Nash Equilibria of Repeated Games with Observable Payoff Vectors

We study a model of repeated games with imperfect monitoring where the payoff vector is observable. In this situation, any profitable deviation is detectable by all the players but the identity of the deviator may be unknown. We design collective punishments directed against the set of potential deviators. A particular class of signals is studied for which a characterization of the set of equilibrium payoffs is obtained. Journal of Economic Literature Classification Numbers: C73.     

Merging, Reputation, and Repeated Games with Incomplete Information

We relate and unify several results that appeared in the following domains: merging of probabilities, perturbed games and reputation phenomena, and repeated games with incomplete information. Journal of Economic Literature Classification Numbers: C72, D83.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

Cost Structures and Nash Play in Repeated Cournot Games

This paper reports an experiment designed to assess the effects of a rotation in the marginal cost curve on convergence in a repeated Cournot triopoly. Increasing the cost curve's slope both reduces the serially-undominated set to the Nash prediction, and increases the peakedness of earnings. We observe higher rates of Nash equilibrium play in the design with the steeper marginal cost schedule, but only when participants are also rematched after each decision. Examination of response patterns suggests that the treatment with a steeper marginal cost curve and with a re-matching of participants across periods induces the selection of Nash Consistent responses.     

On the Number of Pure Strategy Nash Equilibria in Random Games

How many pure Nash equilibria can we expect to have in a finite game chosen at random? Solutions to the above problem have been proposed in some special cases. In this paper we assume independence among the profiles, but we allow either positive or negative dependence among the players' payoffs in a same profile. We provide asymptotic results for the distribution of the number of Nash equilibria when either the number of players or the number of strategies increases. We will show that different dependence assumptions lead to different asymptotic results. Journal of Economic Literature Classification Number C72.     

Nash Equilibria in Pure Strategies

We consider an n ‐person non‐zero‐sum non‐cooperative game in normal form, where the strategy sets are some closed intervals of the real line. It is shown that if the pay‐off functions are continuous on the whole space and if for each pay‐off function the smallest local maximum in the strategy variable is a global maximum, then the game possesses a pure strategy Nash equilibrium.     

不完全信息合作博弈的值：一个很有说服力的例子

一、导言合作博弈理论提出了各种模型来解释合作的收益如何在参与者之间分配。Shapley值在为可转移效用(TU)博弈定义的公平解组中起十分重要的作用。它给出了大量的公理的和非合作的证明。后来,为了把Shapley值扩展到非可转移效用博弈(NTU)中各种技术被开发出来了。比如Shapley     

Implementation in Economic Environments with Incomplete Information: The Use of Multi-Stage Games

This paper shows that, in economic environments with incomplete information, incentive compatibility and a preference reversal condition are sufficient for implementation in sequential equilibrium.Journal of Economic LiteratureClassification Numbers: C72, D71, D82.     

Renegotiation in Repeated Games with Side-Payments

We consider repeated games with side-payments: players have an endowment of wealth in each period in which transfers can be made. We show that if endowments are large enough and the common discount factor high enough, then a strongly renegotiation–proof equilibrium (SRP) in the sense of Farrell and Maskin exists. As the discount factor goes to 1, the set of SRP payoffs converges to the set of efficient, individually rational payoffs. These results provide a justification for the efficiency principle when agreements are not enforceable. Journal of Economic Literature Classification Numbers: C73, D23, L14.     

Repeated Games with Bounded Entropy

We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer's strategies while the other player is unrestricted. We will show that if the bound η(n), a function of the number of repetitions n, satisfies the condition η(n)/n → γ (n → ∞), then the maxmin value Wⁿ(η(n)) converges to (cav U)(γ), the concavification of the maxmin value of the stage game in which the maximizer's actions are restricted to those with entropy at most γ. A similar result is obtained for the infinitely repeated games. Journal of Economic Literature Classification Numbers: C73, C72.     

Public Information, Private Information, and the Multiplicity of Equilibria in Coordination Games

I study coordination games with incomplete public and private information and relate equilibrium convergence to convergence of higher-order beliefs. As the players' signals become more and more precise, the equilibrium manifold converges to the correspondence of common knowledge equilibria, whenever the variance of the public signal converges to 0 at a rate faster than one half the rate of convergence of the variance of private signals. The same condition also determines the convergence of common p-belief to common knowledge, which leads to a simple intuition for its origin and an immediate generalization of the former results about equilibrium convergence. Journal of Economic Literature Classification Numbers: C72, D82.     

Incomplete Simultaneous Discrete Response Model with Multiple Equilibria

A bivariate simultaneous discrete response model which is a stochastic representation of equilibria in a two-person discrete game is studied. The presence of multiple equilibria in the underlying discrete game maps into a region for the exogenous variables where the model predicts a nonunique outcome. This is an example of an incomplete econometric structure. Economists using this model have made simplifying assumptions to avoid multiplicity. I make a distinction between incoherent models and incomplete models, and then analyse the model in the presence of multiple equilibria, showing that the model contains enough information to identify the parameters of interest and to obtain a well defined semiparametric estimator. I also show that the latter is consistent and     normal. Moreover, by exploiting the presence of multiplicity, one is able to obtain a more efficient estimator than the existing methods.     

Markov Equilibria of Stochastic Games with Complementarities

The existence of Markov equilibria for stochastic games with a continuum of states is a complex issue for which no general result holds as yet. In this article, the problem is solved for a class of stochastic games that satisfy assumptions of complementarity and monotonicity. The proof of existence relies on results from lattice programming. In the Markov equilibria singled out by the Theorem of Existence, the policies and continuation values are increasing and Lipschitz continuous functions of the state variable.Journal of Economic LiteratureClassification Numbers: C62, C73.     

A Blotto game with Incomplete Information

We consider a Blotto game with Incomplete Information. A pure-strategy symmetric monotonic Bayesian equilibrium is found and its properties are discussed.     

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.     

Demonstrating the Possibility of Pareto Inferior Nash Equilibria

This study reviews the status of student outcomes assessment in departments of economics at colleges and universities across the country.     

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.     

A Folk Theorem for Repeated Sequential Games

We study repeated sequential games where players may not move simultaneously in stage games. We introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. The Folk theorem asserts that any feasible payoff vector where every player receives more than his effective minimax value in a sequential stage game can be supported by a subgame perfect equilibrium in the corresponding repeated sequential game when players are sufficiently patient. The results of this paper generalize those of Wen (1994), and of Fudenberg and Maskin (1986). The model of repeated sequential games and the concept of effective minimax provide an alternative view to the Anti–Folk theorem of Lagunoff and Matsui (1997) for asynchronously repeated pure coordination games.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Introduction to Repeated Games with Private Monitoring

We present a brief overview of recent developments in discounted repeated games with (imperfect) private monitoring. The literature explores the possibility of cooperation in a long-term relationship, where each agent receives imperfect private information about the opponents' actions. Although this class of games admits a wide range of applications such as collusion under secret price-cutting, exchange of goods with uncertain quality, and observation errors, it has fairly complex mathematical structure due to the lack of common information shared by players. This is in sharp contrast to the well-explored case of repeated games under public information (with the celebrated Folk Theorems), and until recently little had been known about the private monitoring case. However, rapid developments in the past few years have revealed the possibility of cooperation under private monitoring for some class of games. Journal of Economic Literature Classification Numbers: C72, C73, D43, D82, L13, L41.