Finite approximations of the Sion-Wolfe game

As pointed out by Sion and Wolfe (1957), a non-cooperative game on the unit square need not admit a Nash equilibrium, neither in pure nor in randomized strategies. In this paper, we consider finite approximations of the Sion-Wolfe game. For all parameter constellations relevant for the limit consideration, we characterize the set of Nash equilibria in iteratively undominated strategies. Values of finite approximations of the Sion-Wolfe game are shown to accumulate around three values that do not correspond in a simple way to the majorant and minorant values of the continuous game. To understand why this is happening, we apply the iterated elimination of weakly dominated strategies to the continuous game as well. The existence of ε-equilibrium, however, does not seem to be related to the properties of finite approximations.