2 Two-Population Evolution in the Hawk-Dove Game Recall the Hawk-Dove game, Hawk Hawk Dove -1,-1 6,0 Dove 0,6 3,3 This game has two asymmetric equilibria; (Dove, Hawk) and (Hawk, Dove). It also has a mixed equilibrium in which both players put probability 0.75 on Hawk. Suppose there are two populations, the row population and the column population. Members of the row population only play against members of the column population, and vice versa. (a) Let x and xc denote fraction of Hawk players in the row and column populations respectively. What is the expected payoff of a Hawk-player from population i = {R, C} facing a random opponent from population j i? Note that this quantity will be a function of x; and denote it ; (H, x;)? Similarly, what is (D, x;), i.e. the expected payoff of a Dove-player from population i = {R, C} facing a random opponent from population j i? What is the difference (H, xj) i (D,x;) (b) Suppose that the fraction of H-players in a population grows if H cur- rently earns a strictly higher payoff than D and the fraction of H-players declines 1 (H, x ;) > (H, xj) if H currently earns a strictly lower payoff than D. That is, x; grows if Ti (D, x;) and declines if i (H, xj) < i (D, xj), and stays constant if (D,x;). For what values of x and xc do x and xc grow, decline, stay con- stant? Depict the results in a phase diagram, i.e. in a figure where xR and xc are measured on the axes, and arrows represent directions in which move (as a function of xR and xc). What equilibria seem stable/unstable? and =