An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.     

Approximation results for discontinuous games with an application to equilibrium refinement

We provide approximation results for Nash equilibria in possibly discontinuous games when payoffs and strategy sets are perturbed. We then prove existence results for a new “finitistic” infinite-game generalization of Selten’s (Int J Game Theory 4: 25–55, 1975) notion of perfection and study some of its properties. The existence results, which rely on the approximation theorems, relate existing notions of perfection to the new specification.     

On Cournot-Nash equilibrium distributions for games with differential information and discontinuous payoffs

Summary Mas-Colell's model for equilibrium distributions in large anonymous games is extended by the introduction of an abstract, non-topological notion of players' characteristics. The generality of this model allows us to include as a component of a player's characteristic her/his degree of well-informedness (differential information). This makes it possible to address a generalization of a model recently formulated by Khan-Rustichini. We show that one can remove their rather unnatural compactness condition on the space of admissible decision rules, provided that one allows for decision rules which are randomized. Our utility functions need not be continuous; this particular technical aspect solves an open question of Khan concerning Mas-Colell's original model. Our method of proof is based on the noval observation that Cournot-Nash equilibrium distributions are precisely the solutions of a variational inequality for transition probabilities.     

On the existence of pure-strategy perfect equilibrium in discontinuous games

We provide sufficient conditions for a (possibly) discontinuous normal-form game to possess a pure-strategy trembling-hand perfect equilibrium. We first show that compactness, continuity, and quasiconcavity of a game are too weak to warrant the existence of a pure-strategy perfect equilibrium. We then identify two classes of games for which the existence of a pure-strategy perfect equilibrium can be established: (1) the class of compact, metric, concave games satisfying upper semicontinuity of the sum of payoffs and a strengthening of payoff security; and (2) the class of compact, metric games satisfying upper semicontinuity of the sum of payoffs, strengthenings of payoff security and quasiconcavity, and a notion of local concavity and boundedness of payoff differences on certain subdomains of a player's payoff function. Various economic games illustrate our results.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Essential equilibria of discontinuous games

We introduce spaces of discontinuous games in which games having essential Nash equilibria are the generic case. In order to prove the existence of essential Nash equilibria in such spaces, we provide new results on the Ky Fan minimax inequality. In the setting of potential games, we show that games with essential Nash equilibria are the generic case when their potentials satisfy a condition called weak upper pseudocontinuity that is weaker than upper semicontinuity.     

Understanding some recent existence results for discontinuous games

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (Econometrica 67:1029–1056, 1999) and Carmona (J Econ Theory 144:1333–1340, 2009) and that it is equivalent to a recent result of Barelli and Soza (On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, University of Rochester, Rochester, 2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.     

On strategic stability in discontinuous games

We identify a class of discontinuous normal-form games whose members possess strategically stable sets, defined according to an infinite-game extension of Kohlberg and Mertens’s (1986) equilibrium concept, and show that, generically, a set is stable if and only if it contains a single Nash equilibrium.     

Uniqueness of equilibrium for smooth multistage concave games

Smooth, noncooperative, multistage, concave games are formulated so that a new uniqueness condition—based on the Poincaré-Hopf theorem—can be applied. The new condition is the weakest to appear in the uniqueness literature. The uniqueness subgame perfect equilibrium is obtained and examples are given.     

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of e{\epsilon}-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

Correlated equilibrium existence for infinite games with type-dependent strategies

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. This paper gives a counter-example to the main step in Cotter?s [K. Cotter, Correlated equilibrium in games with type-dependent strategies, J. Econ. Theory 54 (1991) 48-69] argument for correlated equilibrium existence for this class of games, and supplies an alternative proof.     

Greediness and equilibrium in congestion games

We study the class of congestion games for which the set of Nash equilibrium is equivalent to the set of strategy profiles played by greedy myopic players. We show these two coincide iff such games are played over extension-parallel graphs.     

Scale functions in equilibrium selection games

We present here an evolutionary game model, and address the issue of equilibrium selection working with the scale function of a diffusion process describing the dynamics of population processes with mutation modeled as white noise. This model is the same as the one in Foster and Young (1990) but with a different interpretation at the boundaries and with different mutation modelings. First, we justifiably assume that the boundaries of the solution of the stochastic differential equation are absorbing so that the first boundary of the interval [0,1] hit will determine the equilibrium selected. Then, working with the scale function, we obtain for 2×2 symmetric games and different mutation parameters, some new and interesting equilibrium selection results. The aim of this article is to describe another method of approach in evolutionary games with mutation which we believe will prove to be very useful in studying more general normal form games and different mutation modelings.     

The single deviation property in games with discontinuous payoffs

We study equilibrium existence in normal form games in which it is possible to associate with each nonequilibrium point an open neighborhood, a set of players, and a collection of deviation strategies, such that at any nonequilibrium point of the neighborhood, a player from the set can increase her payoff by switching to the deviation strategy designated for her. An equilibrium existence theorem for compact, quasiconcave games with two players is established as an application of a general equilibrium existence result for qualitative games. A new form of the better-reply security condition, called the strong single deviation property, is proposed.