The Strategy of Conflict

With a New Preface by the Author

Thomas C. Schelling

• Publisher: Harvard University Press
• Year: 1981
• ISBN:
  9780674251861
• Topic: Politique et relations internationales
• Subtopic: Politique

A series of closely interrelated essays on game theory, this book deals with an area in which progress has been least satisfactory—the situations where there is a common interest as well as conflict between adversaries: negotiations, war and threats of war, criminal deterrence, extortion, tacit bargaining. It proposes enlightening similarities between, for instance, maneuvering in limited war and in a traffic jam; deterring the Russians and one's own children; the modern strategy of terror and the ancient institution of hostages.

PART II

A REORIENTATION OF GAME THEORY

4

TOWARD A THEORY OF INTERDEPENDENT DECISION

On the strategy of pure conflict — the zero-sum games — game theory has yielded important insight and advice. But on the strategy of action where conflict is mixed with mutual dependence — the nonzero-sum games involved in wars and threats of war, strikes, negotiations, criminal deterrence, class war, race war, price war, and blackmail; maneuvering in a bureaucracy or in a traffic jam; and the coercion of one’s own children — traditional game theory has not yielded comparable insight or advice. These are the “games” in which, though the element of conflict provides the dramatic interest, mutual dependence is part of the logical structure and demands some kind of collaboration or mutual accommodation — tacit, if not explicit — even if only in the avoidance of mutual disaster. These are also games in which, though secrecy may play a strategic role, there is some essential need for the signaling of intentions and the meeting of minds. Finally, they are games in which what one player can do to avert mutual damage affects what another player will do to avert it, so that it is not always an advantage to possess initiative, knowledge, or freedom of choice.

Traditional game theory has, for the most part, applied to these mutual-dependence games (nonzero-sum games) the methods and concepts that proved successful in studying the strategy of pure conflict. The present chapter and the one to follow attempt to enlarge the scope of game theory, taking the zero-sum game to be a limiting case rather than a point of departure. The proposed extension of the theory will be mainly along two lines. One is to identify the perceptual and suggestive element in the formation of mutually consistent expectations. The other (in the following chapter) is to identify some of the basic “moves” that may occur in actual games of strategy, and the structural elements that the moves depend on; it involves such concepts as “threat,” “enforcement,” and the capacity to communicate or to destroy communication.

That game theory is underdeveloped along these two lines may reflect its preoccupation with the zero-sum game. Suggestions and inferences, threats and promises, are of no consequence in the accepted theory of zero-sum games. They are of no consequence because they imply a relation between the two players that, unless perfectly innocuous, must be to the disadvantage of one player; and he can destroy it by adopting a minimax strategy, based, if necessary, on a randomizing mechanism. So the “rational strategies” pursued by two players in a situation of pure conflict — as typified by pursuit and evasion — should not be expected to reveal what kind of behavior is conducive to mutual accommodation, or how mutual dependence can be exploited for unilateral gain.

If the zero-sum game is the limiting case of pure conflict, what is the other extreme? It must be the “pure-collaboration” game in which the players win or lose together, having identical preferences regarding the outcome. Whether they win fixed shares of the total or shares that vary with the joint total, they must rank all possible outcomes identically, in their separate preference scales. (And, to avoid any initial conflict, it has to be evident to the players that the preferences are identical, so that there is no conflict of interest in the information or misinformation that they try to convey to each other.)

What is there about pure collaboration that relates it to game theory or to bargaining? A partial answer, just to establish that this game is not trivial, is that it may contain problems of perception and communication of a kind that quite generally occur in nonzero-sum games. Whenever the communication structure does not permit players to divide the task ahead of time according to an explicit plan, it may not be easy to coordinate behavior in the course of the game. Players have to understand each other, to discover patterns of individual behavior that make each player’s actions predictable to the other; they have to test each other for a shared sense of pattern or regularity and to exploit clichés, conventions, and impromptu codes for signaling their intentions and responding to each other’s signals. They must communicate by hint and by suggestive behavior. Two vehicles trying to avoid collision, two people dancing together to unfamiliar music, or members of a guerrilla force that become separated in combat have to concert their intentions in this fashion, as do the applauding members of a concert audience, who must at some point “agree” on whether to press for an encore or taper off together.

If chess is the standard example of a zero-sum game, charades may typify the game of pure coordination; if pursuit epitomizes the zero-sum game, rendezvous may do the same for the coordination game.

An experiment of O. K. Moore and M. I. Berkowitz provides a nice mixture in which the two limiting cases are both visible.¹ It involves a zero-sum game between two teams, each team consisting of three people. The three members of the team have identical interests but, because of a special feature of the game, cannot behave as a single entity. The special feature is that the three members of each team are separated and can communicate only by telephone and that all six telephones are connected on the same line so that everyone can hear both the other team and his own teammates. No prearrangement of codes is permitted. Between teams we have here a pure-conflict game; among the members of the team we have a pure-coordination game.

If in this game we suppress the “other team” and if the three players simply try to coordinate a winning strategy in a game of skill or chance in the face of communication difficulty, we have a three-person pure-coordination game. Several “games” of this sort have been studied, both experimentally and formally; in fact, there is substantial overlap at this point between the nonzero-sum game and organization or communication theory.²

The experiments reported in Chapter 3 showed that coordinated choice is possible even in the complete absence of communication. Further, they showed that there are tacit bargaining situations in which the conflict of interest in the choice of action may be overwhelmed by the sheer need for concerting on some action; in those situations, the limiting case of pure coordination isolates the essential feature of the corresponding nonzero-sum game.

So we do have, in this coordinated problem-solving, with its dependence on the conveyance and perception of intentions or plans, a phenomenon that brings out an essential aspect of the nonzero-sum game; and it stands in much the same relation to it as the zero-sum game, namely, that of “limiting case.” One is the mixed conflict-cooperation game with all scope for cooperation eliminated; the other is the mixed conflict-cooperation game with the conflict eliminated. In one the premium is on secrecy, in the other on revelation.

It is to be stressed that the pure-coordination game is a game of strategy in the strict technical sense. It is a behavior situation in which each player’s best choice of action depends on the action he expects the other to take, which he knows depends, in turn, on the other’s expectations of his own. This interdependence of expectations is precisely what distinguishes a game of strategy from a game of chance or a game of skill. In the pure-coordination game the interests are convergent; in the pure-conflict game the interests are divergent; but in neither case can a choice of action be made wisely without regard to the dependence of the outcome on the mutual expectations of the players.³

Recall the famous case of Holmes and Moriarty on separate trains, neither directly in touch with the other, each having to choose whether to get off at the next station. We can consider three kinds of payoff. In one, Holmes wins a prize if they get off at different stations, Moriarty wins it if they get off at the same station ; this is the zero-sum game, in which the preferences of the two players are perfectly correlated inversely. In the second case, Holmes and Moriarty will both be rewarded if they succeed in getting off at the same station, whatever station that may be; this is the pure-coordination game, in which the preferences of the players are perfectly correlated positively. The third payoff would show Holmes and Moriarty both being rewarded if they succeed in getting off at the same station, but Holmes gaining more if both he and Moriarty get off at one particular station, Moriarty gaining more if both get off at some other particular station, both losing unless they get off at the same station. This is the usual nonzero-sum game, or “imperfect-correlation-of-preferences” game. This is the mixture of conflict and mutual dependence that epitomizes bargaining situations. By specifying particular communication and intelligence systems for the players, we can enrich the game or make it trivial or provide an advantage to one of the two players in the first and third variants.

The essential game-of-strategy element is present in all three cases: the best choice for either depends on what he expects the other to do, knowing that the other is similarly guided, so that each is aware that each must try to guess what the second guesses the first will guess the second to guess and so on, in the familiar spiral of reciprocal expectations.

A RECLASSIFICATION OF GAMES

Before going further, we can usefully reclassify game situations. The twofold division into zero-sum and nonzero-sum lacks the symmetry that we need and fails to identify the limiting case that stands opposite to the zero-sum game. The essentials of a classification scheme for a two-person game could be represented on a two-dimensional diagram. The values of any particular outcome of the game, for the two players, would be represented by the two coordinates of a point. All possible outcomes of a pure-conflict game would be represented by some or all of the points on a negatively inclined line, those of a pure common-interest game by some or all of the points on a positively inclined line. In the mixed game, or bargaining situation, at least one pair of points would denote a negative slope and at least one pair a positive slope.⁴

We could stay close to traditional terminology, with respect to the strictly pure games, by calling them fixed-sum and fixed-proportions games, getting the unwieldy variable-sum-variable-proportions as the name for all games except the limiting cases. We could also call them perfect-negative-correlation games and perfect-positive-correlation games, referring to the correlation of their preferences with respect to outcomes, leaving for the richer mixed game the rather dull title of “imperfect-correlation game.”

The difficulty is in finding a sufficiently rich name for the mixed game in which there is both conflict and mutual dependence. It is interesting that we have no very good word for the relation between the players: in the common-interest game we can refer to them as “partners” and in the pure-conflict game as “opponents” or “adversaries”; but the mixed relation that is involved in wars, strikes, negotiations, and so forth, requires a more ambivalent term.⁵ In the rest of this book I shall refer to the mixed game as a bargaining game or mixed-motive game, since these terms seem to catch the spirit. “Mixed-motive” refers not, of course, to an individual’s lack of clarity about his own preferences but rather to the ambivalence of his relation to the other player — the mixture of mutual dependence and conflict, of partnership and competition. “Nonzero-sum” refers to the mixed game together with the pure common-interest game. And, because it characterizes the problem and the activity involved, coordination game seems a good name for the perfect sharing of interests.

GAMES OF COORDINATION

While most of this book will be about the mixed game, a brief discussion of the pure coordination game, beyond that of Chapter 3, will help to show that this is an important game in its own right and will identify certain qualities of the mixed game that appear most clearly in the limiting case of pure coordination.

Recall the various pure coordination problems of Chapter 3. Each of them evidently provided some focal point for a concerted choice, some clue to coordination, some rationale for the convergence of the participants’ mutual expectations. It was argued there that the same kind of coordinating clue might be a potent force not only in pure coordination but in the mixed situation that includes conflict; and, in fact, the experiments demonstrated that, in the complete absence of communication, this is certainly true. But there are a number of instances in which pure coordination itself — the tacit procedure of identifying partners and concerting plans with them — is a significant phenomenon. A good example is the formation of riotous mobs.

It is usually the essence of mob formation that the potential members have to know not only where and when to meet but just when to act so that they act in concert. Overt leadership solves the problem; but leadership can often be identified and eliminated by the authority trying to prevent mob action. In this case the mob’s problem is to act in unison without overt leadership, to find some common signal that makes everyone confident that, if he acts on it, he will not be acting alone. The role of “incidents” can thus be seen as a coordinating role; it is a substitute for overt leadership and communication. Without something like an incident, it may be difficult to get action at all, since immunity requires that all know when to act together. Similarly, the city that provides no “obvious” central point or dramatic site may be one in which mobs find it difficult to congregate spontaneously; there is no place so “obvious” that it is evident to everyone that it is obvious to everyone else. Bandwagon behavior, in the selection of leadership or in voting behavior, may also depend on “mutually perceived” signals, when a part of each person’s preference is a desire to be in a majority or, at least, to see some majority coalesce.⁶

Excessively polarized behavior may be the unhappy result of dependence on tacit coordination and maneuver. When whites and Negroes see that an area will “inevitably” become occupied exclusively by Negroes, the “inevitability” is a feature of convergent expectation.⁷ What is most directly perceived as inevitable is not the final result but the expectation of it, which, in turn, makes the result inevitable. Everyone expects everyone else to expect everyone else to expect the result; and everyone is powerless to deny it. There is no stable focal point except at the extremes. Nobody can expect the tacit process to stop at 10, 30, or 60 per cent; no particular percentage commands agreement or provides a rallying point. If tradition suggests 100 per cent, tradition could be contradicted only by explicit agreement; if coordination has to be tacit, compromise may be impossible. People are at the mercy of a faulty communication system that makes it easy to “agree” (tacitly) to move but impossible to agree to stay. Quota systems in housing developments, schools, and so forth, can be viewed as efforts to substitute an explicit game with communication and enforcement for a tacit game that has an undesirably extreme “solution.”

The coordination game probably lies behind the stability of institutions and traditions and perhaps the phenomenon of leadership itself. Among the possible sets of rules that might govern a conflict, tradition points to the particular set that everyone can expect everyone else to be conscious of as a conspicuous candidate for adoption; it wins by default over those that cannot readily be identified by tacit consent. The force of many rules of etiquette and social restraint, including some (like the rule against ending a sentence with a preposition) that have been divested of their relevance or authority, seems to depend on their having become “solutions” to a coordination game: everyone expects everyone to expect everyone to expect observance, so that nonobservance carries the pain of conspicuousness. Clothing styles and motorcar fads may also reflect a game in which people do not wish to be l...