Hawks and Doves

Mixed Strategies and Zero-Sum · Open board

The board shows that some games have no single move worth always playing, only a mix at exact odds. This lesson finds the same mix somewhere nobody calculated it: in a population of animals, where the odds are set by who survives.

Before you start. From Mixed Strategies & Zero-Sum you have the indifference trick: a mix is stable when every move in it earns the same, so nobody gains by leaning one way. From Repeated Games & Cooperation you have Axelrod’s tournament, where strategies were played against a field someone chose. Here nobody chooses anything. Strategies are inherited, and the field is whatever is left alive.

The idea in 30 seconds

In the hawk-dove game, a fighter beats a backer-off, but two fighters both get hurt: it is the chicken game, played by animals.
In evolution, a payoff is fitness: kinds that score above average become a bigger share of the next generation.
An all-dove population lets fighters in, and an all-hawk population lets backers-off in, so the population settles in between.
An evolutionarily stable strategy is one that no rare invader can beat; in hawk-dove it is the hawk share at which both kinds score the same.
That share is the prize divided by the cost of a fight, so the costlier fights are, the rarer fighting becomes.
Replicator dynamics, each kind growing with how far it beats the average, finds that share without anyone calculating it.

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