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The queuing game gives us one example of how the Prisoners' Dilemma can be generalized, and I hope that it provides some insights on some real human interactions. But there is another simple approach to multi-person two-strategy games that is closer to textbook economics, and is important in its own right.
As an example, let us consider the choice of transportation modes -- car or bus -- by a large number of identical individual commuters. The basic idea here is that car commuting increases congestion and slows down traffic. The more commuters drive their cars to work, the longer it takes to get to work, and the lower the payoffs are for both car commuters and bus commuters.
Figure 10-1 illustrates this. In the figure, the horizontal axis measures the proportion of commuters who drive their cars. Accordingly, the horizontal axis varies from a lower limit of zero to a maximum of 1 or 100%. The vertical axis shows the payoffs for this game.The upper (green) line shows the payoffs for car commuters. We see that it declines as the proportion of commuters in their cars increases. The lower, red line shows the payoffs to bus commuters. We see that, regardless of the proportion of commuters in cars, cars have a higher payoff than busses. In other words, commuting by car is a dominant strategy in this game. In a dominant strategy equilibrium, all drive their cars. The result is that they all have negative payoffs, whereas, if all rode busses, all would have positive payoffs. If all commuters choose their mode of transportation with self-interested rationality, all choose the strategy that makes them individually better off, but all are worse off as a result.
This is an extension of the Prisoners' Dilemma , in that there is a dominant strategy equilibrium, but the choice of dominant strategies makes everyone worse off. But it probably is not a very "realistic" model of choice of transportation modes. Some people do ride busses. So let's make it a little more realistic, as in Figure 10-2:
The axes and lines in Figure 10-2 are defined as they were for Figure 10-1. In Figure 10-2, congestion slows the busses down somewhat, so that the payoff to bus commuting declines as congestion increases; but the payoff to car commuting drops even faster. When the proportion of people in their cars reaches q, the payoff to car commuting overtakes the payoff to bus-riding, and for larger proportions of car commuters (to the right of q), the payoff to car commuting is worse than to bus commuting.
Thus, the game no longer has a dominant strategy equilibrium. However, it has a Nash-equilbrium. When a fraction q of commuters drives cars, that is a Nash-equilibrium. Here is the reasoning: starting from q, if one bus commuter shifts to the car, that moves into the region to the right of q, where car commuters are worse off, so (in particular) the person who switched is worse off. On the other hand, starting from q, if one car commuter switches to the bus, that moves into the region to the left of q, where bus commuters are worse off, so, again, the switcher is worse off. No-one can be better off by individually switching from q.
This illustrates an important point: in a Nash-equilibrium, identical people may choose different strategies to maximize their payoffs. This Nash-equilibrium resembles some "supply- and- demand" type equilibria in economics, having been suggested by models of that type, but also differs in some important ways. In particular, it is inefficient, in this sense: if everyone were to ride the bus, moving back to the origin point in Figure 10-2 (as in Figure 10-1), everyone would be better off. As in the Prisoners' dilemma, though, they will not do so when they act on the basis of individual self-interest without coordination.
This example is an instance of "the tragedy of the commons." The highways are a common resource available to all car and bus commuters. However, car commuters make more intensive use of the common resource, causing the resource to be degraded (in this instance, congested). Yet the car commuters gain a private advantage by choosing more intensive use of the common resource, at least while the resource is relatively undegraded. The tragedy is that this intensive use leads to the degradation of the resource to the point that all are worse off.
In general, "the tragedy of the commons" is that all common property resources tend to be overexploited and thus degraded, unless their intensive use is restrained by legal, traditional, or (perhaps) philanthropic institutions. The classical instance is common pastures, on which, according to the theory, each farmer will increase her herds until the pasture is overgrazed and all are impoverished. Most of the applications have been in environmental and resource issues. The recent collapse of fisheries in many parts of the world seems to be a clear instance of "the tragedy of the commons."
All in all, it appears that the Tragedy of the Commons is correctly understood as a multiperson extension of the Prisoners' Dilemma along the lines suggested in Figures 10-1 and 10-2, and, conversely, that the Prisoners' Dilemma is a valuable tool in understanding the many tragedies of the commons that we face in the modern world.
We now turn to a different application of the same technique for generalizing two-person, two-strategy games to many persons.
Roger A. McCain 