The Tragedy of the Commons
The Tragedy of the Commons is a Prisoner’s Dilemma with many players. In its classic formulation, each participant wants to use a common resource. But players, acting in their self-interest, end up over exploiting the resource. This in turn generates an inferior outcome for all. The tragedy is a pressing problem for environmentalists and the global sustainability challenge today. This post illustrates the case with a stylized example of overharvesting.
Overharvest
Let’s assume in this simple economy that there are N fisheries. Each fishery must decide on their level of effort (ei) to commit to harvesting tuna.
Harvest and effort
Now, let’s also assume in this ecology that the total volume of tuna harvest (T) depends on the combined effort of every fishery (E = e1 + e2 + … + eN). We express this assumption, the relationship between total harvest and total effort, as follows:
T = 500E – E2
The expression is hill-shaped (parabolic) for two reasons. Firstly, when the total effort is low, an increase in effort leads to an increase in tuna harvested. Secondly, when total effort exceeds the critical threshold (E = 250), an increase in effort leads to a decline in tuna harvest. Put another way, overfishing hurts tuna populations and harvest volumes by extension. And if effort reaches E = 500, we get extinction.
Egalitarian solution
If this fishing society wanted to maximise its tuna harvest, it would cap total effort at E = 250. Ignoring the cost of fishing for now, allowing ei = 250 / N in effort per fishery might be an egalitarian solution. Such a solution probably requires monitoring and enforcement to ensure compliance, perhaps from the local chieftain in pre-modern times, or some governing body today. But regulation is not cost-free. And people rarely agree on the preferred policy direction. Indeed, what seems fair might be in the eye of the beholder.
Firm payoffs
Let’s assume instead that there is no regulation of or coordination between fisheries. Instead, each fishery claims their harvest based on their effort as a proportion of total effort. That is, if a fishery works relatively harder than their competitors, they will get more of the available tuna. If each fishery faces a personal cost of 50ei for their given level of effort, we can express the payoff (U) of each fishery as follows:
Ui = (ei / E) T – 50ei
Substituting our expression for total harvest (T):
Ui = 500ei – Eei– 50ei
Substituting our expression for total effort (E = e1 + e2 + … + eN):
Ui = 450ei – (e1 + e2 + … + eN)ei
We get:
Ui = 450ei – ei2 – Zei
Where Z = e1 + … + ei-1 + ei+1 … + eN = E – ei
Best response
To understand how each fishery will behave, we derive their best response function. That is, to choose an effort level that maximises their payoffs given the effort everybody else is putting in. The easiest way to do this is with a bit of calculus. We take the derivative of the payoff function with respect to effort and solve for its maxima:
dUi / dei = 0
0 = 450 – 2ei – Z
Best response (ei*):
ei*= 225 – 0.5Z
Notice that if N = 1, then Z = 0 and ei*= 225. A harvest of T = 225 is less than the tuna population’s maximum of T = 250. That is, a monopolist fishery, under the conditions of this game, takes less than the maximum because the fishery’s cost rises with effort. As a result, the payoff maximum arrives before the capacity’s maximum. However, if the cost exhibited economies of scale, then the monopolist fisher might be incentivised to fish in even greater volumes.
Nash equilibrium
Since this is a symmetric game, every fishery has the same best response expression. This means we can also look for a symmetric Nash equilibrium in which every fishery uses the same effort as its best response (E = N ei*).
Under such a strategic profile:
Z = E – ei* = N ei* – ei*
Substitute this into our best response:
ei*= 225 – 0.5(N – 1) ei*
ei* = 450 / (N + 1)
So, in this game, there is a symmetric Nash equilibrium in which every fishery commits ei* = 450 / (N + 1) in effort.
Tragedy of the Commons
We can express total combined effort (E), under this symmetric Nash equilibrium, as:
E = N ei* = 450 N / (N + 1)
Notice, that total effort exceeds the critical threshold if:
E > 250
450 N / (N + 1) > 250
N > 1.25
In other words, this fishing society experiences exploitation if there are two or more fisheries in the game! This strategic situation of overfishing is simplified version of the tragedy in action. Each fishery, in trying to maximise its own payoff, hurts the collective’s output due to overuse of a common resource.
Temperance, lock-in, and tragedy
Ecologist Garrett Hardin brought the Tragedy of the Commons to popular attention in the 20th Century. Like Thomas Malthus before him, Hardin was concerned with destructive effects that self-interest and population explosions would have on the collective. While Malthus concentrated on the relationship between overpopulation and food scarcity, Hardin worried about pollution and ecological damage.
In Hardin’s view, without prohibition, regulation, taxation and other “corrective feedbacks” to “legislate temperance”, tragedy was sure to strike. While society has avoided the food crises that Malthus predicted, as the result of manifold increase in agricultural productivity, the question remains whether nations can overcome the environmental challenge with a mix of technological and political solutions.
“Each man is locked into a system that compels him to increase his herd without limit-in a world that is limited. Ruin is the destination toward which all men rush, each pursuing his own best interest in a society that believes in the freedom of the commons. … The laws of our society follow the pattern of ancient ethics, and therefore are poorly suited to governing a complex, crowded, changeable world.”
Garrett, Hardin. (1968). The Tragedy of the Commons.
Resolving collective dilemmas
Hardin’s paper, however, is not the final word. In Beyond Markets and States, Elinor Ostrom highlights that societies can and do overcome social dilemmas all the time (although this doesn’t mean it’s easy). Indeed, case studies and laboratory experiments show that people cooperate more often than game theory predicts.
The Prisoner’s Dilemma and Tragedy of the Commons assumes that players have no option but to play the game they face. More often than not, there is strong incentive for all players to transform the game into more favourable terms.
For example, players might agree upon monitoring and punishments to incentive compliance. Likewise, groups might trial cooperation in one domain. Once norms and conventions form, they may find it favourable to cooperate elsewhere too. Cooperation, in turn, flourishes with incremental growth in trust and tradition.
Ostrom reminds us that our motivational structures in real life are more complex than what we assume in elementary game theory. People think not only of themselves, but of others too. And the scope of games is rarely fixed. Game theory can account for this, but not without greater complexity.
“Building trust in one another and developing institutional rules that are well matched to the ecological systems being used are of central importance for solving social dilemmas. … The most important lesson for public policy analysis … is that humans have a more complex motivational structure and more capability to solve social dilemmas than posited in earlier rational-choice theory. … To explain the world of interactions and outcomes occurring at multiple levels, we also have to be willing to deal with complexity instead of rejecting it.”
Elinor Ostrom. (2009). Beyond Markets and States: Polycentric Governance of Complex Economic systems.