The n-person war of attrition

The War of Attrition (WA) was one of the earliest examples studied in the use of the theory of games to understand animal behavior (see Maynard Smith (1974)). The setup is that two contestants compete for a prize worth V(V > 0), and the one who is prepared to wait longer collects the prize; both contestants incur a cost equal to the length of time taken to resolve the contest. Symbolically, if E(x,y) denotes the amount gained by a contestant prepared to wait time x when the opponent is prepared to wait y, $$ E\left( {x,y} \right) = \left\{ {\begin{array}{*{20}{c}} {V - y\quad if\;x > y} \\ { - x\quad if\;x < y} \\ \end{array} } \right. $$ (1) with $$ \begin{gathered} E\left( {x,x} \right) = \frac{V}{2} - x\quad x \in \left[ {0,\infty } \right) \hfill \\ y \in \left[ {0,\infty } \right) \hfill \\ \end{gathered} $$ . Such a game has precisely one evolutionarily stable strategy or ESS (Bishop and Cannings (1976)), i.e. a strategy such that if played by a population, no mutant using another strategy can invade. This ESS is to wait for a time x drawn at random from the exponetial distribution with mean V, i.e. density \(\frac{1}{V}\exp \left( { - x/V} \right)\left( {x > 0} \right) \).