Evolutionary stability in the asymmetric war of attrition

It is shown that there are at least two evolutionarily stable strategies, or ESSs, in the war of attrition with a role asymmetry when players make an error with a small but positive probability in implementing their strategy choices. This result proves Maynard Smith's original conjecture that players should choose asymmetric pure strategies in the asymmetric war of attrition. It is also in contrast to the standard non-existence result of an ESS in this game under complete information. To derive this result we discretize the original game and apply the limit ESS condition introduced by Selten. The result is also compared with other evolutionary analyses of this game.