Appendix: The prisoner’s dilemma and fisheries

In the discussion on non-cooperative management of transboundary fishery resources, we discussed briefly one of the most famous of all non-cooperative games, the “Prisoner’s Dilemma”. The game, as we have seen, is also fully relevant to the management of the other three classes of shared fish stocks. The point of the game is that the players are driven inexorably to adopt strategies, which they know to be undesirable. The name, it will be recalled, comes from a story used to illustrate the game, involving two male companions, A and B, who are (justifiably) arrested on suspicion of grand larceny. In our example, cooperation is made impossible by virtue of the fact that the two cannot communicate. If the two could communicate, but thoroughly distrusted one another (dishonour among thieves), so that establishing a true binding agreement between the two was out of the question, we would get the same result.

In any event, each player (A and B) has before him two strategies: to plead guilty, or to plead not guilty. If both were to plead not guilty, they would be released from prison, after serving a minimum sentence. Both, however, are driven to plead guilty, and can look forward, as a consequence, to serving lengthy sentences. We can now show the outcome of, or “solution” to the game in greater detail by setting up a Payoff Matrix. The payoffs in the Matrix are expressed in terms of prison sentences. Consider the following, adapted from Luce and Raiffa (1957):

Suppose that Player B pleads guilty. Player A would clearly be better off pleading guilty. Suppose that Player B pleads not guilty. Player A would, once again, be better off pleading guilty. Regardless of which of the two strategies Player B may adopt, the best strategy for Player A is to plead guilty. Hence, pleading guilty is the dominant strategy for Player A. What holds true for Player A, also hold true for Player B.

Colin Clark, in his book Bioeconomic Modelling and Fisheries Management (Clark, 1985), presents a lucid example of the Prisoner’s Dilemma applied to fisheries. Consider a fishery resource, shared by two countries, in which the costs of harvesting are independent of the size of the biomass, and in which the price for harvested fish and unit fishing effort costs are the same for the two countries, and are both constants. For each country, the net return for each unit of fish harvested is p-c, where p is the price of harvested fish and c the unit cost of harvesting. For the sake of simplicity, let p-c=1.

Let x denote the biomass, and G(x) the growth of the biomass, and thus the sustainable harvest for any given level of x. Suppose that we commence at the global optimal biomass level, i.e. the biomass level at which the global economic returns from the resource will be maximized. Denote that biomass by x*. The global economic return from the resource at x = x* is the present value of the sustainable harvest through time, which can be expressed as: G(x*)/d, where d is the appropriate rate of interest, or discount rate, assumed to be common to the two countries.

One possible harvest policy is simply to deplete the resource. Since harvesting costs are independent of the size of the resource, the resource could be reduced to zero. If, commencing at x = x*, the resource is depleted to zero, the economic return from so doing would be just x*. We assume that x* is positive, which implies, in turn, that x * < G(x*)/d.

Country 1 has two possible strategies: deplete the resource, or conserve it. If Country 1 adopts the deplete strategy, while Country 2 follows the conserve strategy, it is assumed that Country 1 can deplete the resource so quickly that Country 2 receives nothing (and thus ends up as the “goat”). What holds true for Country 1, holds true for Country 2, which faces the same set of strategies.

Finally, we assume that the two countries have equal bargaining strength and harvesting power. Hence, if the two follow the same strategies, they will share the economic returns from the fishery equally.

The Payoff Matrix looks as follows:

If both conserve, each will receive one-half of the present value of the sustainable harvest, i.e.

If both deplete, each will receive

Since

then it follows that, if the two countries could communicate with one another and were prepared to enter into a binding agreement, they would cooperate and we would end up with the resource being conserved.

Suppose, on the other hand, that there is no cooperation, no communication, between the two countries. Assume, to begin with, that

and consider Country 1. If Country 2 should follow the conserve strategy, Country 1 will receive

if it conserves, and x*, if it depletes. If Country 2 should follow the deplete strategy, Country 1 would receive 0, if it follows the conserve strategy, and

if it follows the deplete strategy. Clearly Country 1 should adopt the deplete strategy. What holds true for Country 1, hold true for Country 2, and we end up with a deplete, deplete outcome. This is a perfect Prisoner’s Dilemma case (Clark, 1985, pp. 151153).

Suppose, on the other hand, that

Country 1 would be better off conserving, if Country 2 followed the conserve strategy. It is possible that we would end up with a conserve, conserve outcome. But, such an outcome is decidedly unstable. Suppose that Country 1, guessing that Country 2 will conserve, adopts the conserve strategy, but is then proven wrong. Country 2 depletes, with the result that Country 1 is left with 0, and is indeed the “goat”.

There is, in the theory of games, a famous criterion for selecting strategies in noncooperative games, which is particularly applicable when one’s opponent is both aggressive and unpredictable. It is referred to as the maxmin criterion. The criterion states that one should look at the worst possible outcome from following each strategy, and then compare. Choose the strategy having the least worst outcome.

In the case under discussion,

the Payoff Matrix tells us that the worst outcome for Country 1, if it follows the conserve strategy, is that it will receive 0 (the “goat” outcome). The worst outcome for Country 1, if it follows the deplete strategy, is that it will receive

An application of the maxmin criterion would lead Country 1 to choose the deplete strategy. If Countries 1 and 2 each regard one another as aggressive and unpredictable, we can look forward to a deplete, deplete outcome. We might refer to this as the imperfect Prisoner’s Dilemma case (Clark, 1985, ibid.; Bacharach, 1976).