S&R Fig02.20

Solution of the hawk dove game for **V=2**, **C=10** and fitness starting base **B=4**. The fitness of a hawk for different population mixes is plotted as a black line, that of dove in red. An **ESS** (a stationary **Evolutionarily Stable Strategy**) will exist when hawk and dove fitness values are equal: **Hawks are 20%** and **doves are 80%** of the population.

The classic hawk-dove game is a contest over a shareable resource. Within a species, there are two behavioral strategies, "Hawk" or "Dove". [The game is not a model of competition between species]. "Hawks" first display aggression, which if they meet another hawk escalates into a fight until they either win or are injured (loses). "Doves" first display aggression, but if faced with major escalation, back down. If not faced with such escalation,"Doves" attempt to share the resource.

The resource has a value V, and the damage from losing a fight has a cost C. The payoff matrix shows, for any single encounter:

If a hawk meets a dove, the hawk wins and gets the full value of the resource V
If a hawk meets a hawk, half the time it wins, half the time it loses, so the average outcome to the individual is then V/2 - C/2
If a dove meets a hawk, the dove backs off, and get nothing, value 0
If a dove meets a dove, both share the resource and get V/2

The actual payoff, however, depends on the probability of meeting a hawk or dove, which in turn is a representation of the percentage of hawks and doves in the population when a particular contest takes place. That, in turn, is determined by the results of all of the previous contests. If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an evolutionarily stable strategy (ESS), a mix of the two strategies where the population of hawks is V/C. The population regresses to this equilibrium if any new hawks or doves make a temporary perturbation in the population. The solution of the hawk dove game explains why most animal contests involve only ritual fighting behaviours in contests rather than outright battles. The result does not at all depend on "good of the species" behaviours as suggested by Lorenz, but solely on the implication of actions of so-called selfish genes.

Payoff matrix for hawk/dove game
	meets hawk	meets dove
if Hawk	V/2 - C/2	V
if Dove	0	V/2

Hawk dove