Title: Mixed-Strategy Equilibria in the Nash Demand Game
Abstract: In the Nash Demand Game each of two players announces the share he demands of an amount of money that may be split between them. If the demands can be satisfied, they are; otherwise, neither player receives any money. This game has many pure-strategy Nash equilibria. This note characterizes mixed-strategy Nash equilibria. The condition critical for existence of an equilibrium is that players' sets of possible demands be balanced. Two sets of demands are balanced if each demand in one set can be matched with a demand in the other set such that they sum to one. For Nash's original game, a complete characterization is given of the equilibria in which both players' expected payoffs are strictly positive.