Markovian randomized equilibria for general Markovian Dynkin games in discrete time

Abstract

We study a general formulation of the classical two-player Dynkin game in a discrete time Markovian setting. We identify an appropriate class of mixed strategies -- \textit{Markovian randomized stopping times} -- in which players stop at any given state with a state-dependent probability. One main result is an explicit characterization of Wald-Bellman-type for Nash equilibria based on this notion of randomization. In particular, we derive a novel characterization of randomized equilibria in zero-sum Dynkin games, which we use to (i) establish the existence and explicit construction of Markovian randomized equilibria, (ii) provide necessary and sufficient conditions for the non-existence of pure strategy equilibria, and (iii) construct an example that admits a unique randomized equilibrium but no pure one. We also provide existence and characterization results in the symmetric version of our game. Finally, we establish existence of a characterizable equilibrium in Markovian randomized stopping times for the general game formulation under the assumption that the state space is countable.