The N-Player War of Attrition in the Limit of Infinitely Many Players

The War of Attrition is a classical game theoretic model that was first introduced to mathematically describe certain non-violent animal behavior. The original setup considers two participating players in a one-shot game competing for a given prize by waiting. This model has later been extended to several different models allowing more than two players. One of the first of these \(N\)-player generalizations was due to Haigh and Cannings (Acta Appl Math 14:59–74, 1989) where two possible models are mainly discussed; one in which the game starts afresh with new strategies each time a player leaves the game, and one where the players have to stick with the strategy they chose initially. The first case is well understood whereas, for the second case, much is still left open. There are two main results in this paper. The first concerns the asymptotic behavior of the models as the number of players tend to infinity. In particular, we prove that the time evolution of the models coincide in the limit, thus providing a link between the two in the regime of infinitely many players. Secondly we prove, under certain conditions, existence and uniqueness of an ESS in the second model for any given number of players.