Expected Dynamics of an Intra-Population Imitation Model in the Two-Population Hawk-Dove Game

Consider two populations with the same number of individuals . At each iteration (of time length ), all individuals are randomly matched in pairs made up of one individual from each population to play a symmetric Hawk–Dove game (also called snowdrift or chicken). The payoffs in this game are: a Hawk meeting a Dove gets the highest payoff (3), but if he meets another Hawk they both get the lowest payoff (0). A Dove meeting a Hawk gets 2, and when two Doves meet they both get 1. At the end of each iteration, after all individuals have played the game, one randomly selected player from each population revises her strategy—Hawk or Dove—according to the following rule: "I look at another (randomly selected) individual in my population; if and only if she got a payoff higher than mine, I adopt her strategy". The figure shows a simulation of the proportion of Hawks in each population (in white), its expected dynamics (in dashed red), and the phase plane of the expected dynamics (mean field) in the background.