Evolutionary Game Theory - The War of Attrition Game

The War of Attrition Game

In the Hawk Dove game the resource is sharable, which gives payoffs to both Doves meeting in a pairwise contest. In the case where the resource is not sharable but an alternative resource might be available by backing off and trying elsewhere, pure Hawk or Dove strategies become less effective. If an unshareable resource is combined with a high cost of losing a contest (injury or possible death) both Hawk and Dove payoffs are then further diminished. A safer strategy of lower cost display, bluffing and waiting to win, then becomes viable – a Bluffer strategy. The game then becomes one of accumulating costs, either the costs of displaying or costs of prolonged unresolved engagement. It’s effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets, for all his pains, the same cost as the winner but NO resource. The resulting evolutionary game theory mathematics leads to an optimal strategy of timed bluffing.

This is because in the war of attrition any strategy that is unwavering and predictable is unstable – unstable in the sense that such a strategy will ultimately be displaced by a mutant strategy which will simply rely on the fact that it can best the existing predictable strategy just by investing just an extra small delta of waiting resource to insure that it wins. Therefore only a random unpredictable strategy can maintain itself in a population of Bluffers. The contestants in effect choose an “acceptable cost” to be incurred related to the value of the resource being sought – randomly selected at contest start - effectively making a related random “bid” which becomes part of a mixed strategy (a strategy where a contestant has several, or even many, possible actions in his strategy). This impliments a distribution of bids for a resource of specific value V, where the particular bid made for any specific contest is chosen at random from within that distribution. The distribution (an ESS) can be computed by invoking the Bishop-Cannings theorem, which holds true for any case of mixed strategy ESS. The distribution function in these contests was thus determined by Parker and Thompson to be:

This leads to the result where the cumulative population of quitters of the contest versus for any particular cost m in this "mixed strategy" solution is:

which is shown in the adjacent graph. The intuitive sense that greater values of resource sought leads to greater waiting times is borne out here.

This is exactly what is observed in nature for contests in a number of species, for example between male dung flies contesting for mating sites. The timing of disengagement in these contests follows the exact mathematical curve derived from the evolutionary theory mathematics.