In a stable mixed ESS, we can calculate the optimal ratio of strategy A players to strategy B players. We let p=percentage of strategy A players and q=percentage of strategy B players. The sum of these percentages is equal to 1, comprising the entire population (p+q=1). The fitness of strategy A players equals the chance that the player he encounters will be another A player (chance = p) times the payoff Q, plus the chance that he will play a B player (chance=q) times the payoff R. The fitness of a strategy B player is similarly the chance that he will play another B player(chance = q), times the payoff T, plus the chance he will meet a strategy A player (chance=p), times the payoff S. Thus we have two equations and two unknowns, p and q, for which we can solve. This gives us the expected ratio of strategy A and strategy B players in the given population.
p + q = 1
pQ + qR = pS + qT
The Hawk-Dove Game
The Hawk-Dove Game is a classic example of game theory used in animal behavior. In this model, we have two animals (not necessarily birds) that are capable of choosing from two strategies when in conflict with one another. The animal can choose to be a "hawk" and escalate to a fight or the animal can choose to be a "dove" and peacefully back down. Hawks are always willing to fight, and so if two hawks meet, there will always be a fight. Winners receive the benefit, while losers face the cost of the fight. Doves flee, and so are never involved in a fight. There is no cost to be a dove, only the possibility of receiving no payoff.
In , the benefit for player 1 as a hawk meeting another hawk is the benefit of winning (B) minus the cost of losing the fight (C) divided by two because both hawk players have an equal chance of winning. Half the time player 1 will win and half the time he will lose. Should the hawk meet a dove, the hawk will always win, and so the payoff is just the benefit of winning. If player 1 chooses to be a dove, and meets a hawk, he will lose, and so there is no benefit. However, if player 1 as a dove meets another dove, they will share the benefit, because there is no fight and therefore no cost.
If the benefit of winning is greater than the cost of losing a fight (B>C), then the only evolutionarily stable strategy is to be a pure hawk ESS. Hawks will always do better than doves, because the payoff for being a hawk is greater than that of the dove no matter which strategy his opponent plays. However, if the cost of losing the fight is greater than the benefit of winning (C>B), then the only ESS is to mix up your strategy, playing hawk sometimes and dove sometimes. You can calculate the percentage of time each strategy should be played by calculating p and q as in the previous section, Heading .
Predictions of the Hawk-Dove game lead us to some general conclusions about conflicts of this sort. In a population of mostly hawks, doves will do better than hawks if there is a high fight cost. As the ratio of cost to benefit increases, the population of hawks will decrease. Other strategies can be incorporated into this game. For instance if a challenger wishes to displace the occupant of a territory, the owner will probably play hawk more often than the ESS would call for, because he has already invested energy into his territory. The challenger will play dove more often because he has less to lose.
The Prisoner's Dilemma
The Prisoner's Dilemma is a classic game used in behavioral biology, psychology, and even in business. In this game, we have two partners in crime who are brought into the police station for interrogation. They are isolated from each other immediately and interrogated separately so they do not have the chance to discuss a strategy. Each prisoner has two options, he can cooperate with his partner or he can defect and confess. If both cooperate with each other, neither is caught, both are rewarded, and so there is a high payoff (3). However, if you cooperate and your partner rats you out, you become the sucker and go to jail while he gets away (0). If you defect while your partner keeps silent, the payoff is the highest (5), because you can probably receive immunity for any crime you committed. If you both rat each other out, you may get time off for confessing, but you will both still be punished (1). Given the stated payoffs, the logical action seems to be to defect no matter what your partner does because your payoff is always higher if you defect. This is true when the game is only played once. However, in situations when the game or contest repeats several times, the optimal strategy is to mimic your partner's behavior.