Robust Predictions in Games With Incomplete Information

We analyze games of incomplete information and offer equilibrium predictions that are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action–state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.     

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.     

Games Played Through Agents

We introduce a game of complete information with multiple principals and multiple common agents. Each agent makes a decision that can affect the payoffs of all principals. Each principal offers monetary transfers to each agent conditional on the action taken by the agent. We characterize pure‐strategy equilibria and we provide conditions—in terms of game balancedness—for the existence of an equilibrium with an efficient outcome. Games played through agents display a type of strategic inefficiency that is absent when either there is a unique principal or there is a unique agent.     

Computational Complexity and Communication: Coordination in Two–Player Games

The main contribution of this paper is the development and application of cryptographic techniques to the design of strategic communication mechanisms. One of the main assumptions in cryptography is the limitation of the computational power available to agents. We introduce the concept of limited computational complexity, and by borrowing results from cryptography, we construct a communication protocol to establish that every correlated equilibrium of a two–person game with rational payoffs can be achieved by means of computationally restricted unmediated communication. This result provides an example in game theory where limitations of computational abilities of players are helpful in solving implementation problems. More specifically, it is possible to construct mechanisms with the property that profitable deviations are too complicated to compute.     

Implementation with Near‐Complete Information

Many refinements of Nash equilibrium yield solution correspondences that do not have closed graph in the space of payoffs or information. This has significance for implementation theory, especially under complete information. If a planner is concerned that all equilibria of his mechanism yield a desired outcome, and entertains the possibility that players may have even the slightest uncertainty about payoffs, then the planner should insist on a solution concept with closed graph. We show that this requirement entails substantial restrictions on the set of implementable social choice rules. In particular, when preferences are strict (or more generally, hedonic), while almost any social choice function can be implemented in undominated Nash equilibrium, only monotonic social choice functions can be implemented in the closure of the undominated Nash correspondence.     

Repeated Games with Differential Time Preferences

When players have identical time preferences, the set of feasible repeated game payoffs coincides with the convex hull of the underlying stage- game payoffs. Moreover, all feasible and individually rational payoffs can be sustained by equilibria if the players are sufficiently patient. Neither of these facts generalizes to the case of different time preferences. First, players can mutually benefit from trading payoffs across time. Hence, the set of feasible repeated game payoffs is typically larger than the convex hull of the underlying stage-game payoffs. Second, it is not usually the case that every trade plan that guarantees individually rational payoffs can be sustained by an equilibrium, no matter how patient the players are. This paper provides a simple characterization of the sets of Nash and of subgame perfect equilibrium payoffs in two-player repeated games.     

On Forward Induction

A player's pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. The outcome satisfies forward induction if it results from a weakly sequential equilibrium in which players' beliefs assign positive probability only to relevant strategies at each information set reached by a profile of relevant strategies. We prove that if there are two players and payoffs are generic, then an outcome satisfies forward induction if every game with the same reduced normal form after eliminating redundant pure strategies has a sequential equilibrium with an equivalent outcome. Thus in this case forward induction is implied by decision‐theoretic criteria.     

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.     

Bounded Memory and Biases in Information Processing

Before choosing among two actions with state‐dependent payoffs, a Bayesian decision‐maker with a finite memory sees a sequence of informative signals, ending each period with fixed chance. He summarizes information observed with a finite‐state automaton. I characterize the optimal protocol as an equilibrium of a dynamic game of imperfect recall; a new player runs each memory state each period. Players act as if maximizing expected payoffs in a common finite action decision problem. I characterize equilibrium play with many multinomial signals. The optimal protocol rationalizes many behavioral phenomena, like “stickiness,” salience, confirmation bias, and belief polarization.     

政府监管与企业加入第三方国际环境审计的模糊博弈分析

针对政府环境监管与企业实施环境行为博弈分析中无法对双方的损益情况做出判断的问题,文章结合三角模糊数构建了政府监管和企业加入第三方国际环境审计的博弈模型,分别指出了纯策略纳什均衡和混合策略均衡情况需要满足的条件,讨论了政府与污染环境企业博弈策略的影响因素及作用机制,为政府监管与企业加入第三方国际环境审计顺利实施提供了相关建议。然后通过三角结构元方法对算例进行求解,验证了结论的正确性和可行性,引入三角模糊概念更加符合实际应用情况,拓展了第三方国际环境审计的应用背景。研究结果表明,政府对消费者消费的引导和对污染企业的惩罚对企业是否决定加入第三方国际环境审计起着举足轻重的作用。     

基于策略熵的博弈分析研究

通过建立博弈问题的信息论模型和定义不同类型博弈问题的策略熵,论证了纳什均衡与最大熵之间的内在联系:最大策略熵是纳什均衡的充分必要条件。这表明纳什均衡不仅是利益的均衡同时也是信息均衡的结果。在此基础上,给出了基于熵最大化准则的纳什均衡估计与精炼的方法。     

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.     

Adverse Selection in Competitive Search Equilibrium

We study economies with adverse selection, plus the frictions in competitive search theory. With competitive search, principals post terms of trade (contracts), then agents choose where to apply, and they match bilaterally. Search allows us to analyze the effects of private information on both the intensive and extensive margins (the terms and probability of trade). There always exists a separating equilibrium where each type applies to a different contract. The equilibrium is unique in terms of payoffs. It is not generally efficient. We provide an algorithm for constructing equilibrium. Three applications illustrate the usefulness of the approach, and contrast our results with those in standard contract and search theory.     

The Theory of Global Games on Test: Experimental Analysis of Coordination Games with Public and Private Information

The theory of global games has shown that coordination games with multiple equilibria may have a unique equilibrium if certain parameters of the payoff function are private information instead of common knowledge. We report the results of an experiment designed to test the predictions of this theory. Comparing sessions with common and private information, we observe only small differences in behavior. For common information, subjects coordinate on threshold strategies that deviate from the global game solution towards the payoff‐dominant equilibrium. For private information, thresholds are closer to the global game solution than for common information. Variations in the payoff function affect behavior as predicted by comparative statics of the global game solution. Predictability of coordination points is about the same for both information conditions.     

INFORMATION GATHERING IN ORGANIZATIONS: EQUILIBRIUM,WELFARE, AND OPTIMAL NETWORK STRUCTURE

We model an organization as a game in which all agents share a common decision problem and some level of coordination is necessary between individual actions. Agents have individual private information concerning the task they have to perform, and they share this private information through pairwise channels of communication. We analyze how this communication pattern, modeled by means of a network structure, affects individual behavior and aggregate welfare. In the unique equilibrium of this Bayesian game, each agent's optimal action depends on a properly defined knowledge index that measures how the aggregation of information helps him to infer higher‐order beliefs about other's information after communication. Adding communication channels is not always beneficial for the organization because it can lead to mis‐coordination. We single out the geometry of interagent communication links that the manager could implement in order to improve the organization's performance.     

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.     

Bargaining and Reputation

The paper develops a reputation based theory of bargaining. The idea is to investigate and highlight the influence of bargaining ‘postures’ on bargaining outcomes. A complete information bargaining model a la Rubinstein is amended to accommodate ‘irrational types’ who are obstinate, and indeed for tractability assumed to be completely inflexible in their offers and demands. A strong ‘independence of procedures’ result is derived: after initial postures have been adopted, the bargaining outcome is independent of the fine details of the bargaining protocol so long as both players have the opportunity to make offers frequently. The latter analysis yields a unique continuous‐time limit with a war of attrition structure. In the continuous‐time game, equilibrium is unique, and entails delay, consequently inefficiency. The equilibrium outcome reflects the combined influence of the rates of time preference of the players and the ex ante probabilities of different irrational types. As the probability of irrationality goes to zero, delay and inefficiency disappear; furthermore, if there is a rich set of types for both agents, the limit equilibrium payoffs are inversely proportional to their rates of time preference.     

Experientia Docet: Professionals Play Minimax in Laboratory Experiments

We study how professional players and college students play zero‐sum two‐person strategic games in a laboratory setting. We first ask professionals to play a 2 × 2 game that is formally identical to a strategic interaction situation that they face in their natural environment. Consistent with their behavior in the field, they play very close to the equilibrium of the game. In particular, (i) they equate their strategies' payoffs to the equilibrium ones and (ii) they generate sequences of choices that are serially independent. In sharp contrast, however, we find that college students play the game far from the equilibrium predictions. We then study the behavior of professional players and college students in the classic O'Neill 4 × 4 zero‐sum game, a game that none of the subjects has encountered previously, and find the same differences in the behavior of these two pools of subjects. The transfer of skills and experience from the familiar field to the unfamiliar laboratory observed for professional players is relevant to evaluate the circumstances under which behavior in a laboratory setting may be a reliable indicator of behavior in a naturally occurring setting. From a cognitive perspective, it is useful for research on recognition processes, intuition, and similarity as a basis for inductive reasoning.     

Multiperson Bargaining and Strategic Complexity

We investigate the effect of introducing costs of complexity in the n‐person unanimity bargaining game. As is well‐known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are sufficiently patient and if n & 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to a plausible notion of complexity that captures length of memory, we find that the introduction of complexity costs (lexicographically with the standard payoffs) does not reduce the range of possible allocations but does limit the amount of delay that can occur in any agreement. In particular, we show that in any n‐player game, for any allocation z, an agreement on z at any period t can be sustained as a Nash equilibrium of the game with complexity costs if and only if t≤n. We use the limit on delay result to establish that, in equilibrium, the strategies implement stationary behavior. Finally, we also show that ‘noisy Nash equilibrium’ with complexity costs sustains only the unique stationary subgame perfect equilibrium allocation.     

Selecting Cheap‐Talk Equilibria

There are typically multiple equilibrium outcomes in the Crawford–Sobel (CS) model of strategic information transmission. This paper identifies a simple condition on equilibrium payoffs, called NITS (no incentive to separate), that selects among CS equilibria. Under a commonly used regularity condition, only the equilibrium with the maximal number of induced actions satisfies NITS. We discuss various justifications for NITS, including perturbed cheap‐talk games with nonstrategic players or costly lying. We also apply NITS to other models of cheap talk, illustrating its potential beyond the CS framework.