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. This occurs when **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 two different species]. A Hawk first displays aggression, which if it meets another Hawk escalates into a fight that one wins and the other loses with injury. On the other hand, when an aggressive Hawk encounters a Dove, the Hawk wins. A Dove when it meets a Hawk immediately backs down, else if they encounter another Dove, they cooperate.

These four possible outcomes. To makes this all quantifiable, suppose 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 (V), half the time it loses (C), so the average outcome to the individual is then (V - C) / 2
If a dove meets a hawk, the dove backs off, and get nothing, value 0
If a dove meets a dove, they share the resource and each gets V/2

The actual payoff, however, depends on the proportion of Hawk / Dove encounters, which is determined by the percentage of hawks and doves in the population. That percentage evolves, based on 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 nature) the mathematics ends in an evolutionarily stable strategy (ESS), a mix of the two strategies where the aggregate "good" to the Hawks in the population 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 within-species contests involve only ritual (agonistic) fighting behaviors rather than battles "to the death". The result does not at all depend on "good of the species" behaviors 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 - C) / 2	V
if Dove	0	V / 2

Hawk dove