On Failing to Cooperate When Monitoring Is Private

I consider a repeated prisoners' dilemma where in each period, each player receives an imperfect private signal about his opponent's current action. I show that when players are patient enough, any equilibrium where players use trigger strategies (i.e., do not revert to cooperation once they have started defecting) yields players a value arbitrarily close to the mutual minimax. I also examine the robustness of the result to perturbations of the game. Journal of Economic Literature Classification Numbers: C72.     

Minimax after money-max: why major league baseball players do not follow optimal strategies

Using real-life sports data of Major League Baseball, this paper investigates whether professional players follow the minimax theorem in their strategies. Our empirical results using the 2010 regular-season data show that baseball players do not optimize their strategies: there is a significant difference in their payoffs across strategies, and the sequence of their strategy choices is predictable from their previous actions. Further analysis using individual salaries and key contract variables indicates that a higher salary has a positive impact on following minimax strategies in the regular season. By contrast, a longer contract decreases a player’s incentive to pursue optimal strategies in the postseason. These results have important implications for compensation practices in various fields.     

Moral Hazard and Private Monitoring

We clarify the role of mixed strategies and public randomization (sunspots) in sustaining near-efficient outcomes in repeated games with private monitoring. We study a finitely repeated game, where the stage game has multiple equilibria and show that mixed strategies can support partial cooperation, but cannot approximate full cooperation even if monitoring is “almost perfect.” Efficiency requires extensive form correlation, where strategies can condition upon a sunspot at the end of each period. For any finite number of repetitions, we approximate the best equilibrium payoff under perfect monitoring, assuming that monitoring is sufficiently accurate and sunspots are available. Journal of Economic Literature Classification Numbers: C73, D82.     

Extending Invariant Solutions

In this paper we show that an invariant solution for normal form games can be extended uniquely to an invariant solution for strategic form games. This result has the following consequence for the reduction of a (normal form) game. Suppose that the pure strategies are removed that are payoff equivalent with some (possibly mixed) strategy. Then, if one is concerned with an invariant solution, a further reduction by identifying for each player arbitrary payoff-equivalent strategies is not necessary.Journal of Economic LiteratureClassification Number: C72.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

On the Co-existence of Conventions

Recent stochastic evolutionary models have shown that the most likely outcome when the probability of a mutation is sufficiently small is coordination on the risk-dominant strategy rather than on the payoff-dominant one. This paper looks at the consequences of player movement between locations when there are constraints which limit the number of agents who can reside at each location. If the constraints are tight then the risk-dominance result continues to hold. However, we show that when sufficient movement is possible, the most likely outcome involves a mixed state in which agents at different locations coordinate on different strategies. In the asymmetric case, it is the location with the tighter constraint that coordinates on the payoff-dominant strategy. Journal of Economic Literature Classification Numbers: C72, C73, D83.     

Existence of equilibria in countable games: An algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Waldʼs pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group, the payoffs are functions of the group operation, and mixed strategies are not requested to be σ-additive. As a byproduct we show that if finitely additive mixed strategies are allowed, then Waldʼs game admits an equilibrium. Finally we extend the main results to uncountable games.     

Minimax regret with imperfect ex-post knowledge of the state

We consider decision problems under complete ignorance and extend the minimax regret principle to situations where, after taking an action, the decision maker does not necessarily learn the state of the world. For example, if the decision maker only learns what the outcome is, then all she knows is that the actual state is one of the (possibly several) states that yield the observed outcome under the chosen action. We refer to this situation as imperfect ex-post information. We show that, given a choice between more information and less information, the decision maker prefers the latter. We also extend the framework to encompass the possibility of less than the extreme degree of pessimism that characterizes the minimax regret criterion.     

Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze repeated prisoners' dilemma games with imperfect private monitoring and construct mixed trigger strategy equilibria. Such strategies have a simple representation, where a player's action only depends upon her belief that her opponent(s) are continuing to cooperate. When monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of such games. The efficiency result extends when there are more than two players. It requires that monitoring be sufficiently accurate but does not require very low discounting when a public randomization device is available. Journal of Economic Literature Classification Numbers: C73, D82.     

Purification in the infinitely-repeated prisoners' dilemma

This paper investigates the Harsanyi [Harsanyi, J.C., 1973. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory 2 (1), 1–23]-purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of one-period memory mixed strategy equilibria used by Ely and Välimäki [Ely, J.C., Välimäki, J., 2002. A robust folk theorem for the prisoner's dilemma. Journal of Economic Theory 102 (1), 84–105] in their study of the repeated prisoners' dilemma with private monitoring. We find that any such strategy profile is not the limit of one-period memory equilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable.     

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.     

Subjective Uncertainty over Behavior Strategies: A Correction

In order to model the subjective uncertainty of a player over the behavior strategies of an opponent, one must consider the player's beliefs about the opponent's play at information sets that the player thinks have probability zero. This corregendum uses “trembles” to provide a definition of the convex hull of a set of behavior strategies. This corrects a definition we gave in [E. Dekel, D. Fudenberg, and D. K. Levine, 1999, J. Econ. Theory89, 165-185], which led to two of the solution concepts we defined there not having the properties we intended. Journal of Economic Literature Classification Numbers: C72, D82, C610.     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Mixed Equilibrium in a Downsian Model with a Favored Candidate

This paper examines competition in the standard one-dimensional Downsian model of two-candidate elections, but where one candidate (A) enjoys an advantage over the other candidate (D). Voters' preferences are Euclidean, but any voter will vote for candidate A over candidate D unless D is closer to her ideal point by some fixed distance δ. The location of the median voter's ideal point is uncertain, and its distribution is commonly known by both candidates. The candidates simultaneously choose locations to maximize the probability of victory. Pure strategy equilibria often fail to exist in this model, except under special conditions about δ and the distribution of the median ideal point. We solve for the essentially unique symmetric mixed equilibrium with no-gaps, show that candidate A adopts more moderate policies than candidate D, and obtain some comparative statics results about the probability of victory and the expected distance between the two candidates' policies. We find that both players' equilibrium strategies converge to the expected median voter as A's advantage shrinks to 0. Journal of Economic Literature Classification Numbers: C72, D72.     

The minimum approval mechanism implements the efficient public good allocation theoretically and experimentally

We propose the minimum approval mechanism (MAM) for a standard linear public good environment with two players. Players simultaneously and privately choose their contributions to the public good in the first stage. In the second stage, they simultaneously decide whether to approve the otherʼs choice. Both contribute what they choose in the first stage if both players approve; otherwise, both contribute the minimum of the two choices in the first stage. The MAM implements the Pareto-efficient allocation in multiple solution concepts including backward elimination of weakly dominated strategies (BEWDS), limit logit agent quantal response equilibrium, subgame perfect minimax regret equilibrium, level-k thinking, and diagonalization heuristics. Moreover, the MAM is unique under plausible conditions. Overall, contributions in the MAM experiment averaged 94.9%. Quantifying subjectsʼ responses to the questionnaire reveal heterogeneous reasoning processes and highlight the importance of developing mechanisms that implement the desired social choice outcome in multiple solution concepts.     

Uniform payoff security and Nash equilibrium in compact games

We introduce a condition, uniform payoff security, for games with compact Hausdorff strategy spaces and payoffs bounded and measurable in players’ strategies. We show that if any such compact game G is uniformly payoff secure, then its mixed extension is payoff secure. We also establish that if a uniformly payoff secure compact game G has a mixed extension with reciprocally upper semicontinuous payoffs, then G has a Nash equilibrium in mixed strategies. We provide several economic examples of compact games satisfying uniform payoff security.     

An Evolutionary Interpretation of Mixed-Strategy Equilibria

A convincing interpretation of mixed-strategy equilibria describes them as steady states in a large population in which players use pure strategies but the population as a whole mimics a mixed strategy. I study the conditions under which an evolutionary, stochastic learning process converges to the appropriate distribution over pure strategies in the population. I find that not all mixed equilibria can be justified as the result of an evolutionary process even if the equilibrium is unique. For symmetric 2 × 2 and 3 × 3 games I give necessary and sufficient conditions for convergence, which are related to the concept of an ESS, and forn × ngames I give a sufficient condition.Journal of Economic LiteratureClassification Numbers: C73, D83.     

Axioms for minimax regret choice correspondences

This paper unifies and extends the recent axiomatic literature on minimax regret. It compares several models of minimax regret, shows how to characterize the according choice correspondences in a unified setting, extends one of them to choice from convex (through randomization) sets, and connects them by defining a behavioral notion of perceived ambiguity. Substantively, a main idea is to behaviorally identify ambiguity with failures of independence of irrelevant alternatives. Regarding proof technique, the core contribution is to uncover a dualism between choice correspondences and preferences in an environment where this dualism is not obvious. This insight can be used to generate results by importing findings from the existing literature on preference orderings.     

Evolution and Mixed Strategies

Selten (1980, J. Theoret. Biol., 84, 93–101) showed that no mixed equilibria are evolutionarily stable when players can condition their strategies on their roles in a game. Alternatively, Harsanyi's (1973, Int. J. Game Theory, 2, 1–23) purification argument implies that all mixed equilibria are approximations of strict, and hence evolutionarily stable, equilibria of games with slightly perturbed payoffs. This paper reconciles these results: Approximations of mixed equilibria have high invasion barriers, and hence are likely to persist, when payoff perturbations are relatively important and role identification is relatively noisy. When payoff perturbations are unimportant and role identification is precise, approximations of mixed equilibria will have small invasion barriers and are unlikely to persist. Journal of Economic Literature Classification Numbers: C70, C78.     

Exploitable actions of believers in the “law of small numbers” in repeated constant-sum games

In repeated fixed-pair constant-sum games with unique equilibria in mixed strategies, such as matching pennies, the subgame perfect equilibrium is repeating the stage-game mixed-strategy equilibrium action. In such games rational players avoid strategies that are exploitable, in that current actions either deviate systematically from the equilibrium action probabilities or fail to be serially independent of past actions. I revisit classic experiments and find that subjects’ actions are sometimes exploitable because they are serially dependent. Subjects have difficulty in producing serially independent actions and in recognizing serially dependent sequences due to a bias called local representativeness.