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.

We study an optimal collusion-proof auction in an environment where subsets of bidders may collude not just on their bids but also on their participation. Despite their ability to collude on participation, informational asymmetry facing the potential colluders can be exploited significantly to weaken their collusive power. The second-best auction — i.e., the optimal auction in a collusion-free environment — can be made collusion-proof, if at least one bidder is not collusive, or there are multiple bidding cartels, or the second-best outcome involves a nontrivial probability of the object not being sold. In case the secondbest outcome is not weak collusion-proof implementable, we characterize an optimal collusion-proof auction. This auction involves nontrivial exclusion of collusive bidders — i.e., the object is not sold to any collusive bidder with positive probability.

We identify a self-enforcing collusion protocol (a “bidding ring”) for non-repeated first-price auctions. Unlike previous work on the topic such as that by McAfee and McMillan [1992] and Marshall and Marx [2007], we allow for the existence of multiple cartels in the auction and do not assume that non-colluding agents have perfect knowledge about the number of colluding agents whose bids are suppressed by the bidding ring. We show that it is an equilibrium for agents to choose to join bidding rings when invited and to truthfully declare their valuations to a ring center, and for non-colluding agents to bid straightforwardly. Furthermore, even though our protocol is efficient, we show that the existence of bidding rings benefits ring centers and all agents, both members and non-members of bidding rings, at the auctioneer’s expense. 1

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.

Collusion under imperfect monitoring is explored when firms ’ prices are private information and their quantities are public information; such an information structure is consistent with several recent price-fixing cartels, such as those in lysine and vitamins. For a class of symmetric oligopoly games, it is shown that symmetric equilibrium punishments cannot sustain any collusion. An asymmetric punishment is characterized that does sustain collusion and it has firms whose sales exceed their quotas compensating those firms with sales below their quotas. In practice, cartels could have performed such transfers through sales among the cartel members....if I’m assured that I’m gonna get 67,000 tons [of lysine sales] by the year’s end, we’re gonna sell it at the prices we agreed to and I frankly don’t care what you sell it for. Terrance Wilson of Archer Daniels Midland from the March 10, 1994, meeting of the lysine cartel. And that total for us for the year, calendar year is 68,000; 68,334. 68,334 and our target was 67,000 plus alpha. Almost on target. Mark Whitacre of Archer Daniels Midland from the January 18, 1995, meeting of the lysine cartel. 1 1.