An equilibrium closure result for discontinuous games

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.