1. Leta*be an evolutionarily stable action. Doesa*necessarily weakly dominate every other action? Is it possible that some action weakly dominatesa*?
2. Suppose that * is a mixed ESS, and letabe an action to which * assigns positive probability. Show that if * assigns positive probability to an action different froma, then (a,a) is not a pure strategy Nash equilibrium, and henceais not an ESS.
3. 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 the player who demands less obtains the amount demanded by her opponent, whereas the player who demands more obtainsa, whereais her demand andis 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