Your Move

Prisoner’s Dilemma

by Anatol Rapoport, by Albert M. Chammah
Michigan, 268 pp., $2.95 (paper)

Strategy and Conscience

by Anatol Rapoport
Schocken, 351 pp., $2.95 (paper)

For those who have lost even their elementary algebra, a gentler and more persuasive introduction to game theory than that of the mathematical biologist Anatol Rapoport could hardly be imagined. To know nothing about game theory is to ignore not just an odd corner of recent mathematical enterprise but a way of thinking that claims to bring the turmoil of international conflict in sight at least of the serene infallibility of equations. These books date back as far as 1960 (Fights, Games, and Debates); of the two that have just appeared in paperback, Prisoner’s Dilemma (first published in 1965) is the more technical, Strategy and Conscience (1964) the broader in scope and the more strongly felt in its response to international problems.

Game theory consists in an analysis of the courses open to the participants in a situation of conflicting interests, the model being a game with strictly defined rules and possibilities; and since the methods now available can cope only with a small number of possibilities the gap between analyzable games and real affairs is enormous, unless drastically simplifying assumptions are made about the real affairs. Before mathematics can handle some situations, moreover, the value to each player of the outcome of each complete play of the game must be expressed numerically; even if the gains and losses take intangible forms like loyalty or prestige, they must still be put into an order of preference and the extent of the preference then quantified.

How far anything like psychological accuracy can be reached in this way must be moot, but rough and ready approximations can of course be made. (The danger remains that the roughness of the approximation may be lost to sight in the exactness of the resulting figures—numbers being precise regardless of the confusion and vagueness from which they emerge.) With the game defined and the payoffs quantified, the possibilities open to the players can be specified, and in many types of game the most advantageous (or least disadvantageous) strategy for each player can be rigorously deduced.

At this point the non-mathematician begins to ask, Why do it? The elaborations can become extremely intricate even within the bounds of very simple games: for instance, more than two players can be introduced, with the resulting possibility of alliances and shifts of alliance; or the results of each play of the game may be stated only as a probability, so that risk-taking or gambling enters into the choice of strategy. The mathematical analyses are not, Dr. Rapoport insists, meant to guide us in playing any game or winning in any competitive situation. When he does extrapolate from games to political situations it is always with cautious provisos. Only rarely does anything emerge of such practical relevance as the calculation of a power index (the Shapley index) for each player in a game with many players who are free to form coalitions:

It is commonly agreed, for example, that the president of the United States formally commands one-sixth of …

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Morality and Game Theory July 2, 1970