Perfect-Information Games with Lower-Semicontinuous Payoffs

We prove that every multiplayer perfect-information game with bounded and lower-semicontinuous payoffs admits a subgame-perfect ϵ-equilibrium in pure strategies. This result complements Example 3 in Solan and Vieille [2003], which shows that a subgame-perfect ϵ-equilibrium in pure strategies need not exist when the payoffs are not lower-semicontinuous. In addition, if the range of payoffs is finite, we characterize in the form of a Folk Theorem the set of all plays and payoffs that are induced by subgame-perfect 0-equilibria in pure strategies.