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We study a generic family of two-player continuous-time nonzero-sum stoppinggames modeling a war of attrition with symmetric information and stochasticpayoffs that depend on an homogeneous linear diffusion. We first show that anyMarkovian 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 representingplayer $i$'s stopping intensity, and $S^i$ is a subset of the state space overwhich player $i$ stops with probability $1$. We then prove that, if players areasymmetric, then, in all mixed-strategy Markov-perfect equilibria, the measures$\mu^i$ have to be essentially discrete, and we characterize any suchequilibrium through a variational system satisfied by the players' equilibriumvalue functions. This result contrasts with the literature, which focuses onpure-strategy equilibria, or, in the case of symmetric players, onmixed-strategy equilibria with absolutely continuous stopping intensities. Weillustrate this result by revisiting the model of exit in a duopoly underuncertainty, and exhibit a mixed-strategy equilibrium in which attrition takesplace on the equilibrium path though firms have different liquidation values.

Mixed-Strategy Equilibria in the War of Attrition under Uncertainty