Strong Equilibrium in Congestion Games

[6]introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results.Journal of Economic LiteratureClassification Numbers: C71, C72, D62.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Stable games and their dynamics

We study a class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

Excess payoff dynamics and other well-behaved evolutionary dynamics

We consider a model of evolution in games in which agents occasionally receive opportunities to switch strategies, choosing between them using a probabilistic rule. Both the rate at which revision opportunities arrive and the probabilities with which each strategy is chosen are functions of current normalized payoffs. We call the aggregate dynamics induced by this model excess payoff dynamics. We show that every excess payoff dynamic is well-behaved: regardless of the underlying game, each excess payoff dynamic admits unique solution trajectories that vary continuously with the initial state, identifies rest points with Nash equilibria, and respects a basic payoff monotonicity property. We show how excess payoff dynamics can be used to construct well-behaved modifications of imitative dynamics, and relate them to two other well-behaved dynamics based on projections.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Finding all Nash equilibria of a finite game using polynomial algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article, we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.     

Incentive monotonicity and equilibrium selection in 2×2 matrix games

For the class of 2×2 matrix games with two strict Nash equilibria the paper introduces an equilibrium refinement called incentive monotonicity. It selects the risk-dominant equilibrium if interests are conflicting, while it remains silent in games with common interests. These results suggest that the equilibrium-selection problem might be more difficult in games with common interests, which is certainly the case if risk dominance and payoff dominance go in opposite directions.     

Correlated equilibria, incomplete information and coalitional deviations

This paper proposes new concepts of strong and coalition-proof correlated equilibria where agents form coalitions at the interim stage and share information about their recommendations in a credible way. When players deviate at the interim stage, coalition-proof correlated equilibria may fail to exist for two-player games. However, coalition-proof correlated equilibria always exist in dominance-solvable games and in games with positive externalities and binary actions.     

Equilibria in large games with continuous procedures

We study existence of equilibria in large games where players use boundedly rational procedures. The equilibria are different from Nash equilibria; the difference persists even when players use procedures for which the observations gathered in any period are used to evaluate the payoff from different actions.     

Strategic correlativity and network games

We consider the strategic correlativity principle in strategic form games and potential games, which indicate the relationship between an arbitrary pair of players under correlated equilibria. Importantly, the strategies of a pair of players are positively correlated when their expected payoff functions are supermodular, while negatively correlated when their expected payoff functions are submodular. Furthermore, we extend the strategic correlativity principle to strategic form games and potential games in social networks, and investigate the monotonicity of correlated equilibrium in each player's own degree.     

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.     

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 ‘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.     

EXTERNALITIES AGGREGATION IN NETWORK GAMES

We generalize the results on the monotonicity of equilibria for network games with incomplete information. We show that not only the distinction between strategic complements and strategic substitutes is important in determining the nature of the Bayesian Nash equilibria, but the nature of the statistic itself is also determinant. We show that understanding the underlying forces behind people's choices may be of fundamental importance for a policymaker that wants to incentivize efficient behavior.     

Essential equilibria in normal-form games

A Nash equilibrium x of a normal-form game G is essential if any perturbation of G has an equilibrium close to x. Using payoff perturbations, we show that for games that are generic in the set of compact, quasiconcave, and generalized payoff secure games with upper semicontinuous sum of payoffs, all equilibria are essential. Some variants of this result are also established.     

Competitive environments and protective behavior

The class of two-person competition games is introduced and analyzed. For any game in this class the set of Nash equilibria is convex and all Nash equilibria lead to the same payoff vector. Competition games are compared to other competitive environments such as unilaterally competitive games and rivalry games. Moreover, protective behavior within competitive environments is analyzed. For matrix games it is known that protective strategies profiles exactly correspond to proper equilibria. It is shown that this result can be extended to the class of unilaterally competitive games.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.