General Markovian randomized equilibrium existence and construction in zero-sum Dynkin games for diffusions

TL;DR

This paper provides a comprehensive solution for zero-sum Dynkin games with diffusions, constructing approximate Markovian equilibria without ordering restrictions and identifying conditions for pure and randomized Nash equilibria.

Contribution

It introduces a novel approach to construct global epsilon-Nash equilibria in Markovian randomized stopping times without ordering conditions on payoffs.

Findings

Explicit construction of epsilon-Nash equilibria for the game.

Conditions established for existence of pure and randomized Nash equilibria.

Illustrative examples demonstrating equilibrium identification.

Abstract

One of the most classical games for stochastic processes is the zero-sum Dynkin (stopping) game. We present a complete equilibrium solution to a general formulation of this game with an underlying one-dimensional diffusion. A key result is the construction of a characterizable global $\epsilon$-Nash equilibrium in Markovian randomized stopping times for every $\epsilon > 0$. This is achieved by leveraging the well-known equilibrium structure under a restrictive ordering condition on the payoff functions, leading to a novel approach based on an appropriate notion of randomization that allows for solving the general game without any ordering condition. Additionally, we provide conditions for the existence of pure and randomized Nash equilibria (with $\epsilon=0$). Our results enable explicit identification of equilibrium stopping times and their corresponding values in many cases, illustrated by several examples.