# Topics3. In international relations, states are sometimes assumed to be concerned about how well the

Topics3. 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, 1 and 2, and each secretly decides how much of the dollar to demand. If the sum of the demands is less than a dollar, each 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 d1 for player 1 and d2 for player 2 suchthat the strategy profile (d1,d2) is Nash and 1 gets x and player 2 gets 1-x when both players follow their strategies.(a) Verify that .25 for player 1 and .75 for 2 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 otherplayer. Suppose in particular that if 1 receives a monetary payoff of m1 and 2 receives amonetary payoff of m2, then 1â€™s utility to this outcome is u  m  2(m  m ) . (Note that 11211â€™s utility increases as its monetary payoff, m1, increases and decreases as the differencebetween 2â€™s payoff and its own increases. This last part formalizes the assumption that 1cares about how well it does compared to 2.) 2â€™s utility is given by u  m  2(m  m ) . 2212(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?

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