Your Move

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 the whole Congress in matters of passing legislation: the difference between the simple majority needed to pass or defeat legislation and the two-thirds majority needed to override the president’s veto…. Luce and Rogow have investigated the results of applying the Shapley index to the distribution of power in the United States tri-cameral legislature (House, Senate, and president) under the assumption that certain legislators always vote with their party and others are always free to defect. The results depend on the proportion of the die-hards and the mugwumps, on whether the president’s party is in the majority or in the minority, and on whether the president himself is a die-hard or a mugwump. One interesting conclusion is the following: the president’s power is greatest when his own party controls about 55 per cent of Congress. [Fights, Games, and Debates]

A good many of us can do no more than skim Dr. Rapoport’s mathematics with trustful detachment, waiting for the outcome which, unlike many other mathematicians, he does try heroically to express in ordinary language. The outcome is likely to be sound, we argue, not because we can check the reasoning ourselves but because our society has its specialized subgroup of mathematicians who have reason to check one another’s work. (Indeed, Morton D. Davis in Game Theory: A Nontechnical Introduction, Basic Books, 222 pp., $6.95, pounces gleefully on a small error of Rapoport’s.) Yet we would not want to be offered the conclusions without seeing something of the argument that produces them even if its intricacies are too teasing to follow. Dr. Rapoport is ideal in giving both the mathematical reasoning and a rough translation.

Besides this, he makes no secret of being passionately concerned with international politics. Part of Fights, Games, and Debates and a very large part of Strategy and Conscience deal directly with the conflict between the United States and Russia, trying to give a fair picture of their differences of conviction and pleading for toleration and ideological coexistence. But here what is so striking, especially in the latter book, is that the preceding exposition of game theory contributes nothing to the practical discussion. Being an authority in this mathematical field may of course have given Dr. Rapoport confidence to develop and publish his views on Russo-American relationships—views that are often especially sensitive and perceptive because of his personal knowledge of both cultures—but game theory provided no essential intellectual step toward reaching them, nor can its rigorous procedures put them to the test. Rapoport makes no false claims to the contrary.

We have spelled out the severe and, in our opinion, insuperable limitations of game theory as a prescriptive theory of rational decision in conflict situations. Wherein, then, does its usefulness lie?… We shall argue that the usefulness of game theory is somewhat akin to the usefulness of psychology and also, incidentally, that the usefulness of psychology is not that of a “know-how” science, whatever the imagined uses of psychology may be…. If the manipulative applications of psychology are discounted, there still remains a tremendous potential of psychology as a science which can enhance man’s insight into himself. This value tends to be ignored in a climate where manipulative techniques are held to be the most valuable product of knowledge.

Like psychology, game theory can be a source of ideas which lead to insights—insights into the nature of conflict based on the interplay of decisions.

[Two-Person Game Theory]

On the other hand, Dr. Rapoport makes a positive contribution to practical thinking by refusing to countenance the kind of mathematical approach which reshapes the complex psychopolitical world to fit the outline of game theory and “rationally” assesses for military and political purposes the outcome of mutual efforts at mass destruction. With the amiability that he can never quite relinquish he is prepared to concede that the strategists may have decent human feelings like the rest of us in their private lives, but in their specialized role they are anathema to him. While Kahn thinks about the unthinkable, Rapoport insists on thinking about the unfeeling and maintains that a “rationality” which ignores human feeling about the destruction of human beings is fatally flawed. From the writings of nuclear age strategists, he suggests, one gets the impression of a revival of the formal military strategy that had its golden age in eighteenth-century Europe:

…one finds in the writings of contemporary strategists a deliberate striving to rehabilitate was as a normal event among civilized nations. The re-establishment of high intellectual content in military strategy doubtless serves this purpose. In my opinion, the tremendous interest aroused by game theory is in no small measure due to the climate in which the rehabilitation of war, or at least of the sophisticated power struggle, was undertaken…. Since rationality in conflict enjoys extremely high prestige in our day when “realism” and “tough-mindedness” are extolled as evidence of sophistication and maturity, game theory can indeed be sold as a useful science.

But the genuine game theorists, he adds,

…eschew claims to the effect that a formal training in game theory will soon become as necessary to a military strategist as a formal training in physics is necessary to an engineer. Sanguine prognoses for game theory as an adjunct to military science are usually traceable to the military profession and to its fringes, not to the game theoreticians.

However, “conscience” need not stand by itself against “strategy.” Rapoport argues that the strategists’ apparent rationality conceals false assumptions: they have oversimplified the international situation not merely by shedding details and complications but by bringing it under the wrong rubric in game theory. To them it is a “zero-sum” game, one in which any gain by one player must involve a corresponding loss to the other. In the theory of such games two further assumptions are necessary if mathematical analysis is to be possible: one is that each player is fully rational (that he fully understands the game and will always make the best possible move), the other that his motives are entirely unmixed, that he wants to do the best he can for himself with complete disregard for the effect on the other player.

Perhaps an all-out war for survival is something like a zero-sum game, but it would be a travesty of peacetime international relations, bad as they are, to suppose that they are represented by the same model. If all-out war came it would not be a continuation of diplomacy by other means, but a switch to the zero-sum game from a different type of competition, one in which some elements of good will for the opponent are retained and annihilation is not the object.

From this point two paths open out, only one of which Dr. Rapoport explores. He proceeds to another type of game in which cooperation between the players can result in modest gains to each, though if one player “defects” when the other cooperates he will, on that play of the game at any rate, greatly increase his winnings at the cost of the other player. Its paradigm is the anecdote of Prisoner’s Dilemma: two prisoners charged jointly with a crime are held incommunicado and neither can be convicted unless at least one confesses; if only one confesses he is freed and given a reward while the other gets a heavier penalty than he would have had if both had confessed.

For experimental purposes the procedure takes the form of two people (students) playing against each other for money and not allowed to communicate. If, in any one play of the game, they each make the “cooperative” move they will both win money (from the experimenter’s fund); if they both simultaneously “defect” they will both lose money (their own); and if one defects when the other cooperates the defector will win money with a corresponding loss to the cooperator. By arbitrary variations in the payoffs the temptation to defect and the inducement to cooperate can be made great or small. In considering parallel situations in real life we should have to say, I suppose, that the experimenter’s fund from which both can gain corresponds to an expanding market which business competitors can share or natural resources which can be cooperatively developed to the mutual benefit of two nations.

The experimental findings are naturally of less interest to Dr. Rapoport than the mathematical problems of stating the intricately varied sequences of play that may develop. There are, for instance, interesting “martyr” runs in which one player will persist, despite losses, in playing cooperatively against his opponent’s constant defection; sometimes these runs succeed in converting the opponent to cooperation, but failures outnumber successes by about 2 1/2 to 1. However, this is encouraging if we can believe that there are only two-and-a-half times as many nasty people as nice ones—and even some of the baddies had just been converted, when the martyrs gave up their self-sacrificing efforts and defected themselves.

The general finding was that with a run of 300 plays of the game more than half of the pairs of men had by the end “locked in” on a cooperative form of play. Women players were consistently less cooperative, though in the complex situation of this experiment it is not clear why. The game in fact offers endless possibilities in experimental social psychology, although as it becomes widely known there will be increasing difficulty in finding the completely uninformed subjects who are necessary.

Dr. Rapoport’s mathematically sophisticated and experimentally stimulating exploration of the Prisoner’s Dilemma type of game is only one of two paths that diverge at the point where we note the limited relevance for real-life conflict of the zero-sum game. The other seems to lead away from game theory as it now is. It starts with the recognition that there may be good will as well as rivalry between people and between nations. This, it is true, was allowed for by Lewis F. Richardson in his early attempts in the 1930s to quantify and study mathematically various factors contributing to the outbreak and changing phases of war between nations. (Although Rapoport of course sees game theory as effectively stemming from von Neumann’s work in economics, he treats Richardson as the pioneer in mathematical analyses of international conflict.)

Richardson, however, saw good will only at the opposite end of a scale from hostility, something that reduces the quantitative strength of hostility. This scarcely does justice to the divided feelings, the contradictory attitudes, that may occur when we resort to hostility—either nationally or personally—against those for whom we also have liking. The acts of hostility—the prosecution of war, the business deal that damages a likable competitor—can be partly regretted without being moderated. This is what makes human conflict different from whole-hearted aggression against natural obstacles, the stubborn tree root, the boulder blocking a path. We are willynilly a social species and perversely, in spite of all we know, have some liking for each other and some regret at ending all possibility of companionship. Although attempts at genocide and the extermination of categories of a population may be made during psychopathological phases of a nation’s history, they are rationalized at the time and they engender shame afterward. The English take no satisfaction in what Cromwell did at Drogheda and the Russians and Germans have their sources of uneasiness too. In spite of its appearance of uninhibited violence the curious institution of war has for many centuries been regarded by both sides as a step toward establishing peaceful coexistence on satisfactory terms.

This kind of view may seem sheer nonsense when we think of the cruelty and destructiveness perpetrated daily by individual hoodlums and highly civilized states. But these things happen because there seem to be plenty of people to spare and we forget the essential value of each. If the last American and the last Russian happened to meet after the holocaust each would be glad to see a human face. It is not that aggression is unnatural: some struggle to preserve biological existence is automatic, and from this there is a gradual escalation, through the protecting of property and friends, with policing, defense forces, deterrents, and retaliations, up to the final annihilation, without any logical stopping point. But liking and mutual dependence are also fundamental biological patterns and they too escalate by imperceptible stages toward utter devotion and total self-sacrifice in the interest of someone else, equally without a logical stopping place. Peter’s early attempt at quantification came to grief, though his question “Lord, how oft shall my brother sin against me, and I forgive him? till seven times?” must have seemed a reasonable one to put to the teacher who had taken a scourge to the moneylenders in the temple.

It is the conflict within the competitor which game theory, as far as I can follow, has not dealt with. A concern for the other person is of course quite different from the enlightened self-interest that allows the competitors in Prisoner’s Dilemma to cooperate for their common advantage. The element of regret at damaging one’s opponent makes most real-life conflict, paradoxically, less ruthless than games. In a game you can, with a clear conscience if you stick to the rules, try to maximize your gains in complete disregard of your opponent’s losses. This is because the players, in spite of competing, are engaged jointly in a cooperative enterprise: putting their skill or strength or intelligence to the test of a worthy opponent (sometimes with the added variety of luck, the dummy player).

This testing of oneself is an inclusive motivation, over and above the wish to win; hence the fact that people who more often lose than win continue to play and that most of us find little pleasure in winning against a beginner who makes an unworthy opponent. But in real life the good will and compassion which are natural in the overwhelming majority of us persistently interfere with the single-minded infliction of damage on others. We need not go all the way with Kropotkin’s views on the extent of mutual aid in animal species, but the more recent work of ethologists on the innate behavioral mechanisms by which aggression is inhibited before fatal injuries have been inflicted suggests that complete ruthlessness, if we ever achieve it, is one of our cultural attainments. Game theory implicitly carries the ruthlessness of true games into real life.

None of these problems will prevent it from having its contemporary success and we can be sure that its vocabulary—utilities, equilibrium points, mixed strategies, Pareto optimal, minimax play, and so forth—will find a place in the repertory of every business consultant under forty. For his purposes, Dr. Morton Davis’s book is excellent, contenting itself with the usual view that “players don’t look to the game theorists for moral principles; they already have their own. All they ask for is to find a strategy that will suit their purpose, selfish or otherwise.” It is Rapoport’s distinction that he feels disquiet about this orthodox position. For one thing, he doubts whether game theory will in practice help anyone to compete more successfully. More profoundly, and I think rightly, he suspects that the apparent neutrality of the theory conceals moral assumptions of its own.