Two medieval city-states, Simancas and Toro, are located near each other. Each city-state is controlled by a totalitarian prince, so each can be represented as a single player. Call the prince of Simancas player 1, and let the prince of Toro be called player 2. The land surrounding each city-state can be divided among two uses: forested land for deer hunting, and cleared land for growing wheat. Each city-state has five units of land. At the outset, all of the land is forested.
Each city-state i (where i = 1, 2) must make two decisions: how much land to clear for growing wheat, gi ∈ [0, 5], and how many hounds to raise for hunting deer, hi ∈ [0, ). All decisions are made simultaneously. Payoffs depend on the total quantity of forested land in both city-states (deer roam freely across borders) and the number of hounds raised in both city-states. The deer harvest for city-state i is increasing in its own number of hounds but decreasing in the other city-state’s number of hounds. Specifically, the deer harvest in city-state i is max{0, 2hi − hj}(10 − gi − gj), where j denotes the other city-state. Here, the “maximum” operator is needed to ensure that the harvest is never negative. The wheat-growing results for each city-state, on the other hand, depend only on its own quantity of cleared land. Specifically, the wheat harvest in city-state i is 6gi. Raising hounds and clearing land are both costly. Suppose the cost to citystate i is g2 i + 2h2 i . Summing up, the payoff for city-state 1 is
(a) Show that the strategy (gi , hi) = (0, 0) is dominated for each city-state i.
(b) Show that any strategy with hi > 5 is dominated for each city-state i.
(c) Show that (g1 , h1) = (g2 , h2) = (1, 4) is not efficient
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