Driving Forces Behind Informal Sanctions

This paper investigates the driving forces behind informal sanctions in cooperation games and the extent to which theories of fairness and reciprocity capture these forces. We find that cooperators' punishment is almost exclusively targeted toward the defectors, but the latter also impose a considerable amount of spiteful punishment on the cooperators. However, spiteful punishment vanishes if the punishers can no longer affect the payoff differences between themselves and the punished individual, whereas the cooperators even increase the resources devoted to punishment in this case. Our data also discriminate between different fairness principles. Fairness theories that are based on the assumption that players compare their own payoff to the group's average or the group's total payoff cannot explain the fact that cooperators target their punishment at the defectors. Fairness theories that assume that players aim to minimize payoff inequalities cannot explain the fact that cooperators punish defectors even if payoff inequalities cannot be reduced. Therefore, retaliation, i.e., the desire to harm those who committed unfair acts, seems to be the most important motive behind fairness‐driven informal sanctions.     

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.     

Reputation in Continuous‐Time Games

We study reputation dynamics in continuous‐time games in which a large player (e.g., government) faces a population of small players (e.g., households) and the large player's actions are imperfectly observable. The major part of our analysis examines the case in which public signals about the large player's actions are distorted by a Brownian motion and the large player is either a normal type, who plays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. We find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type. By contrast, when it is common knowledge that the large player is the normal type, every public equilibrium of the continuous‐time game is payoff‐equivalent to one in which a static Nash equilibrium is played after every history. Finally, we examine variations of the model with Poisson signals and multiple behavioral types.     

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).     

The Farsighted Stable Set

Harsanyi (1974) criticized the von Neumann–Morgenstern (vNM) stable set for its presumption that coalitions are myopic about their prospects. He proposed a new dominance relation incorporating farsightedness, but retained another feature of the stable set: that a coalition S can impose any imputation as long as its restriction to S is feasible for it. This implicitly gives an objecting coalition complete power to arrange the payoffs of players elsewhere, which is clearly unsatisfactory. While this assumption is largely innocuous for myopic dominance, it is of crucial significance for its farsighted counterpart. Our modification of the Harsanyi set respects “coalitional sovereignty.” The resulting farsighted stable set is very different from both the Harsanyi and the vNM sets. We provide a necessary and sufficient condition for the existence of a farsighted stable set containing just a single‐payoff allocation. This condition roughly establishes an equivalence between core allocations and the union of allocations over all single‐payoff farsighted stable sets. We then conduct a comprehensive analysis of the existence and structure of farsighted stable sets in simple games. This last exercise throws light on both single‐payoff and multi‐payoff stable sets, and suggests that they do not coexist.     

Fast Equilibrium Selection by Rational Players Living in a Changing World

We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk‐dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2×2 game. The payoff matrix changes according to a random walk. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk‐dominant action. History dependence emerges in some cases when frictions remain positive.     

Further thoughts on the minimax principle

A two-person non-zero sum game may have two distinct solutions: the maximin strategy and the equilibrium point. The choice of solution depends on the players' objectives and it can be shown that, although strategies may be found that are better than the maximin solution in terms of payoff, restraint and co-operation are needed for players to achieve such benefits.     

Linear programming approach to solve interval-valued matrix games

Matrix game theory is concerned with how two players make decisions when they are faced with known exact payoffs. The aim of this paper is to develop a simple and an effective linear programming method for solving matrix games in which the payoffs are expressed with intervals. Because the payoffs of the matrix game are intervals, the value of the matrix game is an interval as well. Based on the definition of the value for matrix games, the value of the matrix game may be regarded as a function of values in the payoff intervals, which is proven to be non-decreasing. A pair of auxiliary linear programming models is formulated to obtain the upper bound and the lower bound of the value of the interval-valued matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. By the duality theorem of linear programming, it is proven that two players have the identical interval-type value of the interval-valued matrix game. Also it is proven that the linear programming models and method proposed in this paper extend those of the classical matrix games. The linear programming method proposed in this paper is demonstrated with a real investment decision example and compared with other similar methods to show the validity, applicability and superiority.     

Truthful Equilibria in Dynamic Bayesian Games

This paper characterizes an equilibrium payoff subset for dynamic Bayesian games as discounting vanishes. Monitoring is imperfect, transitions may depend on actions, types may be correlated, and values may be interdependent. The focus is on equilibria in which players report truthfully. The characterization generalizes that for repeated games, reducing the analysis to static Bayesian games with transfers. With independent private values, the restriction to truthful equilibria is without loss, except for the punishment level: if players withhold their information during punishment‐like phases, a folk theorem obtains.     

Efficiency in Games With Markovian Private Information

We study repeated Bayesian games with communication and observable actions in which the players' privately known payoffs evolve according to an irreducible Markov chain whose transitions are independent across players. Our main result implies that, generically, any Pareto‐efficient payoff vector above a stationary minmax value can be approximated arbitrarily closely in a perfect Bayesian equilibrium as the discount factor goes to 1. As an intermediate step, we construct an approximately efficient dynamic mechanism for long finite horizons without assuming transferable utility.     

Belief‐Free Equilibria in Games With Incomplete Information

We define belief‐free equilibria in two‐player games with incomplete information as sequential equilibria for which players' continuation strategies are best replies after every history, independently of their beliefs about the state of nature. We characterize a set of payoffs that includes all belief‐free equilibrium payoffs. Conversely, any payoff in the interior of this set is a belief‐free equilibrium payoff. The characterization is applied to the analysis of reputations.     

Gambling Reputation: Repeated Bargaining With Outside Options

We study the role of incomplete information and outside options in determining bargaining postures and surplus division in repeated bargaining between a long‐run player and a sequence of short‐run players. The outside option is not only a disagreement point, but reveals information privately held by the long‐run player. In equilibrium, the uninformed short‐run players' offers do not always respond to changes in reputation and the informed long‐run player's payoffs are discontinuous. The long‐run player invokes inefficient random outside options repeatedly to build reputation to a level where the subsequent short‐run players succumb to his extraction of a larger payoff, but he also runs the risk of losing reputation and relinquishing bargaining power. We investigate equilibrium properties when the discount factor goes to 1 and when the informativeness of outside options diffuses. In both cases, bargaining outcomes become more inefficient and the limit reputation‐building probabilities are interior.     

A Folk Theorem for Asynchronously Repeated Games

We prove a Folk Theorem for asynchronously repeated games in which the set of players who can move in period t, denoted by I_t, is a random variable whose distribution is a function of the past action choices of the players and the past realizations of I_τ's, τ=1, 2,…,t−1. We impose a condition, the finite periods of inaction (FPI) condition, which requires that the number of periods in which every player has at least one opportunity to move is bounded. Given the FPI condition together with the standard nonequivalent utilities (NEU) condition, we show that every feasible and strictly individually rational payoff vector can be supported as a subgame perfect equilibrium outcome of an asynchronously repeated game.     

Bayesian Learning,Smooth Approximate Optimal Behavior,and Convergence to ε‐Nash Equilibrium

In this paper, I construct players' prior beliefs and show that these prior beliefs lead the players to learn to play an approximate Nash equilibrium uniformly in any infinitely repeated slightly perturbed game with discounting and perfect monitoring. That is, given any ε > 0, there exists a (single) profile of players' prior beliefs that leads play to almost surely converge to an ε‐Nash equilibrium uniformly for any (finite normal form) stage game with slight payoff perturbation and any discount factor less than 1.     

Competition for a Majority

We define the class of two‐player zero‐sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.     

模糊合作对策的Shapley值

考虑合作对策中支付函数是模糊数的情形,利用模糊数学相关理论,对Shapley提出的三条公理进行拓广,并构造了模糊Shapley值.针对局中人在合作完成后需要对具体的联盟收益进行分配的情况,文中利用构造的模糊Shapley值隶属函数给出了确定的收益分配方案.最后将该方法应用到动态联盟伙伴企业收益分配的实例中.     

Recursive Methods in Discounted Stochastic Games: An Algorithm for δ→ 1 and a Folk Theorem

We present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public (but not necessarily perfect) monitoring when the limiting set of (long‐run players') equilibrium payoffs is independent of the initial state. This is the case, for instance, if the Markov chain induced by any Markov strategy profile is irreducible. We then provide conditions under which a folk theorem obtains: if in each state the joint distribution over the public signal and next period's state satisfies some rank condition, every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting.     

Strategic Experimentation with Exponential Bandits

We analyze a game of strategic experimentation with two‐armed bandits whose risky arm might yield payoffs after exponentially distributed random times. Free‐riding causes an inefficiently low level of experimentation in any equilibrium where the players use stationary Markovian strategies with beliefs as the state variable. We construct the unique symmetric Markovian equilibrium of the game, followed by various asymmetric ones. There is no equilibrium where all players use simple cut‐off strategies. Equilibria where players switch finitely often between experimenting and free‐riding all yield a similar pattern of information acquisition, greater efficiency being achieved when the players share the burden of experimentation more equitably. When players switch roles infinitely often, they can acquire an approximately efficient amount of information, but still at an inefficient rate. In terms of aggregate payoffs, all these asymmetric equilibria dominate the symmetric one wherever the latter prescribes simultaneous use of both arms.     

Measuring Inequity Aversion in a Heterogeneous Population Using Experimental Decisions and Subjective Probabilities

We combine choice data in the ultimatum game with the expectations of proposers elicited by subjective probability questions to estimate a structural model of decision making under uncertainty. The model, estimated using a large representative sample of subjects from the Dutch population, allows both nonlinear preferences for equity and expectations to vary across socioeconomic groups. Our results indicate that inequity aversion to one's own disadvantage is an increasing and concave function of the payoff difference. We also find considerable heterogeneity in the population. Young and highly educated subjects have lower aversion for inequity than other groups. Moreover, the model that uses subjective data on expectations generates much better in‐ and out‐of‐sample predictions than a model which assumes that players have rational expectations.     

Rushes in Large Timing Games

We develop a continuum player timing game that subsumes standard wars of attrition and pre‐emption games, and introduces a new rushes phenomenon. Payoffs are continuous and single‐peaked functions of the stopping time and stopping quantile. We show that if payoffs are hump‐shaped in the quantile, then a sudden “rush” of players stops in any Nash or subgame perfect equilibrium. Fear relaxes the first mover advantage in pre‐emption games, asking that the least quantile beat the average; greed relaxes the last mover advantage in wars of attrition, asking just that the last quantile payoff exceed the average. With greed, play is inefficiently late: an accelerating war of attrition starting at optimal time, followed by a rush. With fear, play is inefficiently early: a slowing pre‐emption game, ending at the optimal time, preceded by a rush. The theory predicts the length, duration, and intensity of stopping, and the size and timing of rushes, and offers insights for many common timing games.