Two-player Nonzero–sum Stopping Games in Discrete Time by Eran Shmaya

We prove that every two-player nonzero–sum stopping game in discrete time admits an ε-equilibrium in randomized strategies for every ε > 0. We use a stochastic variation of Ramsey's theorem, which enables us to reduce the problem to that of studying properties of ε-equilibria in a simple class of stochastic games with finite state space. 1. Introduction. The following optimization problem was presented by Dynkin (1969). Two players observe a realization of two real-valued processes (x n) and (R n). Player 1 can stop whenever x n ≥ 0, and player 2 can stop whenever x n < 0. At the first stage θ in which one of the players stops, player 2 pays player 1 the amount R θ and the process terminates. If no player ever stops, player 2 does not pay anything. A strategy of player 1 is a stopping time µ that satisfies {µ = n} ⊆ {x n ≥ 0} for every n ≥ 0. A strategy ν of player 2 is defined analogously. The termination stage is simply θ = min{µ, ν}. For a given pair (µ, ν) of strategies, denote by