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, \(g_{i} \in[0,5]\), and how many hounds to raise for
hunting deer, \(h_{i} \in[0, \infty)\). 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 \(\max \left\\{0,2
h_{i}-h_{j}\right\\}\left(10-g_{i}-g_{j}\right)\), 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 \(6 g_{i}\). Raising hounds and clearing land are
both costly. Suppose the cost to citystate \(i\) is \(g_{i}^{2}+2 h_{i}^{2}\).
Summing up, the payoff for city-state 1 is \(u_{1}\left(g_{1}, h_{1}, g_{2},
h_{2}\right)=\max \left\\{0,2
h_{1}-h_{2}\right\\}\left(10-g_{1}-g_{2}\right)+6 g_{1}-g_{1}^{2}-2
h_{1}^{2}\), and the payoff for city-state 2 is
$$
u_{2}\left(g_{1}, h_{1}, g_{2}, h_{2}\right)=\max \left\\{0,2
h_{2}-h_{1}\right\\}\left(10-g_{2}-g_{1}\right)+6 g_{2}-g_{2}^{2}-2 h_{2}^{2}
.
$$
(a) Show that the strategy \(\left(g_{i}, h_{i}\right)=(0,0)\) is dominated for
each city-state \(i\).
(b) Show that any strategy with \(h_{i}>5\) is dominated for each city-state
\(i\).
(c) Show that \(\left(g_{1}, h_{1}\right)=\left(g_{2}, h_{2}\right)=(1,4)\) is
not efficient.
