The term “Tragedy of the Commons” has come to signify environmental problems of overuse that arise when scarce resources are treated as common property. 4 A game-theoretic illustration of this issue can be developed by assuming that two herders decide how many sheep to graze on the village commons. The problem is that the commons is quite small and can rapidly succumb to overgrazing.
In order to add some mathematical structure to the problem, let qibe the number of sheep that herder i = 1, 2 grazes on the commons, and suppose that the per-sheep value of grazing on the commons (in terms of wool and sheep-milk cheese) is
This function implies that the value of grazing a given number of sheep is lower the more sheep are around competing for grass. We cannot use a matrix to represent the normal form of this game of continuous actions. Instead, the normal form is simply a listing of the herders’ payoff functions
The Nash equilibrium is given by the pair (q*1 ,q*2)that satisfies Equations 8.15 and 8.16simultaneously. Taking an algebraic approach to the simultaneous solution, Equation 8.16can be substituted into Equation 8.15, which yields
upon rearranging, this implies q*1 = 40. Substituting q*1 = 40 into Equation 8.17 implies q*2 = 40 as well. Thus, each herder will graze 40 sheep on the common. Each earns a payoff of 1,600, as can be seen by substituting q*1 = q*2 = 40 into the payoff function in Equation 8.13. Equations 8.15 and 8.16 can also be solved simultaneously using graphical methods. Figure 8.4 plots the two best responses on a graph with player 1’s action on the horizontal axis and player 2’s on the vertical axis. These best responses are simply lines and so are easy to graph in this example. (To be consistent with the axis labels, the inverse of Equation 8.15 is actually what is graphed.) The two best responses intersect at the Nash equilibrium E1. The graphical method is useful for showing how the Nash equilibrium shifts with changes in the parameters of the problem. Suppose the per-sheep value of grazing increases for the first herder while the second remains as in Equation 8.11, perhaps because the first herder starts raising merino sheep with more valuable wool. This change would shift the best response out for herder 1 while leaving 2’s the same. The new intersection point (E2 in Figure 8.4), which is the new Nash equilibrium, involves more sheep for 1 and fewer for 2. The Nash equilibrium is not the best use of the commons. In the original problem, both herders’ per-sheep value of grazing is given by Equation 8.11. If both grazed only 30 sheep then each would earn a payoff of 1,800, as can be seen by substituting q1 ¼ q2 ¼ 30 into Equation 8.13. Indeed, the “joint payoff maximization” problem