Question

There are n herders who share a common pasture. The pasture can support mn cattle without degrading. The ith herder has a choice of two strategies: graze m cattle or graze m + 1 cattle. Each cow that is grazed brings a profit ? > 0 to its herder. Each herder who grazes m + 1 cattle imposes a cost c > 0 on the community because of the degradation of the pasture. The cost is shared equally by the n herders. Assume . [10 pts. each]

a.Show that for each herder, the strategy of grazing m + 1 cattle strictly dominates the strategy of grazing m cattle.

b.Find the payoffs to each herder of the following two outcomes: (i) Every herder grazes m + 1 cattle, (ii) Every herder grazes m cattle. Which outcome gives a higher payoff to each herder?

Answer 1

a.

Let us assume two herders A and B. Let p be the profit to a herder for each cow grazed. The payoff matrix can be constructed as follows:

Herder A Herder B

	m cattle	m+1 cattle
m cattle	mp, mp	mp-c/2, mp+p-c/2
m+1 cattle	mp+p-c/2, mp -c/2	mp+p-c/2, mp+p-c/2

Now let us compare the payoffs for each choice a herder makes.

When herder A chooses to graze m cattle, the payoff to herder B is higher when he grazes m+1 cattle, as mp+p-c/2 > mp (assuming p-c/2 >0)

When herder A chooses to graze m+1 cattle, the payoff to herder B is higher when he grazes m+1 cattle, as mp+p-c/2 > mp-c/2

For both strategies chosen by herder A, it is profitable for herder B to graze m+1 cattle.

Similarly,

When herder B chooses to graze m cattle, the payoff to herder A is higher when he grazes m+1 cattle, as mp+p-c/2 > mp (assuming p-c/2 >0)

When herder B chooses to graze m+1 cattle, the payoff to herder A is higher when he grazes m+1 cattle, as mp+p-c/2 > mp-c/2

For both strategies chosen by herder B, it is profitable for herder A to graze m+1 cattle.

Hence grazing m+1 cattle strictly dominates the strategy of grazing m cattle.

b.

(i) Every herder grazes m+1 cattle.

Given there are n herders and cost is shared between n herders

the payoff to each herder = (m+1)*p - c/n=mp +p -c/n

ii) Every herder grazes m cattle

Payoff to each herder=mp

Grazing m+1 cattle gives a higher payoff to each herder provided p-c/n>0