Confused about question 3 and 4

3. Let a* be an evolutionarily stable action. Does 11* necessarily weakly dominate every other action? Is it possible that some action weakly dominates (1*? 4. Suppose that (1* is a mixed E33, and let a be an action to Which (1* assigns positive probability. Show that if 01* assigns positive probability to an action different om a, then (a, a) is not a pure strategy Nash equilibrium, and hence a is not anESS. 5. Pairs of members of the same population engage in the following game. Each player has three actions, corresponding to demands of 1, 2, or 3 units of payoff. If both players in a pair make the same demand, each player obtains her demand. Otherwise we player who demands less obtains the amount demanded by her opponent, Whereas the player who demands more obtains 6a, where H is her demand and 6 is a positive number less than 1/3. Find the set of pure strategy symmetric equilibria of the game, and the set of pure evolutionarily stable strategies. What happens if each player has n actions, corresponding to demands of 1, 2, ..., Vr units ofpayofand O 0): Tortoise Hare Tortois e c, 0 2,1 Hare 1,-2 0'0 a. For what values of c are tortoises tter than hares if t, the proportion of racers who are tortoises, is 0.5? b. For what values of c are tortoises tter than hares if t is 0.1? c. lfc : 1, will a single hare successfully invade a population of pure tortoises? Explain why or w y not. d. In terms of r, how large must 0 be for tortoises to be tter than hares? e. In terms of a, what is the leve of t in a mixed strategy equi ibrium'.' For what Values of c wi l such an equilibrium exist? Will this equilibrium be evolutionarily stable? 7. Find al the E555, in pure and mixed strategies, of the followinggame: A B C A 0,0 2,1 0,0 B 1,2 0,0 0,0 C 0,0 0,0 1,1