Wars of Attrition with Private Budgets

Abstract

This paper studies wars of attrition between two players who have known values for the prizes but private budget constraints. Budgets limit the length that players can stay in the game. We examine the impact of budgets on the equilibrium outcomes. In the case where players' private budget is either high or low, we characterize all weak perfect Bayesian equilibria and select a unique equilibrium that survives a natural refinement. In equilibrium, high budget players always fight, which is reminiscent of the behavioral type players who commit to play a predetermined action in the reputation literature. Consequently, low budget players, regardless of the value of the prize, never concede with certainty as long as they have not depleted their budget. We then study a generalization with multiple homogeneous prizes, where we obtain an equivalence result between sequential and simultaneous allocations of prizes through wars of attrition: the equilibrium outcomes are the same in both formats. Finally, we extend the analysis to the case with a continuum of budget levels and show that our main results continue to hold.