Markovian randomized equilibria for general Markovian Dynkin games in discrete time

TL;DR

This paper introduces Markovian randomized stopping strategies for discrete-time Markovian Dynkin games, providing explicit equilibrium characterizations, existence results, and examples illustrating the role of randomization in equilibrium solutions.

Contribution

It develops a novel class of Markovian randomized strategies, characterizes Nash equilibria explicitly, and proves existence and non-existence results for pure and mixed strategies in discrete-time Dynkin games.

Findings

Explicit characterization of randomized Nash equilibria

Existence of Markovian randomized equilibria in general games

Examples of games with only randomized equilibria

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.