Oddly enough, the evolutionarily stable proportion of hawk-types (5/6) is also equal to the equilibrium probability in the mixed-strategy Nash equilibrium of the game if the animals were choosing their strategies rationally. This is not a coincidence. To calculate the equilibrium probabilities in the mixed-strategy Nash equilibrium, we look for probabilities where players are just indifferent between the hawk and dove strategies. In equilibrium their expected values from both strategies are equal.
