Discontinuous stochastic games

We prove for a large class of stochastic games with discontinuous payoffs, a stationary Markov perfect equilibrium exists under the condition of “continuation payoff security.” This condition is easy to verify and holds in many economic games. Roughly, a game belongs to this class if for any action/state profiles and continuation payoff, a player can identify another action at the current stage with the payoff not much worse than her current one, even if other players perturb actions slightly. As an illustrative application of the equilibrium existence result, we provide a stochastic dynamic oligopoly model of firm entry, exit, and price competitions.