of refusing to pay for collective undertakings in the sure knowledge that the United States will pay anyway. As Mancur Olson put it,

Once a smaller member has the amount of the collective good he gets free from the largest member, he [sometimes] has more than he would have purchased for himself, and has no incentive to obtain any of the collective good at his own expense. In small groups with common interests there is accordingly a surprising tendency for the "exploitation" of the great by the small.10

The Theory of The prisoner's dilemma is a special case of a general theory of games.11 The Games theory discusses the situation of intelligent people facing other intelligent people in games of tic-tac-toe, poker, chess, cartels, exchanges, collective bargaining, price wars, business mergers, politics, extortions, kidnappings, and wars. Invented in 1928 by the famous mathematician John von Neumann and brought to the attention of economists in 1944 by von Neumann and Oskar Morgenstern in their astonishing Theory of Games and Economic Behavior, the theory gave early promise of solving the problem of bargaining among small groups, such as oligopolists. Like monopolistic competition and other approaches to the problem, it has not fulfilled the promise.

Cultivated for its own sake as a metaphor of social life, however, it must be judged a great intellectual success. At present its main use in general economics is in fact metaphorical. To say that the formation of a cartel is "just like" a prisoner's dilemma game or that a nuclear arms race between the Soviet Union and the United States is "just like" a two-person negative-sum game is to state the essence of the situations with persuasive elegance.

The very notion that we are "playing games" with other people is enlightening.

So, too, are the notions of the negative, positive, or zero sumness of the game. In a zero-sum game my loss is your gain, as in the neighborhood poker game or in the distributing of the gains from trade as viewed by medieval towns. Gary's winnings are John's losings; what Venice gained in the trade with the East in the Middle Ages Genoa lost, or so the Venetians and Genoese believed. It is apparent that any constant sum of spoils (or negative spoils, damages) will give the same results. That is, as long as Venice and Genoa are fighting over a fixed pie, it does not matter whether or not the total size of the pie is called zero (Venice gets what Genoa loses) or 100 (Venice gets 80, Genoa 20, both gaining over zero). The alternative to such a constant-sum game is a variable sum, in which the size of the pie to be divided does vary with how the players act. The sum of the payoffs from the arms race, for example, depends on which solutions are chosen. If the bombs are not used, each country loses only the cost of making the bombs; if they are used, each loses its entire population. A more cheerful example, and the focus of much attention in the theory, is exchange. Exchange is "positive sum"; that is, both parties gain or, at worst, do

10 Olson, The Logic of Collective Action, p. 35 (italics in original).

11 A brief idea of the theory can be gotten from William J. Baumol, Economic Theory and Operations Analysis, 4th ed. (Englewood Cliffs, N.J.: Prentice-Hall, 1977), Chapter 18. A more leisurely and thorough review is Morton D. Davis, Game Theory: A Nontechnical Introduction (New York: Basic Books, 1970). The original book of 1944 by von Neumann and Morgenstern mentioned in the text following (Princeton, N.J.: Princeton University Press, 1947) is highly technical on the whole, but also highly readable in places.

not lose. If two parties to bargaining over the exchange of steel pipes for natural gas fail to reach any agreement because they disagree over the price, the mutual benefit does not materialize. That is, the size of the pie in total varies with the bargaining strategies of the parties,- the game is variable-sum.

As one might expect, the variability introduces complications of threat and bluff that are not present in the simpler case. As was noted, the theory does not literally "solve" the problem of bargaining games. Game theory merely provides the economist with a rich harvest of metaphor: coalitions, the core (mutual benefit), imputations (the prices agreed to), side payments (bribes), maximum strategy (avoiding the worst that people or nature can do to you), saddle points (when such avoidance implies the same strategy for both players), mixed strategies (flipping coins to keep one's behavior from being predictable), and other wonders. It does not, alas, solve the oligopoly problem.

Summary The theory of games is the ultimate response to the principle of outsmarting.

In reaction to the unattractiveness of supposing oligopolists to be outsmarted easily, economists have developed in it a virtual theory of outsmarting. The goal of a set of competing oligopolists is ultimately to eliminate competition. They play a game of cooperation and defection that may or may not have monopoly as its outcome. The approach is more attractive to economists than are Cournotesque approaches because, as George Stigler put it,

A satisfactory theory of oligopoly cannot begin with assumptions concerning the way in which each firm views its interdependence with its rivals. If we adhere to the traditional theory of profit-maximizing enterprises, then behavior is no longer something to be assumed but rather something to be deduced.12

As usual, however, the game theoretic approach to the theory of monopoly is more useful in applications than in reaching a satisfactory theory of oligopoly. The uses of the prisoner's dilemma alone justify the journey: The prisoner's dilemma is the very model of the social problem. Like a cocktail party or a line at a bank, a cartel is a little society facing the problem of defection from a mutually advantageous arrangement. The single member of an oil cartel ordered to cut back its output for the common good, like the single subject of a government ordered to pay taxes for the common good, has an incentive to cheat, to receive the benefit of a high cartel price or a wealthy society without paying the price in lower output or higher taxes.

The society must either find ways to punish free riders or accept collapse. The history of cartels, unions, and not a few governments is written in the algebra of the prisoner's dilemma.

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