We show that the folk theorem holds generically for the repeated two-player game with private monitoring if the support of each player’s signal distribution is sufficiently large. Neither cheap talk communication nor public randomization is necessary.

The communication revelation principle (RP) of mechanism design states that any outcome that can be implemented using any communication system can also be implemented by an incentive-compatible direct mechanism. In multistage games, we show that in general the communication RP fails for the solution concept of sequential equilibrium (SE). However, it holds in important classes of games, including single-agent games, games with pure adverse selection, games with pure moral hazard, and a class of social learning games. For general multistage games, we establish that an outcome is implementable in SE if and only if it is implementable in a canonical Nash equilibrium in which players never take codominated actions. We also prove that the communication RP holds for the more permissive solution concept of conditional probability perfect Bayesian equilibrium.

Motivated by recent concerns surrounding the use of third-party pricing algorithms by competing firms, we study repeated Bertrand competition where market demand or the cost of serving the market is observed by an intermediary (or “algorithm”) that selectively discloses demand or cost information to maximize firms’ collusive profit. We show that an upper censorship disclosure policy is optimal, which leads to price rigidity and supra-monopoly prices in some states. Improving the algorithm’s accuracy reduces expected consumer surplus whenever it does so under monopoly pricing. When the state is positively correlated over time, the algorithm discloses more information when recent demand was lower or costs were higher. The analysis extends to a generalized model that accommodates product differentiation and capacity constraints. We relate our findings to recent antitrust cases.

We study how discounting and monitoring jointly determine whether cooperation is possible in repeated games with imperfect (public or private) monitoring. Our main result provides a simple bound on the strength of players' incentives as a function of discounting, monitoring precision, and on‐path payoff variance. We show that the bound is tight in the low‐discounting/low‐monitoring double limit, by establishing a public‐monitoring folk theorem where the discount factor and the monitoring structure can vary simultaneously.

We study anonymous repeated games where players may be “commitment types” who always take the same action. We establish a stark anti-folk theorem: if the distribution of the number of commitment types satisfies a smoothness condition and the game has a “pairwise dominant” action, this action is almost always taken. This implies that cooperation is impossible in the repeated prisoner's dilemma with anonymous random matching. We also bound equilibrium payoffs for general games. Our bound implies that industry profits converge to zero in linear-demand Cournot oligopoly as the number of firms increases.

We study the repeated prisoner’s dilemma with random matching, a canonical model of community enforcement with decentralized information. We assume that (1) with small probability, each player is a “bad type” who never cooperates, (2) players observe and remember their partners’ identities, and (3) each player interacts with others frequently but meets any particular partner infrequently. We show that these assumptions preclude cooperation in the absence of explicit communication but that introducing within-match cheap talk communication restores cooperation. Thus, communication is essential for community enforcement.

It is conventional wisdom that transparency in cartels—monitoring of competitors’ prices, sales, and profits—facilitates collusion. However, in several recent cases cartels have instead worked to preserve the privacy of their participants’ actions and outcomes. Toward explaining this behavior, we show that cartels can sometimes sustain higher profits when actions and outcomes are observed only privately, because better information can hinder collusion by helping firms devise more profitable deviations from the collusive agreement. We provide conditions under which maintaining privacy is optimal for cartels that follow a market-segmentation strategy.

We prove the folk theorem for discounted repeated games with anonymous random matching. We allow non‐uniform matching, include asymmetric payoffs, and place no restrictions on the stage game other than full dimensionality. No record‐keeping or communication devices—including cheap talk communication and public randomization—are necessary.

Cartels and bidding rings are often facilitated by intermediaries, who recommend prices/bids to firms and can impose penalties (such as reverting to competitive behavior in future interactions) if these recommendations are not followed. Motivated by such cases, we study correlated equilibria in first-price procurement auctions with complete information, where bidders who disobey their recommendations are penalized. Cartel-optimal profit is greater when more information about submitted bids is disclosed at auction and when the maximum penalty is larger. When only the winner’s identity is disclosed (or the winner’s identity and bid), cartels do not benefit from mediation. Our main result characterizes the cartel-optimal equilibrium with two symmetric bidders when both bids are disclosed. The optimal equilibrium involves extensive randomization and displays tied bids and high winning bids with positive probability, even when the maximum penalty is very small. The stationary mediation schemes we consider are always more profitable for the cartel than bid rotation.

Do mergers help or hinder collusion? This article studies the stability of the vitamin cartels in the 1990s and presents a repeated-games approach to quantify “coordinated effects” of a merger. We use data and direct evidence from American courts and European agencies to show the collusive incentive of the short-lived vitamin C cartel was likely to be negative when it actually collapsed in 1995, whereas the incentives of the long-lived cartels (vitamins A and E, and beta carotene) were unambiguously positive until the prosecution in 1999. Simulations suggest some mergers could have prolonged the vitamin C cartel, but others could have further destabilized it, because both the direction and magnitude of coordinated effects depend not only on the number of firms but also on their cost asymmetry.