Mixed-Strategy Equilibria in the War of Attrition under Uncertainty

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 (µ i , S i ), where µ 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 tops with probability 1. We then prove that, if players are asymmetric, then, in all mixed-strategy Markov-perfect equilibria, the measures µ 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.