Mixed-Strategy Equilibria in the War of Attrition under Uncertainty

TL;DR

This paper analyzes mixed-strategy equilibria in a stochastic war of attrition game, revealing that asymmetric players' strategies are essentially discrete and characterizing equilibria through a variational system, contrasting with prior pure or continuous strategies.

Contribution

It introduces a novel representation of Markovian mixed strategies using measures and subsets, and characterizes asymmetric equilibria with discrete measures via a variational system.

Findings

Asymmetric players' mixed strategies are essentially discrete.

Equilibrium characterized by a variational system for value functions.

Illustration with a duopoly exit model showing attrition with different liquidation values.

Abstract

We study a generic family of two-player continuous-time nonzero-sum stopping games modeling a war of attrition with symmetric information and stochastic payoffs that depend on an homogeneous linear diffusion. We first show that any Markovian mixed strategy for player $i$ can be represented by a pair $(\mu^i,S^i)$, where $\mu^i$ is a measure over the state space representing player $i$'s stopping intensity, and $S^i$ is a subset of the state space over which player $i$ stops with probability $1$. We then prove that, if players are asymmetric, then, in all mixed-strategy Markov-perfect equilibria, the measures $\mu^i$ have to be essentially discrete, and we characterize any such equilibrium through a variational system satisfied by the players' equilibrium value functions. This result contrasts with the literature, which focuses on pure-strategy equilibria, or, in the case of symmetric players, on mixed-strategy equilibria with absolutely continuous stopping intensities. We illustrate this result by revisiting the model of exit in a duopoly under uncertainty, and exhibit a mixed-strategy equilibrium in which attrition takes place on the equilibrium path though firms have different liquidation values.