The war of attrition with noisy players

We first consider the Nash equilibria for the two-player normal-form war of attrition, which is equivalent to a second-price all-pay auction. When there is a limit to the maximum effort (for example, a budget constraint), and for low enough prize values, there exists a symmetric mixed-strategy equilibrium under equal prize values. However, the only equilibria when prize values differ involve one player choosing zero effort (conceding immediately). Non-degenerate mixed-strategy equilibria under different prize values reemerge when there is no maximum effort. These equilibria have perverse comparative static properties: an increase in one player’s value leaves that player’s bid distribution unaffected and raises the other player’s bid (in the sense of first-degree stochastic domination). Given these quirks of the Nash equilibrium, we then describe the (boundedly-rational) logit equilibrium for the game. This equilibrium is unique and symmetric when players have the same prize values. Moreover, the player with the higher prize value exerts more effort (in the sense of first-degree stochastic dominance). The paper concludes with a discussion of some experimental evidence for a related game, that supports the qualitative predictions of the logit equilibrium model