Belief‐Free Equilibria in Repeated Games

We introduce a class of strategies that generalizes examples constructed in two‐player games under imperfect private monitoring. A sequential equilibrium is belief‐free if, after every private history, each player's continuation strategy is optimal independently of his belief about his opponents' private histories. We provide a simple and sharp characterization of equilibrium payoffs using those strategies. While such strategies support a large set of payoffs, they are not rich enough to generate a folk theorem in most games besides the prisoner's dilemma, even when noise vanishes.     

The Folk Theorem for Games with Private Almost‐Perfect Monitoring

We prove the folk theorem for discounted repeated games under private, almost‐perfect monitoring. Our result covers all finite, n‐player games that satisfy the usual full‐dimensionality condition. Mixed strategies are allowed in determining the individually rational payoffs. We assume no cheap‐talk communication between players and no public randomization device.     

A Partial Folk Theorem for Games with Unknown Payoff Distributions

Repeated games with unknown payoff distributions are analogous to a single decision maker's “multi‐armed bandit” problem. Each state of the world corresponds to a different payoff matrix of a stage game. When monitoring is perfect, information about the state is public, and players are sufficiently patient, the following result holds: For any function that maps each state to a payoff vector that is feasible and individually rational in that state, there is a sequential equilibrium in which players experiment to learn the realized state and achieve a payoff close to the one specified for that state.     

基于小世界网络的重复囚徒困境博弈

讨论了SWN(Smallworldnetwork)及其数量特征,就经济过程中的长期交易关系建立了重复囚徒困境博弈的SWN模型。研究了由网络结构参数p的变化而引起的博弈局中人的策略变化倾向以及对网络性能的影响。     

Repeated Games with Private Monitoring: Two Players

We investigate two‐player infinitely repeated games where the discount factor is less than but close to unity. Monitoring is private and players cannot communicate. We require no condition concerning the accuracy of players' monitoring technology. We show the folk theorem for the prisoners' dilemma with conditional independence. We also investigate more general games where players' private signals are correlated only through an unobservable macro shock. We show that efficiency is sustainable for generic private signal structures when the size of the set of private signals is sufficiently large. Finally, we show that cartel collusion is sustainable in price‐setting duopoly.     

Repeated Games Where the Payoffs and Monitoring Structure Are Unknown

This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type‐contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.     

Games with Imperfectly Observable Actions in Continuous Time

This paper investigates a new class of two‐player games in continuous time, in which the players' observations of each other's actions are distorted by Brownian motions. These games are analogous to repeated games with imperfect monitoring in which the players take actions frequently. Using a differential equation, we find the set ℰ(r) of payoff pairs achievable by all public perfect equilibria of the continuous‐time game, where r is the discount rate. The same differential equation allows us to find public perfect equilibria that achieve any value pair on the boundary of the set ℰ(r). These public perfect equilibria are based on a pair of continuation values as a state variable, which moves along the boundary of ℰ(r) during the course of the game. In order to give players incentives to take actions that are not static best responses, the pair of continuation values is stochastically driven by the players' observations of each other's actions along the boundary of the set ℰ(r).     

Weakly Belief‐Free Equilibria in Repeated Games With Private Monitoring

Repeated games with imperfect private monitoring have a wide range of applications, but a complete characterization of all equilibria in this class of games has yet to be obtained. The existing literature has identified a relatively tractable subset of equilibria. The present paper introduces the notion of weakly belief‐free equilibria for repeated games with imperfect private monitoring. This is a tractable class which subsumes, as a special case, a major part of the existing literature (the belief‐free equilibria). It is shown that this class can outperform the equilibria identified by the previous work.     

The Role of Information in Repeated Games With Frequent Actions

We show that in repeated interactions the avenues for effective provision of incentives depend crucially on the type of information players observe. We establish this conclusion for general repeated two‐player games in which information arrives via a continuous‐time stationary process that has a continuous multidimensional Brownian component and a Poisson component, and in which the players act frequently. The Poisson jumps can be used to effectively provide incentives both with transfers and value burning, while continuous Brownian information can be used to provide incentives only with transfers.     

Cooperation in Repeated Games When the Number of Stages is not Commonly Known

It is shown that an exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners' dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be approximated by a subgame perfect equilibrium of a finitely repeated version of that game. The sense in which the departure from common knowledge is small is as follows: (I) With probability one, the players know T with precision ±K. (ii) With probability 1 −ε, the players know T precisely; moreover, this knowledge is mutual of order εT. (iii) The deviation of T from its finite expectation is exponentially small.     

Bargaining,Reputation, and Equilibrium Selection in Repeated Games with Contracts

Consider a two‐person intertemporal bargaining problem in which players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds. This can alternatively be viewed as an infinitely repeated game wherein players can offer one another enforceable contracts that govern play for the rest of the game. Theory is silent with regard to how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a specific asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent's play. In equilibrium, rational players initially choose a behavioral type to imitate and a war of attrition ensues. How much should a player try to get and how should she behave while waiting for the resolution of bargaining? In both respects she should build her strategy around the advice given by the “Nash bargaining with threats” (NBWT) theory developed for two‐stage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payoff by imitating a behavioral type with the following simple strategy: in every period, ask for (and accept nothing less than) that player's NBWT share and, while waiting for the other side to concede, take the action Nash recommends as a threat in his two‐stage game. The results suggest that there are forces at work in some dynamic games that favor certain payoffs over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two‐sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payoffs as you go.     

Deterrence and Risk Preferences in Sequential Attacker–Defender Games with Continuous Efforts

Most attacker–defender games consider players as risk neutral, whereas in reality attackers and defenders may be risk seeking or risk averse. This article studies the impact of players' risk preferences on their equilibrium behavior and its effect on the notion of deterrence. In particular, we study the effects of risk preferences in a single‐period, sequential game where a defender has a continuous range of investment levels that could be strategically chosen to potentially deter an attack. This article presents analytic results related to the effect of attacker and defender risk preferences on the optimal defense effort level and their impact on the deterrence level. Numerical illustrations and some discussion of the effect of risk preferences on deterrence and the utility of using such a model are provided, as well as sensitivity analysis of continuous attack investment levels and uncertainty in the defender's beliefs about the attacker's risk preference. A key contribution of this article is the identification of specific scenarios in which the defender using a model that takes into account risk preferences would be better off than a defender using a traditional risk‐neutral model. This study provides insights that could be used by policy analysts and decisionmakers involved in investment decisions in security and safety.