The War of Attrition under Uncertainty: Theory and Robust Testable Implications

We study a symmetric-information war of attrition in which the players' rewards depend on exogenous market conditions evolving as a homogeneous linear diffusion process. Using that a Markovian strategy can be represented by a stopping region along with an intensity measure of stopping, we fully characterize mixed-strategy Markov-perfect equilibria through a variational system for the players' value functions. When players are asymmetric, in any such equilibrium each player randomizes at a discrete set of thresholds for market conditions. As a result, players may alternatively find themselves in a position of strength or weakness on the equilibrium path. Delayed concessions occur because a player currently in a position of weakness can hope for market conditions to eventually turn in his favor. In the standard duopoly model of exit under uncertainty, the firms' stock prices and their return volatilities comove negatively over the attrition region and exhibit patterns documented by technical analysis.