  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.
