Existence of Equilibria for Discontinuous Games in General Topological Spaces with Binary Relations

Ji-Cheng Hou

doi:10.1155/2018/9518478

Abstract

We provide several results on the existence of equilibria for discontinuous games in general topological spaces without any convexity structure. All of the theorems yielding existence of equilibria here are stated in terms of the player’s preference relations over joint strategies.

Highlights

We provide several results on the existence of equilibria for discontinuous games in general topological spaces without any convexity structure
When ⪰i can be represented by a payoff function ui : X → R, the game G = (Xi, ui)i∈I introduced by Nash in [3] is a special case of G = (Xi, ⪰i)i∈I
Nash [3] proved that an (Nash) equilibrium of the game exists if the set Xi of pure strategies of player i is a compact convex subset of an Euclidean space, and if payoff function ui of player i is continuous andconcave in xi, for each i ∈ I

Summary

Introduction

Nash equilibrium is a fundamental concept in the theory of games and the most widely used method of predicting the outcome of a strategic interaction in almost all areas of economics as well as in business and other social sciences. Baye et al [4], Yu [9], Tan et al [10], Zhang [11], Lignola [12], Nessah and Tian [13, 14], Kim and Lee [15], Hou [16], Chang [17], and Tian [10] and others investigated the existence of pure strategy Nash equilibrium for discontinuous and/or non-quasi-concave games with finite or countable players by using the approach to consider a mapping of individual payoffs into an aggregator function (the aggregator function U : X × X → R is defined by U(y, x) = ∑i∈I ui(yi, x−i) for each (x, y) ∈ X × X.), which is pioneered by Nikaido and Isoda [18] To use these results, one must analyze the aggregator function.

Preliminaries

Existence of Symmetric Pure Strategy Nash Equilibria

Existence of Dominant Strategy Equilibria