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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.