While the Prisoner’s Dilemma is the best-known example in game theory, it is also one of the simplest, and its simplicity does place some limits on its application.
The Prisoner’s Dilemma begins with a story of interrogation. For this discussion, we may instead recall the Water Game from Chapter 2, where it is shown in normal form as Table 2.1.
Eastland knows that it cannot influence Westria’s strategy choice, and conversely. Instead, each one chooses his best response to the strategy choice made by the other. This defines Nash equilibrium. Moreover, in this case, the best response is “Divert” regardless of the other agent’s strategy choice. That means “Divert” is a dominant strategy: by definition, if a strategy is the best response to any strategy choice made by the other agent or agents, it is a dominant strategy. Thus, when both agents choose the dominant strategy “Divert,” we have a dominant strategy equilibrium, which is a particularly simple instance of a Nash equilibrium. A dominant strategy equilibrium can be defined as a Nash equilibrium in which each agent has a dominant strategy.
Nevertheless, if both agents were to choose “Don’t” in Game 2.1, both would be better off, with net payoffs of 0 rather than -1. We may borrow terminology from welfare economics and say that the strategy pair “Don’t, Don’t” Pareto-dominates the pair “Divert, Divert.” A strategy vector S1 Pareto-dominates strategy vector S2 if no agent is worse off with S1 than with S2 and at least one agent is better off with S1 than with S2. Together, these observations define Game 2.1 as a social dilemma (Dawes, 1980). Generally, a social dilemma is a game in which (1) there is a dominant strategy equilibrium indicated by strategies S2, and (2) there is vector of strategies S1, such that each component of S1 differs from the corresponding component of S2 and S1 Pareto-dominates S2. Social dilemmas are usually also treated as being symmetrical (so that interchanging any two agents would leave the payoff table unchanged).
A social dilemma model such as Game 2.1 predicts that, in the absence of some public intervention, the dominant strategy equilibrium, S2, will occur. Since it is Pareto-dominated by a different set of decisions, S1, this outcome is inefficient. Decisions S1 are said to constitute a cooperative solution and, in a symmetrical game such as this, the payoffs correspond to the Shapley value and nucleolus in particular. (These will be discussed in more detail in Chapter 8. Note also McCain 2013, Chapters 1, 5, 6.) Public policies may then be advocated that move individual decisions toward the efficient set S1. In this way, social dilemmas capture the principles that seem to underlie a number of major problems of modern societies and public policy, but they may not be very good descriptions of the real world. The symmetry that is usually assumed in social dilemma models is one example. In most real-world applications, there is likely to be some lack of symmetry among the agents. Since the problem (inefficiency) arises in symmetrical models, however, we can be assured that it does not arise because of the lack of symmetry in the real world; thus asymmetry is a complication but not an underlying cause of the problem. This is a valuable point that might be missed if the simplified, symmetrical model were not considered. All the same, for some practical applications, it may be necessary to reintroduce some asymmetry in a model with heterogeneous agents.
Social dilemmas can be generalized to a large number of players following Schelling (1978) and Moulin (1982 p. 92 et seq.). Think of a large number of people living in the watershed of a lake, each of whom may act
Figure 4.1 A social dilemma with large N
so as to pollute the lake or, at some cost, refrain from pollution. Suppose there are N agents, N very large, each of whom must make the same absolute choice of strategies “Don’t” or “Pollute.” The overall amount of pollution will depend on the proportion of the population that choose “pollute;” so that the payoffs to both strategies will depend on the same proportion. Borrowing terminology from the theory of differential games, we can describe the proportion of agents who choose “Pollute” as a state variable for the game. In this usage, a state variable is a variable that is sufficient to determine the payoffs of the different strategies without any other information (such as information on the specific strategy choices of individual agents, for example).
This model is illustrated by Figure 4.1. We see that the payoff to “Pollute” lies above the payoff to “Don’t,” regardless of the proportion of the group who choose “pollute” as their strategy. The diagram illustrates visually that this is an N-person social dilemma. If any group of players chooses “Don’t,” they are not choosing their best response to the strategies chosen by the others. The dominant strategy equilibrium corresponds to the rightward extreme of the diagram, the case in which every agent chooses “Pollute.”
The N-person social dilemma model can also be interpreted to be consistent with imperfectly rational behavior, if it is interpreted in an evolutionary sense (see, for example, Aumann 1997). Suppose that individuals usually act with inertia, simply choosing the same strategy over and over, but from time to time, at random, they experiment with reversing their strategies. If the reversal leads to an increase in the net payoff they persist, and if not they return to the previous strategy. This trial-and-error learning process is one of random variation and directed selection of strategies, a simple evolutionary process. Biologists, having borrowed the concept of Nash equilibrium from game theory, define an evolutionarily stable strategy (ESS) as a Nash equilibrium that is stable under an evolutionary dynamics. The dominant strategy equilibrium for this model is an ESS. Thus the conclusion does not depend on the assumption of perfect rationality.1
An appropriately generalized social dilemma model can account for many instances in which inefficiencies persist in the presence of human decisions that successfully seek self-regarding benefits, whether through perfect rationality or through trial-and-error learning. As this example and the Water Game suggest, environmental economics is largely built of social dilemma models. The production of a public good is another instance of a social dilemma. All in all, social dilemma models are powerful diagnostic and explanatory tools for problems of social inefficiency.