Editing Evolutionary game theory (section)

===War of attrition===
{{main|War of attrition (game)}}

In the hawk dove game the resource is shareable, which gives payoffs to both doves meeting in a pairwise contest. Where the resource is not shareable, but an alternative resource might be available by backing off and trying elsewhere, pure hawk or dove strategies are 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 further diminished.  A safer strategy of lower cost display, bluffing and waiting to win, is then viable – a bluffer strategy. The game then becomes one of accumulating costs, either the costs of displaying or the costs of prolonged unresolved engagement. It is effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets the same cost as the winner but no resource.<ref name=SelfishGene>{{harvp|Dawkins|2006|pp=76-78}}</ref> The resulting evolutionary game theory mathematics lead to an optimal strategy of timed bluffing.<ref>{{cite book |author-link=John Maynard Smith |last=Maynard Smith |first=John |date=1982 |title=Evolution and the Theory of Games |isbn=978-0-521-28884-2 |page=[https://archive.org/details/evolutiontheoryg00jmsm/page/n35 28]|title-link=Evolution and the Theory of Games |publisher=Cambridge University Press }}</ref>

[[File:Attrition graph.jpg|thumb|300px| [[War of attrition (game)|War of attrition]] for different values of resource. Note the time it takes for an accumulation of 50% of the contestants to quit vs. the value (V) of resource contested for.]]

This is because in the war of attrition any strategy that is unwavering and predictable is unstable, because it will ultimately be displaced by a mutant strategy which relies on the fact that it can best the existing predictable strategy by investing an extra small delta of waiting resource to ensure 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, effectively making a random bid as part of a mixed strategy (a strategy where a contestant has several, or even many, possible actions in their strategy). This implements a distribution of bids for a resource of specific value V, where the bid for any specific contest is chosen at random from that distribution.  The distribution (an ESS) can be computed using the [[Bishop-Cannings theorem]], which holds true for any mixed-strategy ESS.<ref>{{cite book |author-link=John Maynard Smith |last=Maynard Smith |first=John |date=1982 |title=Evolution and the Theory of Games |isbn=978-0-521-28884-2 |page=[https://archive.org/details/evolutiontheoryg00jmsm/page/n40 33]|title-link=Evolution and the Theory of Games |publisher=Cambridge University Press }}</ref> The distribution function in these contests was determined by Parker and Thompson to be:

:<math>p(x)=\frac{e^{-x/V}}{V}.</math>
 
The result is that the cumulative population of quitters for any particular cost m in this "mixed strategy" solution is:

:<math>p(m)=1- e^{-m/V},</math>
   
as shown in the adjacent graph.  The intuitive sense that greater values of resource sought leads to greater waiting times is borne out. This is observed in nature, as in male dung flies contesting for mating sites, where the timing of disengagement in contests is as predicted by evolutionary theory mathematics.<ref>{{cite journal | last1=Parker | last2=Thompson | year=1980 | title=Dung Fly Struggle: a test of the War of Attrition | journal=Behavioral Ecology and Sociobiology | volume=7 | issue=1 | pages=37–44 | doi=10.1007/bf00302516| s2cid=44287633 }}</ref>