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Behavioral Ecology

Game Theory

Problems

Problems

In animal conflicts, such as territory or food source defense, the success of one animal's chosen strategy depends on the strategies of the other individuals involved. Because of this, animal conflicts can be modeled after simple games, such as "Rock, Paper, Scissors," in which the outcome of the game depends on which strategies each player chooses. The benefit or detriment to each player is measured quantitatively, usually as the number of offspring (fitness), or net energy gain in calories.

Evolutionarily Stable Strategies

In a two player game where both players have a choice of two strategies, we can calculate which is the best strategy, or the optimal frequency with which two play both strategies. In the four variations of the game pictured below, we have player 1 and player 2, each choosing between strategies A and B.

Figure %: Evolutionarily Stable Strategies (ESS)
By convention, the payoffs are shown for the player on the left side of the matrix. If both players choose strategy A, the payoff is Q. If both players choose strategy B, the payoff is T. If player 1 chooses strategy A and his opponent chooses strategy B, player 1's payoff is R. If player 1 chooses strategy B and his opponent chooses strategy A, player 1's payoff is S. Notice that in this last circumstance, player 2's payoff would be R, not S. Also note that though the name's for the payoffs remain the same, the value of those payoffs differe across the four variations.

In the situation described on the left side of , payoff Q is better than payoff S, and payoff R is better than payoff T. In this case, player 1 should always choose strategy A, because no matter which strategy his opponent chooses, strategy A will provide a better payoff. This situation is a pure evolutionarily stable strategy (ESS). All players will choose strategy A, and evolution will select for only those players who do choose strategy A.

In the situation described by the right side of the figure, because S>Q and T>R, player 1 should always choose strategy B for the same reasons listed above. This is also a pure ESS.

In part c, the situation becomes a bit more difficult. Since Q>S and T>R, there is no clear strategy that will always be best. Player 1's success will depend on which strategy his opponent chooses. This is known as an unstable mixed ESS. The best strategy for player 1 is to always do what his opponent does. If both players choose strategy A, player 1 gets the better payoff of Q. If they both choose strategy B, player 1 receives the better payoff of T. Evolution will result in the frequency of played strategies moving toward what the majority of the population was already doing.

In part d, where S>Q and R>T, the best strategy for player 1 is to always do the opposite of what the opponent does. This is a stable mixed ESS. It is this last situation in which we are most interested, because the other three will always eventually result in one strategy being played constantly. A stable mixed ESS is the only situation in which two strategies can be maintained. The two strategies in a game can be played by different individuals who always play the same strategy (a polymorphic population), or both strategies may be played by any individual in a population (polymorphic individuals).

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

Figure %: 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

Figure %: Prisoner's Dilemma Payoff Matrix

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.

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