Finite Approximations of the Sion–Wolfe Game

Sion and Wolfe [Sion M, Wolfe P (1957) On a game without a value. Contributions to the Theory of Games III, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ), 299–306.] presented a two-person zero-sum game on the unit square without a value. In the present paper, we analyze finite-grid approximations of the Sion–Wolfe game. We find that, as the number of grid points tends to infinity and the payoff function approaches that of the infinite game, the limiting value of finite approximations may lie within, on the boundary of, or even outside the interval defined by the lower and upper values of the infinite game. Although these discrepancies can be explained, our findings underscore the need for great care, even in the case of two-person zero-sum games, when using finite approximations for the analysis of infinite games.