War of Attrition with Individual Differences on RHP

Abstract

The fact that there always exists various kinds of almost continuous mutations for any animal population implies that players in competitions can never be perfectly symmetric in any sense. To develop a model to fit this reality, we consider war of attrition games in which players have continuously different resource holding potential (RHP). The RHP of each opponent is not known in our settings.Pure ESS functions and Nash equilibria are obtained under sufficiently rotational conditions as unique solutions of certain differential equations among the class of Lebesgue measurable functions. They are normal in that a higher RHP induces a longer attrition time, which implies that a player with greater RHP always wins. This model includes as the limit the conclusions of Maynard Smith (1974,J. theor. Biol.47, 209–221) and Normanet al. (1977,J. theor. Biol.65, 571–578), which did not consider individual differences in RHP. Our results suggest that, by changing each player's qualitative differences to continuous quantitative differences, some of the mixed ESS solutions previously found in discrete games may degenerate into pure ESS functions. Moreover, we found that the smaller the individual differences of RHP, the smaller is the mean pay-off of most individuals as well as the total pay-off of the population.