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.     

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.     

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.     

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).     

Fear of Miscoordination and the Robustness of Cooperation in Dynamic Global Games With Exit

This paper develops a framework to assess how fear of miscoordination affects the sustainability of cooperation. Building on theoretical insights from Carlsson and van Damme (1993), it explores the effect of small amounts of private information on a class of dynamic cooperation games with exit. Lack of common knowledge leads players to second guess each other's behavior and makes coordination difficult. This restricts the range of equilibria and highlights the role of miscoordination payoffs in determining whether cooperation is sustainable or not. The paper characterizes the range of perfect Bayesian equilibria as the players' information becomes arbitrarily precise. Unlike in one‐shot two‐by‐two games, the global games information structure does not yield equilibrium uniqueness.     

Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information

This paper analyzes a class of games of incomplete information where each agent has private information about her own type, and the types are drawn from an atomless joint probability distribution. The main result establishes existence of pure strategy Nash equilibria (PSNE) under a condition we call the single crossing condition (SCC), roughly described as follows: whenever each opponent uses a nondecreasing strategy (in the sense that higher types choose higher actions), a player's best response strategy is also nondecreasing. When the SCC holds, a PSNE exists in every finite‐action game. Further, for games with continuous payoffs and a continuum of actions, there exists a sequence of PSNE to finite‐action games that converges to a PSNE of the continuum‐action game. These convergence and existence results also extend to some classes of games with discontinuous payoffs, such as first‐price auctions, where bidders may be heterogeneous and reserve prices are permitted. Finally, the paper characterizes the SCC based on properties of utility functions and probability distributions over types. Applications include first‐price, multi‐unit, and all‐pay auctions; pricing games with incomplete information about costs; and noisy signaling games.     

Efficiency in Repeated Games Revisited: The Role of Private Strategies

Most theoretical or applied research on repeated games with imperfect monitoring has focused on public strategies: strategies that depend solely on the history of publicly observable signals. This paper sheds light on the role of private strategies: strategies that depend not only on public signals, but also on players' own actions in the past. Our main finding is that players can sometimes make better use of information by using private strategies and that efficiency in repeated games can be improved. Our equilibrium private strategy for repeated prisoners' dilemma games consists of two states and has the property that each player's optimal strategy is independent of the other player's state.     

Existence of Equilibrium in Single and Double Private Value Auctions1

We show existence of equilibria in distributional strategies for a wide class of private value auctions, including the first general existence result for double auctions. The set of equilibria is invariant to the tie‐breaking rule. The model incorporates multiple unit demands, all standard pricing rules, reserve prices, entry costs, and stochastic demand and supply. Valuations can be correlated and asymmetrically distributed. For double auctions, we show further that at least one equilibrium involves a positive volume of trade. The existence proof establishes new connections among existence techniques for discontinuous Bayesian games.     

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.     

A Theory of Disagreement in Repeated Games With Bargaining

This paper proposes a new approach to equilibrium selection in repeated games with transfers, supposing that in each period the players bargain over how to play. Although the bargaining phase is cheap talk (following a generalized alternating‐offer protocol), sharp predictions arise from three axioms. Two axioms allow the players to meaningfully discuss whether to deviate from their plan; the third embodies a “theory of disagreement”—that play under disagreement should not vary with the manner in which bargaining broke down. Equilibria that satisfy these axioms exist for all discount factors and are simple to construct; all equilibria generate the same welfare. Optimal play under agreement generally requires suboptimal play under disagreement. Whether patient players attain efficiency depends on both the stage game and the bargaining protocol. The theory extends naturally to games with imperfect public monitoring and heterogeneous discount factors, and yields new insights into classic relational contracting questions.     

Tenable Strategy Blocks and Settled Equilibria

When people interact in familiar settings, social conventions usually develop so that people tend to disregard alternatives outside the convention. For rational players to usually restrict attention to a block of conventional strategies, no player should prefer to deviate from the block when others are likely to act conventionally and rationally inside the block. We explore two set‐valued concepts, coarsely and finely tenable blocks, that formalize this notion for finite normal‐form games. We then identify settled equilibria, which are Nash equilibria with support in minimal tenable blocks. For a generic class of normal‐form games, our coarse and fine concepts are equivalent, and yet they differ from standard solution concepts on open sets of games. We demonstrate the nature and power of the solutions by way of examples. Settled equilibria are closely related to persistent equilibria but are strictly more selective on an open set of games. With fine tenability, we obtain invariance under the insertion of a subgame with a unique totally mixed payoff‐equivalent equilibrium, a property that other related concepts have not satisfied.     

Imperfect Monitoring and Impermanent Reputations

We study the long‐run sustainability of reputations in games with imperfect public monitoring. It is impossible to maintain a permanent reputation for playing a strategy that does not play an equilibrium of the game without uncertainty about types. Thus, a player cannot indefinitely sustain a reputation for noncredible behavior in the presence of imperfect monitoring.     

Computing Supergame Equilibria

We present a general method for computing the set of supergame equilibria in infinitely repeated games with perfect monitoring and public randomization. We present a three‐stage algorithm that constructs a convex set containing the set of equilibrium values, constructs another convex set contained in the set of equilibrium values, and produces strategies that support them. We explore the properties of this algorithm by applying it to familiar games.     

Sharp Identification Regions in Models With Convex Moment Predictions

We provide a tractable characterization of the sharp identification region of the parameter vector θ in a broad class of incomplete econometric models. Models in this class have set‐valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous‐move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ_I, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ_I. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method.     

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.     

Inference of Signs of Interaction Effects in Simultaneous Games With Incomplete Information

This paper studies the inference of interaction effects in discrete simultaneous games with incomplete information. We propose a test for the signs of state‐dependent interaction effects that does not require parametric specifications of players' payoffs, the distributions of their private signals, or the equilibrium selection mechanism. The test relies on the commonly invoked assumption that players' private signals are independent conditional on observed states. The procedure is valid in (but does not rely on) the presence of multiple equilibria in the data‐generating process (DGP). As a by‐product, we propose a formal test for multiple equilibria in the DGP. We also implement the test using data on radio programming of commercial breaks in the United States, and infer stations' incentives to synchronize their commercial breaks. Our results support the earlier finding by Sweeting (2009) that stations have stronger incentives to coordinate and air commercials at the same time during rush hours and in smaller markets.     

Efficient Use of Information and Social Value of Information

This paper analyzes equilibrium and welfare for a tractable class of economies (games) that have externalities, strategic complementarity or substitutability, and heterogeneous information. First, we characterize the equilibrium use of information: complementarity heightens the sensitivity of equilibrium actions to public information, raising aggregate volatility, whereas substitutability heightens the sensitivity to private information, raising cross‐sectional dispersion. Next, we define and characterize an efficiency benchmark designed to address whether the equilibrium use of information is optimal from a social perspective; the efficient use of information reflects the social value of aligning choices across agents. Finally, we examine the comparative statics of equilibrium welfare with respect to the information structure; the social value of information is best understood by classifying economies according to the inefficiency, if any, in the equilibrium use of information. We conclude with a few applications, including production externalities, beauty contests, business cycles, and large Cournot and Bertrand games.     

On the performances of Nash equilibria in isolation games

We study the performances of Nash equilibria in isolation games, a class of competitive location games recently introduced in Zhao et al. (Proc. of the 19th International Symposium on Algorithms and Computation (ISAAC), pp. 148–159, 2008). For all the cases in which the existence of Nash equilibria has been shown, we give tight or asymptotically tight bounds on the prices of anarchy and stability under the two classical social functions mostly investigated in the scientific literature, namely, the minimum utility per player and the sum of the players’ utilities. Moreover, we prove that the convergence to Nash equilibria is not guaranteed in some of the not yet analyzed cases.     

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.