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. (JEL C72, C73, D83)

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.

We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”

We consider general Bayesian persuasion problems where the receiver’s utility is single-peaked in a one-dimensional action. We show that a signal that pools at most two states in each realization is always optimal and that such “pairwise” signals are the only solutions under a nonsingularity condition on utilities. Our core results provide conditions under which the induced receiver action is single-dipped or single-peaked on each set of nested signal realizations. We also provide conditions for the optimality of either full disclosure or negative assortative disclosure, where all signal realizations are nested. Methodologically, our results rely on novel duality and complementary slackness theorems. Our analysis extends to a general problem of assigning one-dimensional inputs to productive units, which we call “optimal productive transport.” This problem covers additional applications including matching with peer effects (assigning workers to firms, students to schools, or residents to neighborhoods), robust option pricing (assigning future asset prices to price distributions), and partisan gerrymandering (assigning voters to districts).