Choosing and sharing

Implementing a project, like a nationwide nuclear waste disposal, which benefits all involved agents but brings major costs only to the host is often problematic. In practice, revelation issues and redistributional concerns are significant obstacles to achieving stable agreements. We address these issues by proposing the first mechanism to select the efficient site (the host with the lowest cost) while simultaneously allowing for the implementation of any individually rational division of the whole hosting cost. The subgame perfect Nash equilibria of our simple mechanism are efficient, budget-balanced and individually rational. Moreover, the truthful equilibrium is immune to coalitional deviations.     

EQUILIBRIA IN NON-COOPERATIVE GAMES II: DEVIATIONS BASED REFINEMENTS OF NASH EQUILIBRIUM

In any Nash equilibrium no player will unilaterally deviate. However, many games have multiple Nash equilibria. In this paper, we survey some refinements of Nash equilibria based on the hypothesis that any player may consider a deliberate deviation from a Nash equilibrium vector while expecting other players to respond optimally to this deviation. The concepts studied here differ in the expectations players have about other players' responses to a deviation. This sort of deviations philosophy is predicated on the thought process of players. Therefore, the validity of a particular equilibrium concept to an economic model may depend upon the relevance of the thought process implied by the concept.     

ENDOGENOUS TIMING IN QUALITY CHOICES AND PRICE COMPETITION

We modify the price‐setting version of the vertically differentiated duopoly model by Aoki (2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Our results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. We also investigate the mixed‐strategy equilibrium, revealing that the probability of generating outcomes out of equilibrium is higher than the probability of playing one Nash equilibria in pure strategies. In the alternative case with full market coverage, we show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide. With full market coverage and corner solution, the results show that the game has a unique subgame perfect equilibrium in pure strategies, where the high‐quality firm takes the lead in the quality stage.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

EQUILIBRIA IN NON-COOPERATIVE GAMES I: PERTURBATIONS BASED REFINEMENTS OF NASH EQUILIBRIUM

The concept of Nash equilibrium is widely used to analyse non-cooperative games. However, one of the problems with that concept is that many games have multiple equilibria. Recent work has concentrated on reducing or refining the set of Nash equilibria in some games. In this paper, we survey some equilibrium concepts based on perturbations of strategies that refine the set of Nash equilibria. We discuss the pros and cons of each concept and its relationship to the others by the use of numerous examples and intuition. It is hoped that this survey will enable the economist to consider the relevance of a particular equilibrium concept to a given economic model of interest. Journal of Economic Literature Classification Number: C72.     

Homotopy methods to compute equilibria in game theory

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.     

Efficient communication in the electronic mail game

The literature on the electronic mail game shows that players' mutual expectations may lock them into requiring an inefficiently large number of confirmations and confirmations of confirmations from one another. This paper shows that this result hinges on the assumption that, with the exception of the first message, each player can only send a message when receiving an immediately preceding message. We show that, once this assumption is lifted, equilibria involving confirmations of confirmations no longer pass standard refinements of the Nash equilibrium, and are no longer evolutionary stable.     

Learning by replicator and best-response: the importance of being indifferent

This paper compares two learning processes, namely those generated by replicator and best-response dynamics, from the point of view of the asymptotics of play. We base our study on the intersection of the basins of attraction of locally stable pure Nash equilibria for replicator and best-response dynamics. Local stability implies that the basin of attraction has positive measure but there are examples where the intersection of the basin of attraction for replicator and best-response dynamics is arbitrarily small. We provide conditions, involving the existence of an unstable interior Nash equilibrium, for the basins of attraction of any locally stable pure Nash equilibrium under replicator and best-response dynamics to intersect in a set of positive measure. Hence, for any choice of initial conditions in sets of positive measure, if a pure Nash equilibrium is locally stable, the outcome of learning under either procedure coincides. We provide examples illustrating the above, including some for which the basins of attraction exactly coincide for both learning dynamics. We explore the role that indifference sets play in the coincidence of the basins of attraction of the stable Nash equilibria.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Voluntary Contributions to Multiple Public Projects

The problem of financing a set of discrete public goods (facilities, projects) by private contributions is studied. The corresponding cooperative game, the realization game , is shown to be convex. For the noncooperative setting we study a realization scheme that induces a strategic game. This contribution game is shown to be a generalized ordinal potential game ; a best–response in the contribution game implies a best response in a coordination game in which the payoff to all players is the utilitarian collective welfare function, i.e., the sum of the utility functions of the players. Strategy profiles maximizing utilitarian welfare are strong Nash equilibria of the contribution game. Each strong Nash equilibrium corresponds in a natural way with a core element of the realization game, and vice versa. Moreover, each strong Nash equilibrium is coalitional proof.     

Coordination need not be a problem

In a game of common interest there is one action vector that all players prefer to every other. Yet there may be multiple Pareto-ranked Nash equilibria in the game and the “coordination problem” refers to the fact that rational equilibrium play cannot rule out Pareto-dominated equilibria. In this paper, I prove that two elements — asynchronicity and a finite horizon — are sufficient to uniquely select the Pareto-dominant action vector (in subgame perfect equilibrium play). Asynchronicity may be exogenously specified by the rules of the game. Alternatively, in a game where players choose when to move, asynchronicity may emerge as an equilibrium move outcome.     

Endogenous timing in the presence of non-monotonicities

The aim of this paper is to generalize the endogenous timing game proposed by Hamilton and Slutsky (Games and Economic Behavior, 1990, 2, pp. 29–46) by allowing the payoff or the marginal payoff of a player to become non-monotonic with respect to the strategy of the opponent. We propose a taxonomy of the subgame-perfect Nash equilibria based on the characteristics of the payoff functions proposed by Eaton (Canadian Journal of Economics, 2004, 37, pp. 805–29). We determine under which conditions of the initial payoff functions commitment has a social value and when the simultaneous-move Nash equilibrium is commitment robust and discuss its Pareto efficiency.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

Bayesian games with unawareness and unawareness perfection

Applying unawareness belief structures introduced in Heifetz et al. (Games Econ Behav 77:100–121, 2013a), we develop Bayesian games with unawareness, define equilibrium, and prove existence. We show how equilibria are extended naturally from lower to higher awareness levels and restricted from higher to lower awareness levels. We apply Bayesian games with unawareness to investigate the robustness of equilibria to uncertainty about opponents’ awareness of actions. We show that a Nash equilibrium of a strategic game is robust to unawareness of actions if and only if it is not weakly dominated. Finally, we discuss the relationship between standard Bayesian games and Bayesian games with unawareness.     

Cycling in a stochastic learning algorithm for normal form games

In this paper we study a stochastic learning model for 2×2 normal form games that are played repeatedly. The main emphasis is put on the emergence of cycles. We assume that the players have neither information about the payoff matrix of their opponent nor about their own. At every round each player can only observe his or her action and the payoff he or she receives. We prove that the learning algorithm, which is modeled by an urn scheme proposed by Arthur (1993), leads with positive probability to a cycling of strategy profiles if the game has a mixed Nash equilibrium. In case there are strict Nash equilibria, the learning process converges a.s. to the set of Nash equilibria.     

On the uniqueness of Nash equilibria

Although the use of Nash equilibria in economic models has been increasing, little attention has been paid to the uniqueness of these equilibria. This paper presents relatively weak conditions that are sufficient for the existence of a globally unique Nash equilibrium.     

CORRELATED EQUILIBRIA,GOOD AND BAD: AN EXPERIMENTAL STUDY*

We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of Nash equilibrium. Specifically, we examine the conditions under which subjects playing the game of Chicken will condition their behavior on private third‐party recommendations drawn from publicly announced distributions. We find that when recommendations are given, behavior differs from both a mixed‐strategy Nash equilibrium and behavior without recommendations. In particular, subjects typically follow recommendations if and only if (1) those recommendations derive from a correlated equilibrium and (2) that correlated equilibrium is payoff‐enhancing relative to the available Nash equilibria.     

Equilibrium selection in experimental cheap talk games

In the past, many refinements have been proposed to select equilibria in cheap talk games. Usually, these refinements were motivated by a discussion of how rational agents would reason in some particular cheap talk games. In this paper, we propose a new refinement and stability measure that is intended to predict actual behavior in a wide range of cheap talk games. According to our Average Credible Deviation Criterion (ACDC), the stability of an equilibrium is determined by the frequency and size of credible deviations. ACDC organizes the results from several cheap talk experiments in which behavior converges to equilibrium, even in cases where other criteria do not make a prediction.     

Evolutionary Stability in the Investment-Opportunism-Retaliation Game

Can economically efficient outcomes be obtained and sustained in the absence of externally enforced property rights? We study the evolutionary properties of a game that exhibits two well-defined Nash equilibria: one generates an inefficient outcome while the other set generates an efficient outcome supported by the potential for retaliation. Although standard forward-looking refinements eliminate the efficient equilibrium, neither equilibrium type satisfies strict evolutionary stability criteria. However, both types of equilibrium define strategies that are neutrally stable, which makes them vulnerable to drift in dynamic environments. We conduct computer simulation experiments in which players learn adaptively via a tournament selection mechanism called sophisticated experimentation. Our simulations demonstrate that while the system spends a disproportionately high proportion of time in the inefficient equilibrium set, the efficient equilibrium is pervasive as the system drifts back and forth between the equilibrium sets, never settling on one or the other.