Private Monitoring in Auctions

Abstract

We study collusion in repeated first-price auctions under the condition of minimal information release by the auctioneer. In each auction a bidder only learns whether or not he won the object. Bidders do not observe other bidders' bids, who participates or who wins in case they are not the winner. We show that for large enough discount factors collusion can nevertheless be supported in the infinitely repeated game. While there is a unique Nash equilibrium in public strategies, in which bidders bid competitively in every period, there are simple Nash equilibria in private strategies that support bid rotation. Equilibria that either improve on bid rotation or satisfy the requirement of Bayesian perfection, but not both, are only slightly more complex. Our main result is the construction of perfect Bayesian equilibria that improve on bid rotation. These equilibria require complicated inferences off the equilibrium path. A deviator may not know who has observed his deviation and consequently may have an incentive to use strategic experimentation to learn about the bidding behavior of his rivals.

Repeated Contests with Asymmetric Information

“…4 In this game, a pure strategy consists of two functions x In addition to strategies, we also have to consider the beliefs of the con- 4 There are several other ways of dealing with the problem; all lead to similar conclusions. (i) Following Blume and Heidhues (2006), one could allow additional efforts such as 0 + , which is identical to 0 except that 0 + wins, with probability one, against 0. This approach will be used for the analysis of the perfectly discriminating contest in section 5 below.…”

“…Lemma 1 Given beliefs µ a < 1 and µ b < 1, there is a unique equilibrium in the second round of the game, where the efforts of the high valuation types are given by equation (4). Expected utility of a high valuation type is…”

Impossibility of Collusion under Imperfect Monitoring with Flexible Production

“…Unfortunately equilibria with private strategies are not understood well enough to show that our impossibility result holds for the entire class of such equilib- 33 Blume and Heidhues (2006) show that, by conditioning on their past actions, players can improve upon PPE in a game with private signals.…”

Dynamic Coordination Via Organizational Routines

“…Specifically, any optimal outcome of a common interest game can be supported by a strategy profile σ that is an essentially perfect Bayesian equilibrium (EPBE) (see Blume and Heidhues [2006] for a the formal definition and detailed discussion of EPBE) 5 , i.e. one can partition the set of all histories into relevant and irrelevant histories so that σ is optimal after all relevant histories regardless of play after irrelevant histories.…”