On equilibria for discontinuous games: Nash approximation schemes

For discontinuous games Simon and Zame (1990) introduced a new approach to the existence of equilibria. To obtain the required continuity, they convert the original discontinuous payoff function into an upper semicontinuous payoff correspondence; graphically, this corresponds to a “vertical interpolation” to close the discontinuity gap at the discontinuity points. The resulting payoff indeterminacy, in the form of endogenous sharing rules (i.e., measurabe selections of the payoff correspondence), is an essential feature of their model. The mixed equilibrium existence result obtained by Simon and Zame (1990) generalizes Glicksberg’s (1952) existence result for Nash equilibria. This paper proposes to view Simon and Zame’s “vertical interpolation” as the limit of a sequence of standard (nonvertical) continuous interpolations across the discontinuity. In other words, we propose to approximate the upper semicontinuous payoff correspondence directly by means of a sequence of continuous payoff functions. To each of these Glicksberg’s existence result applies, which yields a sequence of mixed Nash equilibria. The weak limit of this sequence is the equilibrium of Simon and Zame (1990). However, our approach goes beyond existence, because the approximate Nash equilibria can often be easily computed in actual examples (most notably, with the aid of purification methods). This does not only provide a new interpretation of the endogenous sharing rule as a certain conditional expectation of the payoff vector, but the precise information gathered about it in terms of the approximate Nash equilibria and their payoff values is, as we show, of considerable help in the actual computations.