The structure of the Nash equilibrium sets of standard 2-player games

In this paper I study a class of two-player games, in which both players’ action sets are [0,1] and their payoff functions are continuous in joint actions and quasi-concave in own actions. I show that a no-improper-crossing condition is both necessary and sufficient for a finite subset A of <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$[0,1]\times [0,1]$</EquationSource> </InlineEquation> to be the set of Nash equilibria of such a game. Copyright Springer-Verlag Berlin/Heidelberg 2005