Mixed Markov-Perfect Equilibria in the Continuous-Time War of Attrition

TL;DR

This paper proves the existence of mixed-strategy Markov-perfect equilibria in a continuous-time war of attrition model with a linear diffusion state, using topological fixed-point methods.

Contribution

It introduces a novel proof technique for equilibrium existence in continuous-time stochastic games with randomized strategies.

Findings

Existence of mixed-strategy equilibrium in the model

No pure-strategy equilibrium in some cases

Application of topological fixed-point theorem

Abstract

We prove the existence of a Markov-perfect equilibrium in randomized stopping times for a model of the war of attrition in which the underlying state variable follows a homogenous linear diffusion. The proof uses the fact that the space of Markovian randomized stopping times can be topologized as a compact absolute retract, which in turn enables us to use a powerful fixed-point theorem by Eilenberg and Montgomery. We illustrate our results with an example of a war of attrition that admits a mixed-strategy Markov-perfect equilibrium but no pure-strategy Markov-perfect equilibrium.