A dominance solvable global game with strategic substitutes

Global games emerged as an approach to equilibrium selection. For a general setting with supermodular payoffs, unique selection of equilibrium has been obtained through iterative elimination of strictly dominated strategies. For the case of global games with strategic substitutes, uniqueness of equilibrium has not been proved by iterative elimination of strictly dominated strategies, making the equilibrium less appealing. In this work we provide a condition for dominance solvability in a simple three-player binary-action global game with strategic substitutes. This opens an unexplored research agenda on the study of global games with strategic substitutes.     

Changes in risk and strategic interaction

In this paper we study the comparative statics of Nth degree stochastic dominance shifts in a large class of non-cooperative games. We consider symmetric equilibria as well as asymmetric equilibria in which the risk changes are idiosyncratic and not necessarily of the same stochastic order. Furthermore, we establish conditions for risk changes to produce multiplier effects on equilibrium strategies. Finally, we evaluate the comparative statics of stochastic dominance shifts in supermodular games, which may feature multiple equilibria and non-convex strategy sets.     

An existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy

We establish an existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy. Instead of the traditional approach which depends on Kakutani’s fixed point theorem, we employ the theories of aggregative games and best reply potential games. We show that, if there exists a representative consumer, under some conditions on preferences and production technologies, the profit maximization game is a (pseudo) best reply potential game. Hence, the existence of the equilibria is proved independently of the well known convex-valued assumption on the best responses. Although our assumptions result in the additive separability on a utility function of a representative consumer, the existence of increasing returns and indivisible productions can be allowed. In our model, it is shown that the game played by firms exhibits strategic substitutes whether the products of firms are substitutes or complements, and this plays an important role for the existence of the equilibria.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

On the relationship between robustness to incomplete information and noise-independent selection in global games

This note demonstrates that a symmetric 3×3 supermodular game may fail to have any equilibrium robust to incomplete information. Since the global game solution in symmetric 3×3 supermodular games is known to be independent of the noise structure, this result implies that a noise-independent selection in global games may not be a robust equilibrium. Our proof reveals that the assumption in global games that the noise errors are independent of the state imposes a non-trivial restriction on incomplete information perturbations.     

Convergence to competitive equilibria and elimination of no-trade (in a strategic market game with limit prices)

The strategic market games literature contains many results that predict Walrasian equilibria in the competitive limit. However, they usually come at the expense of ad hoc assumptions that rule out “pathological” no trade equilibria. This paper studies a strategic market game with limit prices. The set of Nash equilibrium allocations of this game converges to the set containing all competitive equilibria and no-trade, when players are replicated. Moreover, two rounds of iterated deletion of weakly dominated strategies eliminate the no-trade equilibria. Hence, replication paired with two rounds of iterated dominance gives a clean prediction of competitive equilibrium.     

Robust games: theory and application to a Cournot duopoly model

In this paper, the robust game model proposed by Aghassi and Bertsimas (Math Program Ser B 107:231–273, 2006) for matrix games is extended to games with a broader class of payoff functions. This is a distribution-free model of incomplete information for finite games where players adopt a robust-optimization approach to contend with payoff uncertainty. They are called robust players and seek the maximum guaranteed payoff given the strategy of the others. Consistently with this decision criterion, a set of strategies is an equilibrium, robust-optimization equilibrium, if each player’s strategy is a best response to the other player’s strategies, under the worst-case scenarios. The aim of the paper is twofold. In the first part, we provide robust-optimization equilibrium’s existence result for a quite general class of games and we prove that it exists a suitable value \(\epsilon \) such that robust-optimization equilibria are a subset of \(\epsilon \)-Nash equilibria of the nominal version, i.e., without uncertainty, of the robust game. This provides a theoretical motivation for the robust approach, as it provides new insight and a rational agent motivation for \(\epsilon \)-Nash equilibrium. In the last part, we propose an application of the theory to a classical Cournot duopoly model which shows significant differences between the robust game and its nominal version.     

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Equilibrium selection in games: the mollifier method

This paper introduces an embedding of a Nash equilibrium into a sequence of perturbed games, which achieves continuous differentiability of best responses by mollifying them over a continuously differentiable density with compact support (window size). Along any sequence with shrinking window size, the equilibria are single-valued whenever the function has a regular Jacobian and the set of equilibria where it is singular has measure zero. We achieve a further reduction of the equilibrium set by inserting within the embedding a procedure that eliminates the strict interior of equilibrium sets. The mollifier can be approximated consistently using kernel density regression, and we sketch a non-stationary stochastic optimization algorithm that uses this approximation and converges with probability one to an equilibrium of the original game.     

Multiplicity of mixed equilibria in mechanisms: A unified approach to exact and approximate implementation

We characterize full implementation of social choice sets in mixed-strategy Bayesian equilibrium. Our results concern both exact and virtual mixed implementation. For exact implementation, we identify a strengthening of Bayesian monotonicity, which we refer to as mixed Bayesian monotonicity. It is shown that, in economic environments with at least three agents, mixed Bayesian implementation is equivalent to mixed Bayesian monotonicity, incentive compatibility and closure. For implementing a social choice function, the case of two-agents is also covered by these conditions and mixed Bayesian monotonicity reduces to Bayesian monotonicity. Following parallel steps, mixed virtual implementation is shown to be equivalent to mixed virtual monotonicity, incentive compatibility and closure. The key condition, mixed virtual monotonicity, is argued to be very weak. In particular, it is weaker than Abreu–Matsushima’s measurability, thereby implying that: (1) virtual implementation in mixed Bayesian equilibrium is more permissive than virtual implementation in iteratively undominated strategies, and (2) non-regular mechanisms are essential for the implementation of rules in that gap.     

MACROECONOMIC POLICY EVALUATION AND OPTIMAL CONTROL THEORY: A CRITICAL REVIEW OF SOME RECENT DEVELOPMENTS

Abstract. This survey addresses the recent literature on the application of optimal control theory and game theory to macroeconomic policy evaluation and design. This literature focuses on strategic interactions between governments and private agents engaged in dynamic non-cooperative games and emphasizes such issues as precommitment, credibility and reputation as important and endemic aspects of optimal policy design. A dominant theme is the problem of dynamic inconsistency and the inferiority of an equilibrium in single-stage full-information games without commitment. This may be alleviated in repeated games involving reputation effects and threat strategies designed to coerce rivals into pursuing particular strategies and to sustain a sequential equilibrium. The type and characteristics of a sequential equilibrium also depend importantly upon the information structure conditioning players'beliefs under uncertainty. Included here are separating and pooling equilibria, involving notions of learning, signalling and information revelation. Additional dimensions exist in stochastic systems which introduce other forms of uncertainty. Optimal control and game theory have also been applied to international macroeconomic policy design, with emphasis on the potential costs and benefits of non-cooperative and cooperative behaviour between countries and problems of international policy coordination.     

Non-cooperative games on hyperfinite Loeb spaces

We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations—asymptotic implementability, homogeneity and measurability—and argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so represented, the limit of an existing approximate equilibrium may disappear in the limit game. Thus, games on hyperfinite Loeb spaces constitute the ‘right' model even if one is primarily interested in capturing the asymptotic nature of large but finite game-theoretic phenomena.     

Global analysis and economics IV : Finiteness and stability of equilibria with general consumption sets and production

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between and the set of Pareto Optima.     

Global analysis and economics III: Pareto Optima and price equilibria

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let ? be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between ? and the set of Pareto Optima.     

Uniqueness conditions for strongly point-rationalizable solutions to games with metrizable strategy sets

This paper introduces the iterative solution concept of strong point-rationalizability as a strengthening of standard point-rationalizability by stipulating that any two players have identical beliefs about the strategy choices of their common opponents. By adopting and generalizing the contraction-property approach of Moulin [Moulin, H., 1984. Dominance solvability and cournot stability. Mathematical Social Sciences 7, 83–102] and Bernheim [Bernheim, B.D., 1984. Rationalizable strategic behavior. Econometrica 52, 1007–1028] we derive sufficient conditions for the existence of unique strongly point-rationalizable strategies in strategic games with best response functions. These uniqueness results are derived under fairly general assumptions and are especially useful for complete and bounded, as well as finite strategy sets. For games with monotonic individual best response functions, equivalence between a unique strongly point-rationalizable and a unique point-rationalizable solution in the standard sense is established. Furthermore, the existence of unique fixed points is proven under conditions that generalize for bounded metric spaces an established fixed point theorem by Bonsall [Bonsall, F.F., 1962. Lectures on Some Fixed Point Theorems of Functional Analysis. Tata Institute of Fundamental Research, Bombay] and Smart [Smart, D.R., 1974. Fixed Point Theorems. Cambridge University Press, Cambridge, London]. Possible extensions of our findings to games with multi-valued best response correspondences are discussed.     

Functional sunspot equilibria

Consider a one step forward looking model where agents believe that the equilibrium values of the state variable are determined by a function whose domain is the current value of the state variable and whose range is the value for the subsequent period. An agent’s forecast for the subsequent period uses the belief, where the function that is chosen is allowed to depend on the current realization of an extrinsic random process, and is made with knowledge of the past values of the state variable but not the current value. The paper provides (and characterizes) the conditions for the existence of sunspot equilibria for the model described.     

Inefficiency and complementarity in sharing games

A sharing game is a very simple device for partially reconciling an organization’s goal with the interests of its members. Each member chooses an action, bears its cost, and receives a share of the revenue which the members’ actions generate. A (pure-strategy) equilibrium of the game may be inefficient: surplus (revenue minus the sum of costs) may be less than maximal. In a previous paper, we found that for a wide class of reward functions, no one squanders at an inefficient equilibrium (spends more than at an efficient profile) if the revenue function has a complementarity property. In the present paper, we examine the “opposite” of the complementarity property (Substitutes) and we study a class of finite games where squandering equilibria indeed occur if Substitutes holds strongly enough. Squandering equilibria play a key role when one traces the effect of technological improvement on a sharing game’s surplus shortfall. We then turn to the question of choice among reward functions in a principal/agents setting. We find that if we again assume complementarity then strong conclusions can be reached about the reward functions preferred by “society”, by the players (agents), and by the principal.     

Inference in a synchronization game with social interactions

This paper studies inference in a continuous time game where an agent’s decision to quit an activity depends on the participation of other players. In equilibrium, similar actions can be explained not only by direct influences but also by correlated factors. Our model can be seen as a simultaneous duration model with multiple decision makers and interdependent durations. We study the problem of determining the existence and uniqueness of equilibrium stopping strategies in this setting. This paper provides results and conditions for the detection of these endogenous effects. First, we show that the presence of such effects is a necessary and sufficient condition for simultaneous exits. This allows us to set up a nonparametric test for the presence of such influences, which is robust to multiple equilibria. Second, we provide conditions under which parameters in the game are identified. Finally, we apply the model to data on desertion in the Union Army during the American Civil War, and find evidence of endogenous influences.     

Delay,multiplicity, and non-existence of equilibrium in unanimity bargaining games

We consider a class of perfect information bargaining games with unanimity acceptance rule. The proposer and the order of responding players are determined by the state that evolves stochastically over time. The probability distribution of the state in the next period is determined jointly by the current state and the identity of the player who rejected the current proposal. This protocol encompasses a vast number of special cases studied in the literature. We show that subgame perfect equilibria in pure stationary strategies need not exist. When such equilibria do exist, they may exhibit delay. Limit equilibria as the players become infinitely patient need not be unique.     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.