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.

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).