3. In international relations, states are sometimes assumed to be concerned about how well they are doing relative to other states. This problem examines that issue. Recall the divide the dollar game discussed in class. There are two players, m If the sum of the deman s is ess t an a do ar, eac player receives what it demanded. If the demands exceed a dollar, each receives nothing. In class we also assumed that each player only cared about its monetary payoff and showed that any division of the dollar was a Nash-equilibrium outcome. That is, given any division of the dollar (x,1 x) there are strategies all for player 1 and d2 for player 2 such that the strategy profile (d1, d2) is Nash and 1 gets x and player 2 gets 1-}: when both players follow their strategies. (a) Verify that .25 for player I and .75 for2 is a Nash-equilibrium outcome by specifying a strategy for each player that produces this outcome and by showing that neither player can benefit by deviating from its strategy given that the other player follows its strategy. Now assume that each player cares in part about how well it does relative to the other player. Suppose in particular that if 1 receives a monetary payoff of mi and 2 receives a monetary payoff of mg, then 1's utility to this outcome is \"1 =ml 203aE m1) . (Note that 1's utility increases as its monetary payoff, m1, increases and decreases as the difference between 2's payoff and its own increases. This last part formalizes the assumption that 1 cares about how well it does compared to 2.) 2'5 utility is given by 112 = mE 201+] m2). (b) Is .25 for 1 and .75 for 2 a Nash equilibrium outcome. Be sure to justify your answer. (c) What set of divisions can be rationalized as Nash-equilibrium outcomes given that the players care about their relative gains