Collusive dominant-strategy truthfulness

We show that collusion and wrong beliefs may cause a dramatic efficiency loss in the Vickrey mechanism for auctioning a single good in limited supply. We thus put forward a new mechanism guaranteeing efficiency in a very adversarial collusion model, where the players can partition themselves into arbitrarily many coalitions, exchange money with each other, and perfectly coordinate their actions. Our mechanism bypasses classic impossibility results (such as those of Green and Laffont, and of Schummer) by providing the players with a richer set of strategies, making it dominant for every coalition C to instruct each of its members to report truthfully not only his own valuation, but also his belonging to C. Our mechanism is coalitionally rational, which implies being individually rational for independent players.     

Starting Small and Commitment

I study a model of a long-term partnership with two-sided incomplete information. The partners jointly determine the stakes of their relationship and individually decide whether to cooperate with or betray each other over time. I characterize the extremal—interim incentive efficient—equilibria. In these equilibria, the partners generally “start small,” and the level of interaction grows over time. The types of players separate quickly. Further, cooperation between “good” types is viable regardless of how pessimistic the players are about each other initially. The quick nature of separation in an extremal equilibrium contrasts with the outcome selected by a strong renegotiation criterion (as studied in Watson (1999, J. Econ. Theory85, 52–90). Journal of Economic Literature Classification Numbers: C72, C73, D74.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

Asymmetric technologies in contests

I consider a contest between two risk-neutral players over a common-value prize, in which one player has a linear cost-of-effort function and the other a strictly convex cost-of-effort function f. I show that if the value of the prize is above (below) a certain threshold level, then the equilibrium aggregate effort in this contest is larger (smaller) than in a contest in which both players are characterized by the strictly convex cost-of-effort function f, and smaller (larger) than in a contest in which both players are characterized by a linear cost-of-effort function. Therefore, in contrast to the general result in the literature, asymmetry in contests can increase competition.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.     

A model of signed network formation with heterogeneous players

This paper proposes a model of formation of signed networks that comprise both positive (friendly/cooperative) and negative (antagonistic) social ties between players who differ intrinsically in strength. Friendships/alliances serve to increase one’s power over their intrinsic strength; in the determination of power it matters not how many allies one has, but who these allies are. The model proposed here is suited to any context where players derive power from friendships/alliances with stronger players and payoffs are extracted from less powerful enemies. The structure of ties across various types of players is studied in Nash equilibria along with a characterization of Nash equilibria under various parametric conditions.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

Cheap talk in games with incomplete information

The paper studies Bayesian games which are extended by adding pre-play communication. Let Γ be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q, that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of Γ, provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in Γ such that for each type t_i of each player i the expected payoff of t_i in q is larger than the expected payoff of t_i in s.     

Are you hiring Johnny Football or Johnny Doe? Uncertain labour quality and the measurement of monopsony in college football

I assess monopsony in the college football labour market as it relates to both typical and elite performers, while also accounting for the uncertain performance quality of prospective players who are in this labour market. Existing studies suggest that the marginal revenue product (MRP) of elite collegiate players later drafted into the National Football League (NFL), significantly exceeds their compensation. These comparisons overstate monopsony rents because schools don’t know ex ante which prospects will become these elite performers. Using financial data spanning 2004–2011 from 114 major schools, I estimate the MRP of players sorted into three ex post quality tiers, and find that while eventual NFL draftees generate revenue that significantly exceeds their compensation, a majority of scholarship players do not. Then, using Rivals.com talent ratings of 6,604 prospective players recruited from 2002–2008, I estimate the probability that prospects will reach each ex post quality tier given their ex ante ratings. MRP estimates are adjusted by these conditional probability estimates to reflect the expected MRP of prospects, which allows more appropriate assessment of monopsony rents. I find that schools possess substantially less monopsony power than unadjusted MRP estimates suggest.     

Partial exposure in large games

In this work we introduce the notion of partial exposure, in which the players of a simultaneous-move Bayesian game are exposed to the realized types and chosen actions of a subset of the other players. We show that in any large simultaneous-move game, each player has very little regret even after being partially exposed to other players. If players are given the opportunity to be exposed to others at the expense of a small decrease in utility, players will decline this opportunity, and the original Nash equilibria of the game will survive.     

Increasing Dominance with No Efficiency Effect

I uncover a new force towards increasing dominance (the property whereby, in dynamic games, the leader tends to increase his or her lead in expected terms). The new effect results from the strategic choice of covariance in races. I assume that players must choose not the amount of resources to spend but how to allocate those resources. I show that, in equilibrium, the laggard chooses a less promising path, in effect trading off lower expected value for lower correlation with respect to the leader. This results in increasing dominance and holds true even if no joint-payoff (or efficiency) effect is present. Journal of Economic Literature Classification Numbers: C7, L1.     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

On Sustaining Cooperation without Public Observations

This paper examines a dynamic game in which each player only observes a private and imperfect signal on the actions played. Our main result is that in a repeated prisoner's dilemma where defections are irreversible (at least for a long enough period of time), patient enough players may achieve almost efficient outcomes. Dealing with models of imperfect private monitoring is difficult because (i) continuation games are games of incomplete information, hence they do not have the same structure as the original game. In particular, continuation equilibria are correlated equilibria. (ii) Players are typically uncertain about their opponents' past observations and actions, and they use their entire own private history to learn about these actions. As a result equilibrium strategies are in general nontrivial and increasingly complex functions of past observations. We bypass these difficulties by looking at correlated equilibria of the original game and find correlated equilibria in which the decision problem faced by each player remains the same over time. Journal of Economic Literature Classification Numbers: C72.     

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

Uniqueness of Stationary Equilibrium Payoffs in the Baron-Ferejohn Model

We consider a multilateral sequential bargaining model in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, we characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, we establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, we show that the payoff to a player is nondecreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, we show that the payoff to a player is nondecreasing in his discount factor. Journal of Economic Literature Classification numbers: C72, C78, D70.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

Imitation and Experimentation in Changing Contests

This paper analyzes the equilibrium play of individuals that are randomly matched to play a contest where the dominant action changes over time. Under myopic decision making, players adopt imitation strategies similar to those observed in evolutionary models with sampling from past play in the population. If the players are patient, equilibrium strategies display elements of experimentation in addition to imitation. If the changes in the environment are infrequent enough, these strategies succeed in coordinating almost all of the players on the dominant action almost all of the time. The myopic rules, on the other hand, result in miscoordination for a positive fraction of time. Journal of Economic Literature Classification Numbers: C73, D83, D84.