Takuya Iimura

A discrete fixed point theorem and its applications

Abstract This paper proves a fixed point theorem on a discrete set which exploits the “contiguous convexity” of the set and the “direction preserving-ness” of correspondence. Then, applying this result, we study a general equilibrium model with indivisible commodities. A set of sufficient conditions for the existence of a Walrasian equilibrium price is given, in terms of properties of the excess demand function. We also apply it to a non-cooperative game on discrete strategy sets, and show some conditions for the existence of a Nash equilibrium point. A new notion of “contiguity” is proposed as a discrete analog of continuity.

Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

In this paper we show that a finite symmetric game has a pure strategy equilibrium if the payoff functions of players are integrally concave due to Favati and Tardella (1990). Since the payoff functions of any two-strategy game are integrally concave, this generalizes the result of Cheng et al. (2004). A simple algorithm to find a pure strategy equilibrium is also provided.

Pure strategy equilibrium in finite weakly unilaterally competitive games

We consider the finite version of the weakly unilaterally competitive game (Kats and Thisse, in Int J Game Theory 21:291–299, 1992) and show that this game possesses a pure strategy Nash equilibrium if it is symmetric and quasiconcave (or single-peaked). The first implication of this result is that unilaterally competitive or two-person weakly unilaterally competitive finite games are solvable in the sense of Nash, in pure strategies. We also characterize the set of equilibria of these finite games. The second implication is that there exists a finite population evolutionarily stable pure strategy equilibrium in a finite game, if it is symmetric, quasiconcave, and weakly unilaterally competitive.

Sperner's lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperners lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.

Binary action games: Deviation properties, semi-strict equilibria and potentials

Abstract For binary action games we present three properties which have in common that they are defined by conditions on marginal payoffs. The first two properties guarantee the existence of a special type of Nash equilibrium called semi-strict Nash equilibrium, for which we also show an algorithm to locate. The third one guarantees the existence of an exact potential, and can realize the aforementioned two properties in a class of exact potential games. The first one guarantees the existence of a generalized ordinal potential. Each symmetric binary action game possesses all the three properties. The results are illustrated by three applications.

Pure Strategy Equilibria in Finite Symmetric Concave Games and an Application to Symmetric Discrete Cournot Games

We consider a finite symmetric game where the set of strategies for each player is a one-dimensional integer interval. We show that a pure strategy equilibrium exists if the payoff function is concave with respect to the own strategy and satisfies a pair of symmetrical conditions near the symmetric strategy profiles. As an application, we consider a symmetric Cournot game in which each firm chooses an integer quantity of product. It is shown, among other things, that if the industry revenue function is concave, the inverse demand function is nonincreasing, and the cost function is convex, then the payoff function of the firm satisfies the conditions and this symmetric game has a pure strategy equilibrium.