The Absence of Attrition in a War of Attrition under Complete Information

TL;DR

This paper demonstrates that introducing stochastic payoffs and heterogeneous exit payoffs in a war of attrition game under complete information leads to the existence of pure strategy equilibria only, highlighting the fragility of mixed strategies.

Contribution

It shows that stochastic payoffs and heterogeneity eliminate mixed-strategy equilibria, contrasting with the canonical model where both pure and mixed equilibria exist.

Findings

Pure strategy equilibria exist under stochastic payoffs and heterogeneity.

Mixed strategies are fragile and do not exist under certain stochastic conditions.

Deterministic payoffs and homogeneous exit payoffs allow both pure and mixed equilibria.

Abstract

We consider a two-player game of war of attrition under complete information. It is well-known that this class of games admits equilibria in pure, as well as mixed strategies, and much of the literature has focused on the latter. We show that if the players' payoffs whilst in "war" vary stochastically and their exit payoffs are heterogeneous, then the game admits Markov Perfect equilibria in pure strategies only. This is true irrespective of the degree of randomness and heterogeneity, thus highlighting the fragility of mixed-strategy equilibria to a natural perturbation of the canonical model. In contrast, when the players' flow payoffs are deterministic or their exit payoffs are homogeneous, the game admits equilibria in pure and mixed strategies.