Prior to the British agricultural revolution in the 17th century, most agriculture in Britain occurred on common land. Consider a common pasture where N individuals graze their cows. The pasture can productively support kN cows. If the total number of cows exceeds kN, the pasture becomes degraded supporting fewer cows.
Suppose that each individual currently grazes k cows. We will assume that each individual has two strategies available
The responsible strategy (R): continue grazing k cows.
The irresponsible strategy (I): graze k + 1 cows.
Each cow in the pasture above the kN limit imposes a cost c > 0 to the community in terms of degradation. This cost is shared equally by all individual. Thus, for every cow above kN in the pasture, each individual experiences a cost of c/N .
Each individual also realizes a profit p for each of their cows that graze in the pasture.
Assume À (a) Let Ti(si, m) denote the payoff to player i when m other players have chosen to graze k +1 cows (m € {0, 1, ...,N 1}). Write out the payoff to player i for each of their own strategy choices, (R and I).
(b) Prove that R is strictly dominated by I.
(c) If every player uses their dominant strategy 1, what is the payoff to each player? How does this compare with the payoff if every player instead plays their dominated strategy, R?
(d) How does this compare with the outcome of the prisoner’s dilemma game?