimal given their beliefs about the consequences of choosing any feasible strategy. In particular, repeated game considerations (e.g., how feedback affects collusion) are ignored in this article) The difference between an NE and an SCE is that in an SCE beliefs are not required to be correct, but instead restricted to be "consistent" with feedback that players obtain about equilibrium outcomes. To illustrate the meaning of "consistent," suppose that a bidder always observes the winning bid in the context of playing a series of identical first price auctions. The definition of SCE then requires her to have correct beliefs about the equilibrium distribution of the winning bid but not necessarily correct beliefs about the equilibrium distribution, say, of the second highest bid. By allowing for incorrect beliefs, the set of SCE contains the set of NE and coincides with it only when players have correct beliefs in every SCE.
The main objective is to answer two related questions. First, how much information about opponents' bids needs to be revealed so that bidders have correct beliefs in an SCE (therefore implying that each SCE is also an NE)? This question is important to the extent that several theoretical and empirical results in the literature are obtained under the assumption that bidders play an NE. Second, what is the effect of different information policies on (self-confirming) equilibrium bidding, with a particular emphasis on the revenue-maximizing information policy? The answers to these questions depend on whether types are independent or correlated (affiliated), valuations are private or interdependent, and bidders are symmetric or asymmetric. (6)
In the private values model, beliefs are always correct in an SCE when the auctioneer reveals at least the two highest bids (implying that the sets of SCE and NE coincide in this case). The reason is that the only uncertainty relevant to bidders concerns their probability of winning as a function of their offers. This probability is determined by the distribution of the maximum of opponents' bids, a statistic that is always observed when the highest two bids are revealed.
In contrast, when only the winning bid is revealed, a bidder will not observe the maximum of opponents' bids when she wins the auction. Hence, in equilibrium, she is only required to have correct beliefs about the probability of winning with offers higher than her equilibrium bid (because she observes the maximum of opponents' bids only in that range of bids). As a result, in an SCE, a bidder may overbid relative to the best response of a bidder who has correct beliefs about the probability of winning for each possible bid. (7)
The previous overbidding result establishes a testable relationship between the (perceived) best response of a player in an SCE and the best response of a player who has correct beliefs, as in an NE. However, it does not necessarily follow that self-confirming equilibrium bidding is more aggressive. The reason is that it is not true, in general, that our previous comparison of best responses suffices to obtain an unambiguous comparison of the set of equilibria. I do show, however, that in the symmetric affiliated private values model of Milgrom and Weber (1982), any symmetric SCE is (weakly) more aggressive than the unique NE when only the winning bid is revealed. Furthermore, I establish existence of non-Nash SCE that are strictly more aggressive and provide strictly higher revenue than the unique NE.
To focus on the new issues that arise when valuations are interdependent, consider the case where the auctioneer reveals at least the two highest bids. In this case, bidders may only have incorrect beliefs about the expected surplus from winning the object. Under the often realistic assumption that bidders only receive feedback about the value of the object when they win it, bidders learn about the rimes they were too optimistic about value, but need not learn they are sometimes too pessimistic. Using a line of reasoning analogous to the one applied for the private values case, it follows that a bidder may now underbid in an SCE relative to the optimal strategy of a bidder with correct beliefs. I then apply the findings on comparative statics in symmetric first price auctions to conclude that when at least the two highest bids are revealed in a symmetric, affiliated model with interdependent valuations, revenues are lower in a symmetric SCE as compared to an NE. When the two highest bids are not revealed, the effect identified in the case of private values affects outcomes in the opposite direction, and equilibria ranking is ambiguous even in a symmetric model.
The results for the symmetric model suggest that in a private value setting, an auctioneer might want to limit the amount of information revealed about submitted bids to either increase revenues or lower procurement costs. (8) When valuations are interdependent, an auctioneer might benefit from providing information about the ex post value of an object. The reason is that when bidders observe the realized value of the object (irrespective of whether they win or lose), in an SCE they must have correct beliefs about the expected value of the object. Thus, the sets of SCE and NE coincide (assuming at least the two highest bids are revealed, as explained above), and there is consequently no underbidding in equilibrium. This information policy could be implemented by requiring the winner to provide information about the revenue obtained from the use of the object. For example, in U.S. offshore oil and gas lease sales, the federal government publishes monthly data on the production of oil and gas (Porter, 1995).
Another implication of assuming that bidders play an SCE is that the Dutch (i.e., descending) and first price auctions are not equivalent, unless only the highest bid is revealed in the latter. Hence, information feedback constitutes an alternative, nonpsychological explanation to the nonequivalence of the Dutch and first price auctions. (9)
Several articles in the experimental literature emphasize that behavior depends on the amount of feedback that players receive as they learn to play the game. (10) In the context of first price auctions, Isaac and Walker (1985), Neugebauer and Selten (2006), and Ockenfels and Selten (2006) show that in experimental settings with independent private values, overbidding occurs when subjects only receive feedback about the winning bid, compared to the case where they receive more feedback about their opponents' bids. The results in this article shed light not only on the forces that may be behind their findings but also suggest new directions for further experimental work. (11)
In a recent article, Jehiel (2007) also looks at feedback in auctions as a design issue. In contrast, he allows for more general auction formats and focuses on a different kind of feedback. In his framework, bidders obtain coarse feedback about past bids, so that, for example, only the aggregate distribution of bids across the various auction formats that the auctioneer has implemented in the past is revealed. (12)
In Section 2, I describe the model and define both NE and SCE. I then present conditions for beliefs to be correct in an SCE in Section 3, and characterize non-Nash SCE behavior in Section 4. In Section 5, I discuss implications for the empirical auction literature and show that SCE behavior imposes testable restrictions on observed bids. I briefly conclude in Section 6, and relegate all proofs to the Appendix.
2. Setup and definition of equilibrium
* Consider the following stage game, which describes a standard first price auction. There are N risk-neutral bidders who simultaneously submit bids for a single object. (13) The bidder who submits the highest bid gets the object and pays her bid; more generally, the winner is chosen randomly from the set of bidders who submit the highest bid. Before submitting their bid, each bidder i receives a signal or type [s.sub.i] [member of] [[[s.bar].sub.i], [[bar.s].sub.i]] [subset] [R.sub.+] . Player i's utility from winning the object is [v.sub.i] - [b.sub.i], where...
|
