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CONFLICT RESOLUTION

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[L.C.]

# The strategy of conflict
Prospectus for a reorientation of game theory

THOMAS C. SCHELLING

*Harvard University*®

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 non-zero-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 a social hierarchy 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 (non-zero-sum games) the methods and concepts that proved successful in

studying the strategy of pure conflict. The present paper attempts 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 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 relationship 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.

® The author is currently with The RAND Corporation, on leave from Harvard University.

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THOMAS C. SCHELLING

I. Mutual Perception and
Suggestive Behavior

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, hav-
ing identical preferences regarding the out-
come. Whether they win fixed shares of the
total or shares that vary with the joint to-
tal, 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 prefer-
ences are identical, so that there is no con-
flict of interest in the information or misin-
formation that they try to convey to each
other.)

What is there about pure collaboration
that relates it to game theory or to bargain-
ing? 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 occurs
in non-zero-sum games. Whenever the
communication structure does not permit
players to divide the task ahead of time ac-
cording to an explicit plan, it may not be
easy to co-ordinate behavior in the course of
the game. Players have to understand each
other, to discover patterns of individual be-
havior that make each player's actions pre-
dictable to the other; they have to test each
other for a shared sense of pattern or regu-
larity and to exploit clichés, conventions,
and impromptu codes for signaling their in-
tentions and responding to each other's sig-
nals. They must communicate by hint and
by suggestive behavior. Two vehicles trying
to avoid collision, two people dancing to-
gether 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 en-
core or taper off together.

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

An experiment of O. K. Moore and M. I.
Berkowitz provides a nice mixture in which
the two limiting cases are both visible. It
involves a zero-sum game between two
teams, each team consisting of three people.
The three members of the team have iden-
tical interests but, because of a special fea-
ture 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 pre-
arrangement of codes is permitted. Between
teams we have here a pure-conflict game;
among the members of the team we have a
pure-co-ordination game (25).

If in this game we suppress the "other
team" and if the three players simply try to
co-ordinate a winning strategy in a game of
skill or chance in the face of communica-
tion difficulty, we have a three-person pure-
co-ordination game. Several "games" of this
sort have been studied, both experimentally
and formally; in fact, there is substantial
overlap at this point between the non-zero-
sum game and organization or communica-
tion theory.¹

The author's experiments, reported in an
earlier issue of this *Journal*, have shown that

¹ An extensive formal analysis of the co-ordination problem is developed by Marschak (22, 24) and with Roy Radner (23). Examples of interesting and relevant empirical work can be found in Bavelas (2), Heise and Miller (14), Mueller (20), and Carmichael, Hogan, and Walter (4).

CONFLICT RESOLUTION

VOLUME II NUMBER 3

THE STRATEGY OF CONFLICT

205

co-ordinated choice is possible even in the
complete absence of communication. Fur-
ther, they showed that there are tacit bar-
gaining situations in which the conflict of
interest in the choice of action may be over-
whelmed by the sheer need for concerting
on some action; in those situations, the lim-
iting case of pure co-ordination isolates the
essential feature of the corresponding non-
zero-sum game (31).

So we do have, in this *co-ordinated problem-solving*, with its dependence on the conveyance and perception of intentions or plans, a phenomenon that brings out an essential aspect of the non-zero-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-co-operation game with all scope for co-operation eliminated; the other is the mixed conflict-co-operation 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-co-ordi-
nation game is a *game of strategy* in the
strict technical sense. It is a behavior situa-
tion 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 pre-
cisely what distinguishes a game of strategy
from a game of chance or a game of skill.
In the pure-co-ordination 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.²

Recall the famous case of Holmes and
Moriarty on separate trains, neither direct-
ly in touch with the other, each having to
choose whether to get off at the next sta-
tion. 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-co-ordination 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 non-zero-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 pro-

² Concerning this point, Kaysen (16) in his review of the Von Neumann and Morgenstern book says: "The theory of such games of strategy deals precisely with the actions of several agents, in a situation in which all actions are interdependent, and where, in general, there is no possibility of what we called parametrization that would enable each agent (player) to behave as if the actions of the others were given. In fact, it is this very lack of parametrization which is the essence of a game." Similar language is used by Luce and Raiffa (21, p. 14): "Intuitively, the problem of conflict of interest is, for each participant, a problem of individual decision making under a mixture of risk and uncertainty, the uncertainty arising from his ignorance as to what the others will do." Their preoccupation is with the conflict, however; the case of coincident preferences among the players they dispose of as trivial (pp. 59, 88), and they deal with such players as a single individual (p. 13).

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THOMAS C. SCHELLING

vide 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 simi-
larly 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 ex-
pectations.

A RECLASSIFICATION OF GAMES

Before going further, we can usefully re-
classify game situations. The twofold divi-
sion into zero-sum and non-zero-sum lacks
the symmetry that we need and fails to
identify the limiting case that stands oppo-
site to the zero-sum game. The essentials of
a classification scheme for a two-person
game could be represented on a two-dimen-
sional diagram. The values of any particular
outcome of the game, for the two players,
would be represented by the two co-ordi-
nates of a point. All possible outcomes of a
pure-conflict game would be represented by
some or all of the points on a negatively in-
clined line, those of a pure common-interest
game by some or all of the points on a posi-
tively 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.³

We could stay close to traditional termi-
nology, 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-cor-
relation games, referring to the correlation
of their preferences with respect to out-
comes, leaving for the richer mixed game

the rather dull title of "imperfect-correla-
tion game."

³ If the nature of the game makes it desirable for a player to use a random device in the choice of his strategy, or feasible for the players to negotiate an enforceable agreement that, like a drawing of lots, depends on a chance mechanism, there may be room for co-operation in the choice of *strategies* even when there is perfect disagreement over the ranking of *outcomes*. In that case the points representing the pure-conflict game must meet the tighter restriction of lying on a straight line, with the two axes measuring the players' "utilities" in the sense now familiar in game theory. This restriction also applies to the pure common-interest game, since players who agree perfectly on the ranking of *outcomes* may not agree on the desirability of, say, one particular point over a fifty-fifty chance between the two points immediately above and below it. Thus "strictly pure" conflict and common-interest games, providing no scope for collaboration in the one case and no scope for disagreement in the other, would have to show the *expected values* of all pertinent mixed (random) strategies lying along the downward-sloping and upward-sloping lines, respectively, with axes measured in "utility units" of the kind mentioned; this in turn means that the points denoting *outcomes* must lie on a *straight line*.

Also, the pure games cannot admit “side pay-
ments.” If one of the partners in a pure common-
interest game threatens to sabotage the effort
unless he is paid—assuming that the communica-
tion and enforcement structure of the game
makes this possible—a conflict of interest is in-
troduced; in effect, the point denoting the pay-
ment of a bribe would appear to the upper left
or lower right of another point or points on the
upward-sloping line, producing the configura-
tion of a mixed game. And if one of the players
in a pure-conflict game can threaten damage or
offer compensation to induce his opponent to
yield in this game, there is scope for bargaining;
there is no longer a relationship of pure conflict,
and the points denoting the threatened damage
or promised compensation would lie off the
downward-sloping line. In other words, all per-
tinent potential outcomes must be allowed for.
(Two simultaneous pure-conflict games, even if
they meet the restriction of straight lines, pro-
vide room for negotiation unless the slopes of the
two lines happen to be identical.)

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THE STRATEGY OF CONFLICT

207

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 *relationship* 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 relationship that is involved in wars, strikes, negotiations, and so forth, requires a more ambivalent term.⁴ In the rest of this paper 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 relationship to the other player—the mixture of mutual dependence and conflict, of partnership and competition. “Non-zero-sum” refers to the mixed game together with the pure common-interest game. And because it characterizes the problem and the activity involved, *co-ordination game* seems a good name for the perfect sharing of interests.

GAMES OF CO-ORDINATION

While most of this paper will be about the mixed game, a brief discussion of the

pure co-ordination game is in order, partly to indicate that it is an important game in its own right, partly to identify certain qualities of the mixed game that appear most clearly in this limiting case.

An illustration of the game—and the fact that people can play it with success—is the following experiment recorded by the author (31, pp. 21–22) in an earlier issue of this *Journal*. Forty-two people were asked to choose heads or tails without any communication among themselves. Each participant was instructed to try to concert with the rest, success being represented by the proportion of the total sample giving the same answer that he did. Evidently on the basis of chance a member of the 42-person sample might have expected to concert his choice successfully with about 20 or 21 other participants.

Thirty-six chose heads. The probability of as many as 36 heads or tails in a sample of 42, on the hypothesis that people could do no better than chance, is less than .0001. In similar fashion respondents were able without communication to concert their choices of points on a map, candidates on a ballot, numbers in a series of numbers, places to meet in a specified city, or a division of a sum into two piles. When instructed, “Write some positive number—if you all write the same number, you win,” out of the infinity of possible numbers to choose, 40 per cent were able to pick the same number. Each of the problems evidently provided some focal point for a concerted choice, some clue to co-ordination, some rationale for the convergence of the participants’ mutual expectations.

In the earlier article it was argued that the same kind of co-ordinating clue—usually a “non-mathematical” clue—might be a potent force not only in pure co-ordination but in the mixed situation that includes conflict; and, in fact, the experiments demon-

⁴ It deserves to be emphasized that non-zero-sum games can as properly be classed under the-ory of partnership as under theory of conflict; and, for providing insight into problems like that of limiting war, there is merit in using words that bring out the common interest of the ad-versaries and the “bargaining process” involved in the military maneuvers themselves. As the au-thor has pointed out in another paper (34), even the problem of surprise attack is logically equiv-alent to a problem in partnership discipline. If *theory of games* has become endowed with a too conflict-oriented connotation, perhaps something like *theory of interdependent decision* would be a neutral term that equally covers the two limit-ing cases as well as the mixed case.

208

THOMAS C. SCHELLING

strated that, in the complete absence of communication, this is certainly true. But there are a number of instances in which pure co-ordination 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 co-ordinating 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.⁵

Excessively polarized behavior may be the unhappy result of dependence on tacit

co-ordination 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.⁶ 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 co-ordination 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, etc., 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 co-ordination 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 re-

⁶ The phenomenon, called "tipping," is analyzed by Grodzins (12). A more innocuous example of explosively convergent expectations, based on tacit communication that has an almost electric quality, is the snicker that ignites an outburst of uncontrollable laughter in a nervous crowd.

⁵ A closely related phenomenon is appreciated by the person who tries to blend into the crowd to avoid being called on to recite, picked on by a bully, or singled out for "election" to some post that everybody wants to escape.

THE STRATEGY OF CONFLICT

209

straint, 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 co-ordination game:
everyone expects everyone to expect every-
one to expect observance, so that non-ob-
servance carries the pain of conspicuous-
ness. Clothing styles and motorcar fads may
also reflect a game in which people do not
wish to be left out of any majority that
forms and are not organized to keep majori-
ties from forming. The concept of *role* in so-
ciology, which explicitly involves the expec-
tations that others have about one's behav-
ior, as well as one's expectations about how
others will behave toward him, can in part
be interpreted in terms of the stability of
"convergent expectations," of the same type
that is involved in the co-ordination game.
One is trapped in a particular role, or by
another's role, because it is the only role
that in the circumstances can be identified
by a process of tacit consent.

A good example might be the *esprit de
corps* (or lack of it) of an army unit or naval
vessel or the value system of a particular
college or fraternity. These are social organ-
isms that are subject to a substantial rate of
replacement but that maintain their own
peculiar identities to an extent that does not
seem to be accounted for by selective or
biased recruitment. The individual charac-
ter of one of these units seems to be largely
a matter of convergent expectations—every-
one's expectation of what everyone expects
of everyone—with the new arrivals' expecta-
tions being molded in time to help mold the
expectations of subsequent arrivals. There
is a sense of "social contract," the particular
terms of which are sensed and accepted by
each incoming generation. I am told that
this persistence of a tradition in a social en-
tity is one of the reasons why the legal iden-
tity of an army division or regiment—its

name and number and history—is often de-
liberately preserved when its strength has
fallen to where abolition might seem indi-
cated: the tradition that goes with the legal
identity of the group is an asset worth pre-
serving for a future buildup. It may be the
same phenomenon that makes it possible to
collect income tax in some countries and not
in others: if appropriate mutual expecta-
tions exist, people will expect evasion to be
on a scale small enough not to overwhelm
the authorities and may consequently pay
up either out of a sense of reciprocated hon-
esty or out of fear of apprehension, thus to-
gether justifying their own expectations.

**Nature of the intellectual process in co-ordination.** It should be emphasized that co-ordination is not a matter of guessing what the “average man” will do. One is not, in tacit co-ordination, trying to guess what another will do in an objective situation; one is trying to guess what the other will guess one’s self to guess the other to guess, and so on ad infinitum. (“Meeting” someone in the personal column of a newspaper is a good example.) The reasoning becomes disconnected from the objective situation, except insofar as the objective situation may provide some clue for a concerted choice. The analogy is not just trying to vote with the majority but trying to vote with a majority when everyone wants to be in a majority and everyone knows it— not to predict Miss Rheingold of 1958 but to buy the stock or real estate that everyone expects everyone to expect everyone to buy. Investment in diamonds may be a perfect example; the greatest example of all may be the monetary role of gold, which can perhaps be explained only as the “solution” of a co-ordination game. (A common household version of the co-ordination game occurs when two people are cut off in a telephone conversation; if they both call back, they only get busy signals.)

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210

THOMAS C. SCHELLING

Consider the game of "name a positive number." Experiments demonstrate that most people, asked just to pick a number, will pick numbers like 3, 7, 13, 100, and 1. But when asked to pick the same number that others will pick when the others are equally interested in picking the same number, and everyone knows that everyone else is trying, the motivation is different. The preponderant choice is the number 1. And there seems to be good logic in this: there is no unique "favored number"; the variety of candidates like 3, 7, etc., is embarrassingly large, and there is no good way of picking the "most favorite" or most conspicuous.

FIG. 1

If one then asks what number, among all positive numbers, is most clearly unique, or what *rule of selection would lead to unambiguous results*, one may be struck with the fact that the universe of all positive numbers has a "first" or "smallest" number.⁷

**Game-theory formulation of the co-ordination problem.** The payoff matrix for a pure co-ordination problem would look something like that in Figure 1. One player chooses a row, the other a column; and they receive the rewards denoted by the numbers contained in the cell where their choices intersect. If to each choice of one player there corresponds a single choice for the other that "wins" for both of them, we can arrange columns so that all the winning cells lie along the diagonal. In those cells there are positive payoffs to

both players, in the rest we can put zeros. (For our present purpose there is nothing lost by letting a single number stand in each cell for the payoff to both players.)

But we must rule out a possible axiom that might seem to be suggested by analogy with other game theories, namely, that (to use the term of Luce and Raiffa) the "labeling" of rows, columns, and players should make no difference to the outcome.⁸

⁷ There is a widely quoted passage in Keynes (17, p. 156) that may be worth repeating in order to point out that, while it deals with exactly the problem dealt with here, its conception of the "solution" is *not* at all the same: "Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preference of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one's judgment, are really the prettiest, nor even those which average opinion genuinely thinks prettiest. We have reached the third degree where we devote our intelligence to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the fourth, fifth, and higher degrees." This class of games demonstrates, incidentally, that the usual correlation between parametric behavior and large numbers does not hold for tacit play. To adapt "parametrically" to the behavior of others requires that their behavior be observable, not conjectural; the non-parametric character of tacit co-ordination remains, no matter how large the number of players.

⁸ Labeling of the *players* is explicitly ruled out by Luce and Raiffa (21, pp. 123-27) in discussing co-operative games, and in effect is ruled out by Nash (26, 27) in his symmetry assumption. Labeling of *strategies* for tacit or explicit non-zero-sum games is implicitly precluded by dealing only with games in normal form, i.e., the abstract version of them as represented by a payoff matrix (which is itself an *analytical* de-

THE STRATEGY OF CONFLICT

211

It is precisely because strategies are “labeled” in some sense—i.e., have symbolic or connotative characteristics that transcend the mathematical structure of the game—that players can rise above sheer chance and “win” these games; and it is for that same reason that these games are interesting and important.

Even the game portrayed in Figure 1, which might seem to have a minimum of symbolic significance attached to rows and columns, is not a hard one to “win,” i.e., for players to do substantially better on than chance would suggest. (If we give that same game an infinite series of rows and columns, it seems to become easier rather than harder. In that case it is formally identical with the game mentioned earlier, “Pick a positive number,” but, because the “labeling” is different, there is less tendency for minorities to congregate at 3, 7, 13, etc.) Just forming the matrix prejudices the choice, since it focuses attention on “first,” “middle,” “last,” and so forth.⁹ If strategies are not given sequential labels, i.e., labels that can be ordered like numbers and alphabets, but are given individual names, and these are not presented in

vice, not part of the game, and hence provides no left-right, upper-lower, or numerical ordering of the actual strategies). A good example in which the labeling of players is the controlling factor is the interrupted telephone call mentioned earlier, with the problem of who should call back and who should wait for the call.

⁹ This point is typical of a number of demonstrations in the author's experiments reported earlier, to the effect that the postulate regarding the “independence of irrelevant alternatives” cannot be credited in the tacit game and, for analogous reasons, should not be expected to hold in the explicit bargaining game. Potential outcomes can be relevant to the co-ordination of choice, though not themselves near to being chosen. For a statement and discussion of this postulate see Luce and Raiffa (21, p. 127).

any particular order, it is the names that must co-ordinate choice.

And here it becomes emphatically clear that the intellectual processes of choosing a strategy in pure conflict and choosing a strategy of co-ordination are of wholly different sorts. At least this is so if one admits the “minimax” solution, randomized if necessary, in the zero-sum game. In the pure-co-ordination game, the player's objective is to make contact with the other player through some imaginative process of introspection, of searching for shared clues; in the minimax strategy of a zero-sum game—most strikingly so with randomized choice—one's whole objective is to avoid any meeting of minds, even an inadvertent one.¹⁰

¹⁰ Randomized strategies may nevertheless be useful to achieve a co-ordinated distribution of votes, say, among a panel of candidates. If a 55 per cent majority exists and knows that it does, among a hundred voters; if two out of six candidates are congenial to it; and if the three candidates polling the largest numbers of votes become the board of directors, there is danger that unco-ordinated polling may concentrate too many votes on the first (or second) majority choice, leaving the minority two winning candidates with 22 votes apiece. But if each member of the majority flips a coin to cast his vote for one of his party's men, the likelihood of one's getting as few as 22 votes is only one chance in six. If the minority, too, lacks an overt means of collaborating and relies on a chance device, the majority's chances are excellent.

A partial randomized strategy may also be used to reduce an area of conflict. Suppose two people sit at North and East sides of a card table, are to move to another card table adjacent that is identically oriented, must choose without communication what seats they will take at the other table, and will win prizes of $1 apiece if they pick adjacent seats. This is an easy co-ordination problem; but let us subvert the incentives, by giving an additional $2 premium to the player who is on the other's right in the event they succeed in sitting next to each other. This game has no equilibrium point; interests do not converge; there is no seating arrangement that would not

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THOMAS C. SCHELLING

To illustrate, suppose that I am to name one card in an ordinary deck of fifty-two and you are to guess which one I name. Traditional game theory gives guidance on how to make my choice on the assumption that I do not want you to outguess me; I can select at random and defy you to have a better than random chance of guessing what I name. But if the game is that I do want you to guess correctly and you know that I will try to pick one that facilitates your guess, the random device can only guarantee to make tacit co-operation impossible. Holmes can *destroy* the labeling of the stations by flipping a coin to decide where to get off the train; and Moriarty has only a fifty-fifty chance of guessing a coin. But in the common-interest version they must somehow *use* the labeling of the stations in order to do better than pure chance; and how to use it may depend more on imagination than on logic, more on poetry or humor than on mathematics. It is noteworthy that traditional game theory does not assign a "value" to this game: how well people can concert in this fashion is something that, though hopefully amenable to systematic analysis, cannot be discovered by reasoning a priori. This corner of game theory is *inherently* dependent on empirical evidence.¹¹

It should particularly be noted that to assert the influence of "labels"—i.e., of the symbolic and connotative details of the

give one an incentive to move. (Each may wish that he could promise to sit on the other's left, but cannot.) A random strategy yields each player a minimax value of $1. But if each decides where he would sit in the pure common-interest game, then flips a coin to see whether he does sit there or sits opposite, the players guarantee that they neither choose the same seat nor sit opposite each other and share equal chances of winning the premium. This is an equilibrium pair of (mixed) strategies, worth an expected value of $2 apiece.

game—and the dependence of the theory on empirical evidence does not involve the question of whether game theory is predictive or normative—concerned with generalizations about actual choice or the strategy of correct choice. The assertion here is *not* that people simply *are* affected by symbolic details but that they *should* be for the purpose of correct play. A normative theory must produce strategies that are at least as good as what people can do without them. More, it must not deny or expunge details of the game that can demonstrably benefit two or more players and that the players, consequently, should not expunge or ignore in their mutual interest. Two

¹¹ Incidentally, in cases like this we need only to consider the question of what *price* players would pay for a bit of co-ordinating information and what different information patterns yield what chances of co-ordinating to find ourselves in the middle of Marschak's *theory of teams* (22, 23, 24).

There is also a version of "prisoners' dilemma" for this game: two accomplices, apprehended before their alibi is prepared and interrogated separately, who must concert the alibis they invent or be revealed in their guilt. A tantalizing variant can be built by supposing that confession carries a lighter sentence than unconfessed guilt; each player has a "minimax" strategy of confession and must not only consider which particular alibi constitutes the *best* alibi strategy but *how good it is* (in terms of likely coincidence with his partner's) and whether they share the decision to try it. The matrix might be:

<table><tr><td>.5</td><td>0</td><td>0</td><td>0</td></tr><tr><td>.5</td><td>.5</td><td>.5</td><td>.5</td></tr><tr><td>.5</td><td>1</td><td>0</td><td>0</td></tr><tr><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>.5</td><td>0</td><td>0</td><td>1</td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td></tr></table>

(Lower left entry in each cell is payoff to player choosing row, upper right to player choosing column.)

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THE STRATEGY OF CONFLICT

213

couples jockeying for space on a dance
floor or two armies jockeying for a truce
line may jointly suffer from decision proc-
esses that are limited to the abstract prop-
erties of the situation.

A particular implication of this general
point is that the game in "normal" (math-
ematically abstract) form is not logically
equivalent to the game in "extensive" (par-
ticular) form, once we admit the logic by
which rational players concert their ex-
pectations of each other. It will be argued
later, and is also argued in the author's
earlier paper in this *Journal* (31), that
these same considerations are powerfully
present in explicit bargaining as well. A
terminological implication of these consid-
erations is that "non-co-operative" is a poor
name for the game of tacit co-ordination;
it is desperately co-operative in its own
peculiar way and is still so when we add
conflict and form the tacit mixed-motive
game. (In separate papers the writer has
attempted to demonstrate that certain solu-
tion concepts familiar in game theory can
be given an interpretation in terms of the
co-ordination concept [32, 33].)

SUGGESTION AND MUTUAL PERCEPTION
IN THE MIXED-MOTIVE GAME

Co-ordination-game theory, while inter-
esting in its own right, is interesting mainly
for the light that it sheds on the nature of
the mixed-motive game. The co-ordination
element shows up most strikingly in a pure-
ly tacit game, in which there is neither
communication nor any sequence of moves
by which the two players accommodate
themselves to each other. An example, sim-
ilar to one of those reported in the earlier
article, would be the following.

One player is "located" in Cincinnati, the
other in San Francisco; they have identical
maps of the United States and are to divide
the country between them. Each is to draw

a line dividing the United States into two
parts; the line may be straight or curved,
related or unrelated to physical or political
landmarks. If the two of them divide the
map differently, neither gets anything; but
if they draw identical division lines on their
maps, they are both rewarded. The reward
for each player depends on what is con-
tained in his piece after the division, i.e.,
the piece that contains the city in which
he is located. Let us leave these rewards
vague; they may depend partly on area,
partly on population, partly on industrial
wealth and agricultural resources, etc., and
may differ somewhat for the two players.
In other words, while all terrain is valuable,
not all parts of the country are equally
valuable, and there is no clear specification
of the valuation formula. (There is con-
sequently no means of selecting a perfectly
symmetrical division of values between the
two players.)

In this game there is a compelling prob-
lem of co-ordination; each player can win
only if he does exactly what the other ex-
pects him to, knowing that the other is
similarly trying to do exactly what is ex-
pected of him. They must jointly find a
line that in some fashion suggests itself to
both of them or appeals to both of them.
Neither can "outsmart" the other without
outsmarting himself.

The experiments reported earlier suggest
that players are by no means helpless when
faced with this kind of game. The game
is nowhere near so "infinitely" difficult as
the infinity of possible division lines might
suggest; some variants of the game are not
difficult at all. But a successful outcome
does depend on the kinds of factors that
are controlling in the pure-co-ordination
game; in fact, some games of this sort are
"won" by the two players' choosing exactly
the same outcome as they would have
chosen if the reward system gave them

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THOMAS C. SCHELLING

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identical, instead of conflicting, interests. The problem is to find some signal or clue or rationalization that both can perceive as the "right" one, with each party prepared to be disciplined by that signal or clue in the event that it appears to discriminate against him. They must find their clues where they can. (If the map they are using happens, for example, to contain an embarrassing richness of clues, making it difficult to single out any particular one, a fairly arbitrary line drawn as a suggestion by the referee, identical on both maps, might have to be accepted as a "mediator," even if it is substantially biased toward one of the players.)

But this co-ordination element, especially in the case without conflict, appears to be essentially related to a communication problem. The pure-co-ordination game not only ceases to be interesting but virtually ceases to be a "game" if the players can concert with certainty, without difficulty, and without cost. The question arises, then, how important the co-ordination element can be in mixed-motive games generally, since many of these take the form of overt bargaining with uninhibited speech.

The pervasiveness of the co-ordination principle arises from two separate considerations. One, which will be elaborated in Part III of this paper, is that tacit bargaining provides an analytical model—perhaps only an analogy but perhaps an identification of the actual psychic and intellectual phenomenon—of the "rational" process of finding agreement in pure bargaining situations, those in which both parties recognize that there is a wide range of outcomes preferable to both of them over no agreement at all. The psychic phenomenon of "mutual perception" that can be verified as real and important in the tacit case has a role to play in the analysis of explicit

bargaining. (Co-ordination of expectations will be the related concept.)

Second, many of the bargaining processes or game situations that we want to analyze are at least partly tacit. In some cases, like maneuvering a car in a traffic jam, speech is physically precluded; in others, like developing a modus vivendi with a neighbor, speech is inhibited in the interests of privacy. Illicit bargaining or diplomatic bargaining that would be embarrassing to both sides if overheard by other countries may be less than fully articulate. If the number of players in a game is large, as it is in the bargaining process that determines the racial border lines between residential areas and professions, there may be no institutional provision for explicit negotiation. In these cases, while speech may be part of the bargaining process, actions are also part of it, and the game is one of "maneuver" rather than just talk.

Furthermore, if there are moves available to the players, so that it is an advantage to get on with the maneuver even while negotiating, and particularly if some maneuvers become visible to the other player only after a time lag, there is no reason to suppose that an instantaneous moritorium on maneuver will reign from the outset; in that case, the game progresses while the talk is going on. If the moves had only symbolic significance, we could include them in the communication process along with speech; but, typically, moves have a tactical significance, leaving the game irreversibly different from what it was before, and typically also their tactical significance raises them above the level of pure speech even in their communication content. One may say and say that a gun is loaded without being able to prove it until he actually shoots; one may say and say that he considers an area strategically important

CONFLICT RESOLUTION

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NUMBER 3

THE STRATEGY OF CONFLICT

215

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and not be believed until he incurs expense or risk in its protection. Thus moves can reveal information about a player's value system or about the choices of action available to him; moves can commit him to certain actions when speech often cannot; and moves can often progress at a speed that is determined unilaterally, not dependent on formalities of agreement at a conference.

In other words, bargaining games quite typically involve a dynamic process of mutual accommodation rather than pure communication culminating in a crystallized agreement. The jockeying for limits in limited war is a perfect example, and we might illustrate it by modifying the parlor game described above.

**An illustrative tacit game.** Suppose our two players with their maps of the United States before them are each given 100 chips and told to play a game as follows.¹² At each "move," each player will distribute five chips among states on his

¹² Since it will be proposed later in this paper that such games have, in fact, a research value, as well as an illustrative value, it should be observed at the outset that there is a special problem of motivating the players in an experimental non-zero-sum game. In a zero-sum game, winning is measured relative to one's immediate adversary, and the intellectual challenge and bilateral competition motivate the player toward the correct (and only) type of winning. But for a mixed-motive game, "winning" must be made to involve one's absolute score, not his score relative to that of the person he plays with; the incentives are distorted if the play is dominated by strictly bilateral competition. So, unless real rewards are given, the game has to be organized as a round robin or some such schedule that involves more than two players in a series of two-person plays, with the final outcome decided by the relative position of one's absolute score. (This is why there are no two-person non-zero-sum parlor games.)

map. The moves are compared, and if the two players have put a chip apiece in the same state, those two chips are removed; if one player has put a chip and the other player three chips in the same state, a chip apiece is removed leaving only two chips representing the one player; etc. They do the same at the next move, again with five chips; this time they have the option of placing their chips on states that are yet uncovered or of placing them on states where there are already chips. If A puts two chips on a state in which B previously put a chip, B's is removed along with one of A's, leaving one of A's chips present to "claim" the state. And so the game goes until the players have used up all their chips; it then continues, and at each move a player may transfer up to five chips from the states in which they are to other states, again with equal numbers of chips being removed from a state in which both players have placed chips. This process goes on until both players have notified the referee that they are willing to terminate the game.

Prizes are now distributed. Each player receives a dollar for every one of his chips still on the board, i.e., for those that were not removed when he "took" a state or "lost" it to the other player. He also gets money for the states that he "possesses," these being the states that he has chips on plus those without chips that are in the area containing his home base that is completely inclosed by states that he does have chips on.

These "rewards" for states possessed are specific dollar values attached to each of the 48 states; they vaguely follow a pattern suggestive of, say, "economic worth" or something of the sort. There is no presumption that the values are the same, or even very closely correlated, for the two players; population may be an important

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THOMAS C. SCHELLING

element in the "values" of the states for one of the players and a comparatively unimportant element in the "values" for the other player. Neither player knows the other player's value system—or perhaps knows just a little about it, such as what elements matter but not how much they matter. Each must learn what he can about the other's value system by observing the other player's moves.

Here we have a mixed-motive game, which progresses by a process of mutual accommodation—a series of moves in the course of which the players suffer damage jointly if their accommodation is poor. They may lose dollars by failing to predict where each other will place his chips during the current move, in those cases where they prefer not to lose dollars fighting over a state. Each loses at least a dollar when one takes a state from the other; and they may lose more than a dollar apiece if the one who loses a state attempts to recapture it by putting more chips on it. And not only do they lose a dollar with each dollar forfeited, but each player has fewer "chips" left from the point of view of claiming states; and they may have to leave some states completely unclaimed between them if they have not enough chips left on the board when the game ends.

Now how do the players "bargain" in this game? One way or another, they do in fact make proposals and counterproposals; they accept, reject, retaliate, and even discover ways of conveying threats and promises. But if we deny them any form of speech, they must convey their intentions and their proposals by their patterns of behavior. Each must be alert to what the other is expressing in his maneuvers, and each must be inventive enough to convey his intentions when he wants them conveyed. If one player badly wants a particular state, because it has especially high

value for him, so that he is willing to stick around and fight it out a long time, losing several dollars to the kitty before the other player gives up, it is better for both players that they realize ahead of time which one wants it most badly. And if a player is really prepared to concede a large portion of the country as a "trade" for some other portion that he badly wants, he must not only make it conspicuously available to the other side but must somehow demarcate its limits by his own pattern of play.

But where do the patterns come from? They are not very richly provided by the mathematical structure of the game, particularly since we have purposely made each player's value system too uncertain to the other to make considerations of symmetry, equality, etc., of any great help. Presumably, they find their patterns in such things as natural boundaries, familiar political groupings, the economic characteristics of states that might enter their value systems, Gestalt psychology, and any clichés or traditions that they can work out for themselves in the process of play.¹³

**Explicit communication.** Now let us change the rules so that the players may talk as much as they please. How different would this make the game? In some respects, it should increase the efficiency of the players; particular trades can be identi-

¹³ If my neighbor's fruit tree overhangs my yard and I pick exactly all the fruit on my side of the line, my neighbor can probably discern what my "proposal" is, and has a good idea of what he has acquiesced in for the future if he does not retali ate. But if, instead, I pick that same amount of fruit from both sides of the line hap- hazardly or pick some amount that is related, say, to the size of my family, he is less likely to perceive just what I have in mind. (He may also be more obliged to resist or retali ate if I pick only part of the fruit on my side of the line than if I pick it all, since I have failed to demarcate the limit of my intentions.)

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fied now that were too complex to make proposals about under the more clumsy system. Perhaps, too, the players can avoid some of the inadvertent clashes of chips on the same state, which cost them dollars. We cannot be sure that they will avoid mutually costly competitive bidding for states, since the advantage of being first on a state is great enough to motivate players to keep playing even while they talk. And they have no way to persuade each other that they mean what they say except by showing it in the way they play. (We let them tell each other how they value the states; but we explicitly make fibs unpunishable, and we provide the players no written evidence of their value systems that they could show each other.)

So the introduction of uninhibited speech may not greatly alter the character of the game, even though the particular outcome is different. The dependence of the two players on conveying their intentions to each other and perceiving the intentions of each other, of behaving in predictable patterns and acquiescing in rules or limits, is much the same as before.

The contrast with a zero-sum game and the peculiarly self-effacing quality of a minimax solution is striking here. With a minimax solution, a zero-sum game is reduced to a completely unilateral affair. One not only does not need to communicate with his opponent, he does not even need to know who the opponent is or whether there is one. A randomized strategy is dramatically anticommunicative; it is a deliberate means of destroying any possibility of communication, especially communication of intentions, inadvertent or otherwise. It is a means of expunging from the game all details except the mathematical structure of the payoff, and from the players all communicative relationships.

In chess it does not matter whether the

pieces look like horses, ecclesiastics, elephants, castles, or hamburger buns; whether the game is called "chess," "civil war," or "real estate"; or whether the squares are distorted to look like political or geographical subdivisions. It does not matter what the players know about each other or whether they speak the same language and have a common culture; nor does it matter who played the game previously and how it came out. (If it did matter, one of the players would be motivated to destroy the influence of these details; and a minimax strategy, randomized if necessary, would destroy it.)

But change the payoff matrix in a chess game, making it a non-zero-sum game that rewards the players not only for the pieces they capture but for the pieces they have left over at the end, as well as the squares they occupy, in such fashion that both players have some interest in minimizing the "gross" capture of pieces with its mutual destruction of value. Make each player uncertain about just what squares and what particular pieces the other player values most. And have moves by the clock, so that neither player can hold up the other player's moves for the sake of talking to him.

Now it may make a difference to the players whether we call the game "war" or "gold rush"; whether the pieces look like horses, soldiers, explorers, or children on an Easter egg hunt; what map or picture is superimposed on the playing board and how the squares are distorted into different shapes; or what background story the players are told before they begin.

We have now rigged the game so that the players must bargain their way to an outcome, either vocally or by the successive moves that they make, or both. They must find ways of regulating their behavior, communicating their intentions, letting themselves be led to some meeting of minds,

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THOMAS C. SCHELLING

tacit or explicit, to avoid mutual destruction of potential gains. The "incidental" details may facilitate the players' discovery of expressive behavior patterns; and the extent to which the symbolic contents of the game—the suggestions and connotations—suggest compromises, limits, and regulations should be expected to make a difference. It should, because it can be a help to both players not to limit themselves to the abstract structure of the game in their search for stable, mutually non-destructive, recognizable patterns of movement. The fundamental psychic and intellectual process is that of participating in the creation of traditions; and the ingredients out of which traditions can be created, or the materials in which potential traditions can be perceived and jointly recognized, are not at all coincident with the mathematical contents of the game.¹⁴

The outcome is determined by the expectations that each player forms of how

¹⁴ A good example is the question whether a clear line can be drawn between atomic and other weapons, the answer to which is reported now to be negative if explosive power is the criterion, the explosive ranges having overlapped. But there is nevertheless a difference if enough people think so, and they undoubtedly do. It is a difference constructed of the pure fabric of expectations: it is a ten years' tradition that atomic weapons are different; people believe so and believe others to believe so, and even those who deny the difference will undoubtedly catch their breath, whenever the next one goes off in a war, in a manner they cannot explain by reference to the force of the explosion. It is a purely conventional difference, like the one that makes imprisonment not a "cruel and unusual" punishment or that makes, say, university representation in Parliament perfectly compatible with English democracy if it has always existed but not if it has to be reinstated after a ten years' lapse. The atomic-weapons difference is also one that, probably, can be deliberately reinforced or deliberately blurred over time, as most traditions can.

the other will play, where each of them knows that their expectations are substantially reciprocal. The players must jointly discover and mutually acquiesce in an outcome or in a mode of play that makes the outcome determinate. They must together find "rules of the game" or together suffer the consequences.

A good example of this problem of communicating intentions is that of getting across, persuasively, an intended pattern of retaliation for particular acts that one proposes to consider "out of bounds." Without full communication, one's ability to convey such a pattern of intentions is dependent not only on the contextual materials available for the formation of bounds and limits but on the capacity of the other player to recognize the formula (Gestalt) of retaliation when he sees a sample of it. Historical and literary precedent, legal and moral casuistry, mathematics and aesthetics, as well as familiar analogues from other walks of life, may constitute the menu from which one has to choose his recognizable pattern of retaliation as well as his interpretation of the other's intended pattern. Even with full verbal communication, the situation may not be greatly different; patterns of action may speak louder than words.

Thus the influence that the suggestive details of a game may have on its outcome and the dependence of the players on what clues and signals the game provides are relevant not merely to the study of how players actually do behave in a non-zero-sum game. It is not being argued that players just do respond to the non-mathematical properties of the game but that they ought to take them into account, hence that even a normative theory—a theory of the strategy of games—must recognize that rational players may jointly take advantage of them. And even when one rational player

THE STRATEGY OF CONFLICT

219

realizes that the configuration of these details discriminates against him, he may also rationally recognize that he has no recourse —that the other player will rationally expect him to submit to the discipline of the suggestions that emanate from the game's concrete details and will take actions that— on pain of mutual damage—assume he will co-operate.

**Communicating subjective information.** The role of "expressive moves" in a mutual-accommodation game of this sort is enhanced by the consideration that in mixed-motive games, in contrast to zero-sum-games that are known to the players to be zero-sum, there is likely to be uncertainty about each other's value system. Moves have an *information* content in the mixed-motive game.

Nor can we set up as a general case the bargaining game in which each side has foreknowledge of the other's preferences. To assume that either knows the "true" payoff matrix of the other is often to make an extraordinary assumption about the institutional arrangements of the game. The reason is that certain elements in a bargaining game are *inherently unknowable* for some of the participants, except when there are special conditions. How can we know how badly the Russians would dislike an all-out war in which both sides were annihilated? We cannot; and the reason we cannot is *not* solely that the Russians are necessarily unwilling that we should know. On the contrary, circumstances may arise in which they are desperate that we should know the truth. But how can they make us know it? How can they make us believe that what they tell us is true? How can the prisoner being tortured for secrets that he really does not know persuade his captors that he does not know them? How could the Chinese, if they were really determined to take For-

mosa at the cost of an all-out war, persuade us that they could not be deterred in any fashion and that any threat on our part would only commit us both to all-out war?¹⁵

In special cases the information can be conveyed. In an artificial game, in which each player's "value system" is contained on cards or chips, he may simply turn them face up (if the rules permit or if he and his adversary can jointly cheat against the referee). In a society that believes absolutely in a superior power that will punish falsehood when asked to do so and that everybody knows everybody else believes in, "cross my heart and hope to die" is a sufficient formula for conveying truth voluntarily. But these are special cases. If we are to have a "general case" it must be one in which there is at least some ignorance of each other's value system, or each other's strategy choices, if only because such facts are inherently unknowable or incommunicable.

Von Neumann and Morgenstern (38) illustrated their *solution* concept for the non-zero-sum game with the example of a seller,

¹⁵ The lack of any means of testing the truth is the very basis of that tantalizing game in which each participant attaches positive value to the other's welfare, as when husband and wife discuss whether or not to go to a movie, each wanting to do whatever the other wants to do and wanting to seem to want it himself, knowing that the other is similarly expressing a preference that represents a guess at what one wants to do, etc. There is also an entire domain of game theory involving interpersonal relations in which the overt revelation or recognition of one's value system itself affects values; my awareness that my neighbor does not like me may cause me small discomfort, as does his awareness of my awareness, but if we are forced to accredit the fact overtly, the pain may be acute. "Social etiquette," remarks Erving Goffman (11, p. 224), "warns men against asking for New Year's Eve dates too early in the season, lest the girl find it difficult to provide a gentle excuse for refusing."

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THOMAS C. SCHELLING

A, prepared to sell his house for any price above 10, and two buyers prepared to pay up to 15 and 25, respectively. (My numbers.) The novel part of the solution was that C might pay B a share of his saving if, through B's staying out of the market, C got the house for less than 15. They proposed—and this limitation was inherent in their concept of *solution*—that the most B might receive from C was 15 — 10 = 5. What is interesting about the information requirement of this solution is not that B's reservation price of 15 is something that he might try to misrepresent but that in the ordinary world he could not convincingly communicate the truth if he wanted to. Not only does the "solution" concept—by its assumption of full information—rule out the intrusion of speculators (unless they genuinely want the house enough to give them a basis for sharing in the solution), but it assumes that C can discern, or B can reveal, a subjective truth, one that D and E (speculators who are attracted by the observation that B makes a pure bargaining profit in connection with an object that he never owns before or after) cannot counterfeit.

There are undoubtedly special cases in which one can suppose that the other player is like one's self in basic values and can consequently estimate the other's values by the simple application of symmetry. But in too many exciting cases one plays an opponent who is a wholly different kind of person. The father of a kidnapped boy will not be very successful in guessing what his own bottom price would be if he had been the kidnapper instead; it may not be easy for a British or French officer introspectively to guess how terrible a penalty would have to be to deter him if he were a Mau Mau or an Algerian terrorist. It is hard for a boy to guess how much he would like

himself if he were the girl that he wants to date, or for the customer in the restaurant to know how much he would dislike a scene if he were the waiter instead.

This is one of the reasons why talk is not a substitute for moves. Moves can in some way alter the game, by incurring manifest costs, risks, or a reduced range of subsequent choice; they have an information content, or *evidence* content, of a different character from that of speech. Talk can be cheap when moves are not (except for the "talk" that takes the form of *enforcible* threats, promises, commitments, and so forth, and that is to be analyzed under the heading of *moves* rather than communication anyway). Mutual accommodation ultimately requires, if the outcome is to be efficient, that the division of gains be in accordance with "comparative advantage"; that is, the things a player concedes should be those that he wants less than the other player, relative to the things he trades for. Each needs, therefore, to communicate his value system with some truth, although each can also gain by deceiving. While one's maneuvers are not unambiguous in their revelation of one's value systems and may even be deliberately deceptive, they nevertheless have an evidential quality that mere speech has not.

The uncertainty that can usually be presumed to exist about each other's value systems also reduces the usefulness of the concept of mathematical *symmetry* as a normative or predictive principle. Mathematical symmetry cannot be perceived if one has access to only half the relevant magnitudes. To the extent that symmetry is helpful to the players in accommodating their movements to each other's, it would tend to be symmetry of a more qualitative sort, of the kind that depends on visible context rather than underlying values.

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## II. Enforcement, Communication, and Strategic Moves

Whenever we speak of deterrence, atomic blackmail, the balance of terror, or an openskies arrangement to reduce the fear of surprise attack; when we characterize American troops in Europe as a trip wire or plate-glass window or propose that a threatened enemy be provided a face-saving exit; when we advert to the impotence of a threat that is so enormous that the threatener would obviously shrink from carrying it out or observe that taxi drivers are given a wide berth because they are known to be indifferent to dents and scratches, we are evidently deep in game theory. Yet formal game theory has contributed little to the clarification of these ideas. The author suggests that non-zero-sum game theory may have missed its most promising field by being pitched at too abstract a level of analysis. By abstracting from communication and enforcement systems and by treating perfect symmetry between players as the general case rather than a special one, game theory may have overshot the level at which the most fruitful work could be done and have defined away some of the essential ingredients of typical non-zero-sum games. Preoccupied with the solution to the non-zero-sum game, game theory has not done justice to some typical game situations or game models and to the "moves" that are peculiar to non-zero-sum games of strategy.

What "model," for example, epitomizes the controversy over massive retaliation? What conditions are necessary for an efficacious threat? What in game theory corresponds to the proverbial situation "to have a bear by the tail"; how do we identify the payoff matrix, the communication system, and the enforcement system that it embodies? What are the tactics by which

pedestrians intimidate automobile drivers, or small countries large ones; and how do we formulate them in game-theoretical terms? What is the information or communication structure, or the complex of incentives, that makes dogs, idiots, small children, fanatics, and martyrs immune to threats?

The precarious strategy of cold war and nuclear stalemate has often been expressed in game-type analogies: two enemies within reach of each other's poison arrows on opposite sides of a canyon, the poison so slow that either could shoot the other before he died; a shepherd who has chased a wolf into a corner where it has no choice but to fight, the shepherd unwilling to turn his back on the beast; a pursuer armed only with a hand grenade who inadvertently gets too close to his victim and dares not use his weapon; two neighbors, each controlling dynamite in the other's basement, trying to find mutual security through some arrangement of electric switches and detonators. If we can analyze the structures of these games and develop a working acquaintance with standard models, we may provide insight into real problems by the use of our theory.

To illustrate, an instructive model is that of 20 men held up for robbery or ransom by a single man who has a gun and six bullets. They can overwhelm him if they are willing to lose six of themselves, if they have a means of deciding which six to lose. They can defeat him without loss if they can visibly commit themselves to a threat to do so, if they can simultaneously commit themselves to a *promise* to abstain from capital punishment, once they have caught him. He can deter their threat if he can visibly commit himself to shoot in disregard of any subsequent threat they might make, or if he can show that he could not believe

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their promise. If they cannot deliver their threat—if, say, he understands only a foreign language—they cannot disarm him verbally. Nor can they make a threat unless they agree on it themselves; so, if he can threaten to shoot any two who talk together, he can deter agreement. If the 20 cannot find a way to divide the risk, there may be no one to go first to carry out the threat, hence no way to make the threat persuasive; and if he can announce a formula for shooting, such as that those who move first get shot first, he can deter them unless they find a way to move together without a “first.” If 14 of the 20 can overpower the remaining 6 and force them to advance, they can demonstrate that they could overwhelm the man; if so, the threat succeeds, and even the 6 “expendables” gain through their own inability to avoid jeopardy. If the 20 could overwhelm the man but have no way of letting him escape, a promise of immunity may be necessary; but if they cannot deny their capacity to identify him and testify against him later, it may be necessary to let him take a hostage. This, in turn, depends on the ability of 19 to enforce their own agreement to protect, by silence, whoever is currently the hostage. . . . And so on. When we have identified the critical ingredients in several games of this sort, we may be in a better position to understand the basis of power of an unpopular despot or of a well-organized dominant minority, or the conditions for successful mutiny.

This section (Part II) of the paper is an attempt to suggest the kinds of typical moves and structural elements that deserve to be explored within the framework of game theory. They include such moves as “threat,” “promise,” “destruction of communication,” “delegation of decision,” and so forth, and such structural elements as the communication and enforcement provisions.

## AN ILLUSTRATIVE MOVE

As an example of a standard “move,” consider the bargaining between a person who has a particular object of art and someone else who is the only person interested in buying it, and suppose that each has a pretty good idea how badly the other wants it. The two are haggling over a price. We have not yet defined a game: “haggling” is not a well-defined concept. But suppose the institutional environment makes it possible for the potential buyer, who knows approximately the minimum price at which the potential seller would part with it, to make a single “final” offer subject to extreme penalty in the event that he should amend the offer. This possibility of invoking a penalty permits the potential buyer to commit himself, in a way that irreversibly changes the character of the game. There remains but a single decision for the potential seller: to sell at the price proposed or to keep the object indefinitely. The possibility of commitment converts an indeterminate bargaining process into a two-move game; one player assumes a commitment, and the other makes a final decision. The game has become determinate.¹⁶

¹⁶ In the real estate example of Von Neumann and Morgenstern referred to earlier (p. 219) buyer B (whose top price is 15) might raise the limit on what he can extract from buyer C (whose top price is 25) if he can find some means to bind himself to buy the house for 20 and keep or destroy it (i.e., not be free to resell it to C for a loss) unless he gets a specified large fraction of, say, 20—P, where P is the ultimate price paid by C. In effect, B changes his own “true” top price, thus raising the limit on what he may extract from C. Of course, D and E may try to do the same; and the first to get properly committed, or the one who can find a means if only one of them can, is the winner. If D, who attaches no personal value to the house, is committed to pay up to 22 for it, he is a bona fide member of the game with a true reservation price of 22; his bona fides is

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This particular move was analyzed at some length in an earlier article (30) and is mentioned here only as a particularly simple illustration of a typical move. Certain points should be noted about it. First, the availability and the efficacy of this move depend on the communication structure of the game and the ability of the player to find a way to commit himself, to "enforce" the commitment against himself. Second, we have allowed the move structure of the game to be asymmetrical; the "winner" is the one who can assume the commitment or, if both can, the one who can do it first. (We can consider the special case of a tie, but we have not, by an assumption of symmetry, made ties a foregone conclusion.)

Third, although we have made the game "determinate" in the sense that we have no difficulty in identifying the "solution," once we have identified which of the two players can first commit himself, it remains a game of strategy. Though the winner is the one who achieves his commitment first, the game is not like a foot race that goes to the fastest. The difference is that the commitment does not automatically win under the rules of the game, either physically or legally. The outcome still depends on the second player, over whom the first player has no direct control. The commitment is a strategic move, a move that induces the other player to choose in one's favor. It constrains the other player's choice by affecting his expectations.

The power to commit one's self in this kind of game is equivalent to "first move." And if the institutional arrangements provide no means for incurring an irrevocable commitment in a legal or contractual sense, one may accomplish the same thing by an

even greater than was B's originally, if the commitment is demonstrable while subjective valuations are not.

irreversible maneuver that reduces his own freedom of choice. One escapes an undesired invitation by commitment when he arranges a "prior" engagement; failing that, he can deliberately catch cold. Luce and Raiffa (21, p. 75) have pointed out that the same tactic can be used by a person against himself when he wants, for example, to go on a diet but does not trust himself. "He announces his intention, or accepts a wager that he will not break his diet, so that later he will *not* be free to change his mind and to optimize his actions according to his tastes at *that* time." The same thing is accomplished by maneuver rather than by commitment when one deliberately embarks on a vacation deep in the wilds without cigarettes.

## THREATS

The distinctive character of a threat is that one asserts that he will do, in a contingency, what he would manifestly prefer not to do if the contingency occurred. The motive behind the threat is to coerce or to deter, to constrain the other player's choice of action. It works by altering the other player's expectation of how the threatening player would react; it is an attempt to alter the threatener's own incentives as they are seen by the party threatened.

One threatens retaliation not because, if the provocative act occurs, one has anything to gain by retaliating; instead one risks having to retaliate in the hope that, by the sheer act of creating the risk, he will deter the act that retaliation is made contingent on. There is consequently a motive to make the threat but not to carry it out. More correctly, there is a motive to bind one's self so that fulfilling the threat is obligatory; but if the threat fails, so that it has to be carried out, the only motive for carrying it out is the obligation that was deliberately incurred earlier (plus any motive arising from the

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likelihood that fulfilment in this case increases the potency of some future threat or other simultaneous threat-i.e., from the evidence-value of the fulfilling action).

The threat is related to the commitment in two ways. First, like the commitment, it is a surrender of choice, a renunciation of alternatives, that makes one worse off than he need be in the event that the tactic fails; the threat and the commitment are both motivated by the possibility that a rational second player can be constrained by his knowledge that the first player has altered his own incentive structure. Both tactics are intended to rig one's own incentive structure so that the other player is left the initiative and will be induced to choose in the first player's favor.

Second, the threat is related to the commitment in that it depends on it; the threat can constrain the other player only insofar as it carries to the other player at least some appearance of obligation. If one is not committed to the threat in any way and cannot even seem to be committed, it is ineffectual. If I threaten to blow us both to bits unless you close the window, you know that I won't unless I have somehow managed to leave myself no choice in the matter.

The threat differs from the commitment, however, in that it makes one's courses of action *conditional* on what the other player does at his next turn; while the commitment fixes one's courses of action, the threat fixes a course of reaction, of response to the other player. The commitment is a means of gaining *first move* in a game in which *first move* carries an advantage; the threat is a commitment to a strategy for *second move*.

A threat can be effective only if the game is one in which the first move is up to the other player or one can force the other to move first. But if one must, in a mechanical sense, move first or simultaneously, he can still force the legal equivalent of "first

move" on the other by attaching his threat to a demand that the other promise in advance how he will behave—if the game has communication and enforcement structures that make promises feasible and that the party to be threatened cannot destroy in advance. The holdup man whose rich victim happens to have no money on him at the time can make nothing of his opportunity unless he can extract a hostage while he awaits payment; and even that will not work unless he can himself find a way to assume a convincing commitment to return the hostage in a manner that does not subject himself to identification or capture.

The fact that some kind of commitment, or at least appearance of commitment, must lie behind the threat and be successfully communicated to the threatened party is in contradiction to another notion that often appears in game theory. This is the notion that a threat is desirable, or admissible, or plausible, only if the reaction threatened would cause worse damage to the threatened party than to the party making the threat. This is the view of Luce and Raiffa (21), who characterize threats by the phrase, "This will hurt you more than it hurts me," explicitly making threats depend on interpersonal utility comparisons. In the event that both players attempt to make plausible threats, they say, the result becomes indeterminate, depending on the "bargaining personalities" of the players; "and to predict what will in fact happen without first having a complete psychological and economic analysis of the players seems foolish indeed."¹⁷

¹⁷ Morton A. Kaplan (19, p. 198), in applying game theory to international relations, also takes the position that "any criterion giving weight to the threat positions of the players involves an interpersonal comparison of utilities." Luce and Raiffa (21, pp. 110–11, 119–20, 143–44) may be led partly to their view that only one of the

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But the issue is both simpler and more precise than that. Consider the left-hand matrix in Figure 2, where Column is assumed to have "first move." Without

players has a "plausible" threat to make, by confining their brief discussion to 2 × 2 matrices. It is impossible to show, with a 2 × 2 matrix, a game in which both players could be interested in making threats. A threat is essentially a credible declaration of a conditional choice for second move. It is profitable only if it yields a better payoff than either first move or second move alone and when one can make the other player move first either actually or by promise. (If second move alone is as good, the threat is unnecessary; and if first move were as good, one needs only an unconditional commitment to his strategy choice, not a commitment to a conditional choice.) But if this preference order holds for one player in a 2 × 2 matrix, it cannot hold for the other player. The actual matrices used by Luce and Raiffa in discussing the point show no "plausible" threat strategy for player No. 2, not because the absolute size of his gains or losses is greater than player 1's but for the much simpler reason that player 2 has no use for a threat. He wins if he moves first; he wins if he moves second; and he wins with simultaneous moves, in the games shown. His only interest in a threatlike declaration would be to forestall his partner's threat; and for that purpose he needs only an unconditional commitment to his preferred strategy-i.e., the legal equivalent of "first move" in advance of his partner's threat. The "threat tactic" of J. F. Nash (27), which applies to bargaining games that have a continuous range of efficient outcomes-or that can be made to, by agreement on the odds in a drawing of lots differs from the threat discussed here, in that the threatener does not demand, on pain of mutual damage, a particular outcome but only some outcome in the efficient range; that is, he shifts the zero point corresponding to "no agreement." The motive for that threat is the expectation of a particular mathematically determinate outcome whose locus is shifted by the shift in the payoffs corresponding to non-agreement. (For further discussion of Nash's theory see Harsanyi [13].) This is the kind of threat assumed by Luce and Raiffa (21, p. 139) in the "asymmetrical" game. The implicit legal structure of the game apparently honors no irrevocable commitments (other-

threats, Column has an easy "win." He chooses strategy I, forcing Row to choose between payoffs of 1 and 0; Row chooses strategy i, providing Column a payoff of 2. But if we allow Row to make a threat, he declares that he will choose strategy ii unless Column chooses II; that is, he gives Column a choice of ii, I or i, II by committing himself to that conditional choice. If Column went ahead and chose I, of course, Row would prefer to choose i; and they both know it. The tactic succeeds only if Column believes that Row must choose ii in the event of I.

Either he does believe this, or he does not. If he does not, the "threat" is nothing

FIG. 2

at all to him; he goes ahead and makes his "best" first move, choosing I. If he does believe that Row must follow a strategy of i, II

wise, first commitment would easily win the game for either player). Each player is subject to the legal "disability" that he can always, by the overt act of explicit agreement with his partner on any outcome, evade his own commitment. This being so, the revocable commitments can only shift the zero point-the "status quo" that will rule unless explicit agreement on some outcome is reached. The "asymmetry" that is present in the particular game shown by Luce and Raiffa is thus a feature of the particular legal system that implicitly prevails. In practice it might correspond, say, to the deliberate incurring of social disapproval on failure to reach agreement, with such disapproval constituting cost or punishment (perhaps asymmetrical between participants) in addition to the cost of non-agreement but with the public not concerned with what the agreement provides as long as some agreement is reached.

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or ii,I, Column prefers 1 to 0 and chooses II. But this is true of any numbers that we might put in the matrix that reflect the same order of preferences. It is true of the right-hand matrix as well. That one dramatizes the essential character of the threat more than the first one, since the penalty on Row of an irrational choice by Column is greater in this case; but for rational play and full information, Row need not worry. Column's preference is clear; and, once Row has given him the pair to choose from—ii,I versus i,II —there is no doubt what Column will do. If I threaten to blow my own brains all over your new suit unless you give me that last slice of toast, you'll give me the toast or not depending on whether you know that I've arranged to have to do so, exactly as if I'd only threatened to throw my scrambled eggs at you.

The issue here is in whether or not we admit that the game has “moves,” i.e., that it is possible for one player or both players to take actions in the course of the game that irreversibly change the game itself—that in some fashion alter the payoff matrix, the order of choices, or the information structure of the game. If the game by its definition admits no moves of any sort, except mutual agreement and refusal to agree, then it may be true that the “personalities” of the players determine the outcome, in the sense that their expectations in a “moveless” game converge by a process that is wholly psychic. But if a threat is anything more than an assertion that is intended to appeal to the other player by power of suggestion, we must ask what more it can be. And it must involve some notion of commitment—real or fake—if it is to be anything.

“Commitment” is to be interpreted broadly here. It includes maneuvers that leave one in such a position that the option of non-fulfilment no longer exists (as when one intimidates the other car by driving too fast to

stop in time), maneuvers that shift the final decision beyond recall to another party whose incentive structure would provide an ex post motive for fulfilment (as when the authority to punish is deliberately given to sadists, or when one shifts his claims and liabilities to an insurance company), and maneuvers that simply “worsen” one's own payoff in the contingency of non-fulfilment so that even the horror of a mutually damaging fulfilment becomes more attractive (as when one arranges for himself to appear a public coward if he fails to fulfil, or when he puts a plate-glass window in front of his wares or stations women and children on the particular bit of territory that he has threatened somewhat implausibly to defend at great cost). A nice everyday example is given by Erving Goffman (11, p. 215), who reminds us that “Salesmen, especially street ‘stemmers,’ know that if they take a line that will be discredited unless the reluctant customer buys, the customer may be trapped by considerateness and buy in order to save the face of the salesman and prevent what would ordinarily result in a scene.”¹⁸

There are, however, some ways in which this notion of commitment to a threat can be usefully loosened. One is to recognize that “firm” commitment amounts to the invocation of some wholly potent penalty, such that one would in all circumstances prefer to carry out what he was committed to. It is a penalty of infinite (or at least of superfluous) size that one voluntarily, irreversibly, and visibly attaches to all patterns of action but the one that he is committed to. This concept can be loosened by supposing that the penalty is of finite size and not nec-

¹⁸ Goffman's paper is a brilliant study in the relation of game theory to gamesmanship and a pioneer illustration of the rich game-theoretic content of formalized behavior structures like etiquette, chivalry, diplomatic practice, and—by implication—public law.

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essarily so large as to be controlling in all cases. In Figure 3, Column will win if he has first move, unless Row can commit himself to i. (Commitment obtains "first move" for Row.) But if commitment means the attachment of a finite penalty to the choice of row ii and we show this in the matrix by subtracting from each of Row's payoffs in ii some finite amount representing the penalty, then the commitment will be effective only if the penalty is greater than 2. Otherwise it is clear to Column that Row's response to II will be ii, in spite of the commitment. In this case the commitment is simply a loss that Row would impose on himself, so he avoids it.

Similarly with a threat. In Figure 4, without threats, the solution is at iii,II whether the rules call for Row to choose first, Column first, or both to choose simultaneously. Either player can win if he can move second and confront the other with a threat.¹⁹ Column would threaten I against iii, Row would threaten i against II. But if the threat is secured by a penalty, the lower limit to any persuasive penalty that Column could

¹⁹ If a player, Column, for example, cannot force first move on Row in a mechanical sense, he can do so in a "legal" sense by threatening to choose I unless Row promises to choose ii. Full analysis in this case requires attention to the penalties on promises as well as on threats. Since the physical and institutional arrangements for promises (i.e., for commitments to the second party) are generally of a quite different nature from those for unilateral commitments (i.e., commitments that the second player cannot himself dissolve), available penalties could differ drastically as between threats and promises—just as, in general, they would differ as between the first and second players. The particular payoffs shown in Fig. 4 would require penalties of at least 1 on a promise by Column or by Row. Note that in the case of a promise extracted by a threat, it is an advantage to the threatener to be able to invoke penalty and a disadvantage to the victim to be able to invoke penalty on his own breach of contract, ie., to be able to comply.

invoke would be 4; any smaller penalty leaves him preferring II to I when Row chooses iii. The lower limit to a persuasive penalty on Row's non-compliance would be 3. If, then, the situation is one in which penalties come in a single "size," a size less than 3 goes unused and the outcome is at iii,II; a size greater than 4 is adequate for either player, and the "winner" is the one who can avail himself of the threat first; a size between 3 and 4 is of use only to Row,

FIG. 3

FIG. 4

who wins. In this latter case the player who would be hurt the more by his own unsuccessful threat is the one who cannot threaten—but only through the paradox that he is incapable of calling a sufficiently terrible penalty on his own head.

Note that the "hurt-more" comparison in this case refers not to whether Row or Column would be hurt more by what Row threatens but to whether Row would be hurt more by having to fulfil his own threat than Column would be hurt if, instead, Column had made his threat. Actually, in the particular payoff matrix shown, Row's successful threat is one that would hurt him more

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in the fulfilment than it would hurt Column, while Column's potential *unsuccessful* threat would hurt him *less* to fulfil than it would hurt Row.

Another loosening of the threat concept is to alter our assumption of rationality. Suppose there is some probability Pr for player R, and some probability Pc for player C, that he will make a mistake or an irrational move, or that he will act in an unanticipated way because the other player is mistaken about the first player's payoffs. This yields us a game in which the possible gains and losses in committing one's self to a threat must take into account the possibility that a fully committed threat will not be heeded. If, then, the potential loss that will ensue from having to carry out the threat is greater for one player than for another, there could be symmetrical circumstances—the P's being equal and the threat penalties equal for the two players—in which one player may find it advantageous to make the threat and the other player not, considering the possibility of “error.” (A somewhat similar calculation may be involved if both players have opportunities for threats and there is danger of simultaneous commitment through the failure of one to observe the other's commitment and to stop in time to save both.)

This modification in the threat concept— in the rationality postulate that underlies it— goes somewhat in the direction of the “hurt-more” criterion. On the whole, though, game theory adds more insight into the strategy of bargaining by emphasizing the striking truth that the threat does *not* depend on the threatener's having less to suffer than the threatened party if the threat had to be carried out rather than by exaggerating the possible truth contained in the intuitive first impression. Threats of war, of price war, of damage suit; threats to make a “scene”; most of the threats of organized society to

prosecute crimes and misdemeanors; and the concepts of extortion and deterrence generally cannot be understood except by denying the utility-comparison criterion. It is indeed the asymmetries in the threat situation, as between the two players, that make threats a rich subject for study; but the relevant asymmetries include those in the communication system, in the enforcibility of threats and of promises, in the speed of commitment, in the rationality of expected responses, and, finally (in some cases), in the relative-damage criterion.

## PROMISES

Enforcible promises cannot be taken for granted. Agreements must be in enforcible terms and involve enforcible types of behavior. Enforcement depends on at least two things—some authority somewhere to punish or coerce and an ability to discern whether punishment or coercion is called for. The postwar discussions of disarmament proposals and inspection schemes indicate how difficult it may be, even if both sides should desperately desire to reach an enforcible agreement or find a persuasive means of enforcement. The problem is compounded when neither party trusts the other and each recognizes that neither trusts the other and that neither can therefore anticipate the other's compliance. Many of the technical problems of arms inspection would disappear if there were some earthly means of making enforcible promises or if the nations of the world all rendered unquestioned allegiance to some unearthly authority. But since non-compliance may be undetectable, promises of compliance could not be enforced even if punishment could be guaranteed. The problem is doubled by the fact that punishment cannot be guaranteed, except such punishment as can unilaterally be meted out by the other party in its act of denouncing the original agreement.

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Furthermore, some seemingly desirable agreements must be left out for being undefinable operationally; agreements not to discriminate against each other will work only if defined in objective terms capable of objective supervision.

Promises are generally thought of as bilateral (contractual) commitments, given against a quid pro quo that is often a promise in return. But there is incentive for a unilateral promise when it provides inducement to the other player to make a choice in the mutual interest. In the left-hand matrix of Figure 5, if choices are to be simultaneous, only a pair of promises can be effective; in the right-hand matrix, Row's promise brings its own reward: Column can safely choose II, yielding superior outcomes for both players. (If, in the left-hand matrix, moves are in turn, the player who chooses

FIG. 5

second must have the power to promise. If the players are themselves to agree on the order of moves and only one of the two can issue promises, they can agree that the other one move first. These promises, in contrast to those for the right-hand matrix, must be conditional on the second player's performance. A unilateral unconditional promise does the trick on the right-hand side but not on the left with moves in turn.) The witness to a crime has a motive for unilateral promise if the criminal would kill to keep him from squealing.²⁰ A nation known to be on the threshold of an absolutely potent surprise-attack weapon may have reason to foreswear it unilaterally—if there is

any possible way to do so—in order to forestall a desperate last-minute attempt by an enemy to strike first while he still has a chance (34).

The exact definition of a promise—e.g., in distinction to a threat—is not obvious. It might seem that a promise is a commitment (conditional or unconditional) that the second party welcomes, one that is mutually advantageous, as in both the games shown in Figure 5. But Figure 6 shows a situation in which Row must couple a threat and a promise; he threatens ii against I and promises i in the event of II. The promise insures Column a payoff of 4 rather than zero, once he has made a choice of II, and in that sense

FIG. 6

it is favorable to him; it does so at a cost of 1 unit to Row. But if Row could not make the promise, Column would win 5; he would because the threat would be ineffectual without the promise, and the threat would not be incurred. A threat of ii against I by itself is no good; it cannot force Column to choose II, since a choice of II leaves him with an outcome at ii,II, zero instead of 1. Row's threat can work only if the promise

²⁰ This notion is celebrated in “Wet Saturday,” by John Collier (5, pp. 171–78), recently reproduced by Alfred Hitchcock on TV. An inadvertent eavesdropper on a murder is ordered at gunpoint to seal his lips by leaving his own fingerprints and other incriminating evidence, so that if the body is found he will be charged with the murder. He should have insisted, however, on fabricating the evidence so as to share the guilt with the actual murderer; as it was, he got badly cheated.

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goes with it; the net effect of the promise is to make the threat work, yielding Column 4 instead of 5, gaining 5 rather than 2 for Row. (One cannot force spies, conspirators, or carriers of social diseases to reveal themselves solely by the *threat* of a relentless pursuit that spares no cost; one must also promise immunity to those that come forward.)²¹

A better definition, perhaps, would make the promise a commitment that is controlled by the second party, i.e., a commitment that the second party can enforce or release as he chooses. But timing is important here. The promise just discussed will work *after* the threat is fully committed; but if the victim of the promise (Column) can renounce the promise in advance, so that Row knows that Column expects zero if he chooses II, the threat itself is deterred. And if the threat and promise are contrived in such a way as to be "legally" inseparable or if they are accomplished by some irreversible maneuver, the definition becomes obscured. (In fact, the definition breaks down whenever the equivalent of a promise is obtained by some irrevocable act rather than by a "legal" commitment.)

Actually, whenever the alternative choices are more than two, threat and promise are likely to be mixed in any "reaction pattern" that one presents to the other. So it is probably best to consider the threat and the promise to be names for different aspects of the same tactic of selective and conditional self-commitment, which in certain simple instances can be identified in terms of the second party's interest.

**Enforcement schemes.** Agreements are unenforcible if no outside authority exists to

enforce them or if non-compliance would be inherently undetectable. The problem arises, then, of finding forms of agreement, or terms to agree on, that provide no incentive to cheat or that make non-compliance automatically visible or that incur the penalties on which the possibility of enforcement rests. While the possibility of "trust" between two partners need not be ruled out, it should also not be taken for granted; and even trust itself can usefully be studied in game-theoretic terms. Trust is often achieved simply by the continuity of the relationship between parties and the recognition by each that what he might gain by cheating in a given instance is outweighed by the value of the tradition of trust that makes possible a long sequence of future agreement. By the same token, "trust" may be achieved for a single discontinuous instance, if it can be divided into a succession of increments.

There are, however, particular game situations that lend themselves to enforceable agreement. One is an agreement that depends on some kind of co-ordination or complementarity. If two people have disagreed on where to meet for dinner; if two criminal accomplices have disagreed on what joint alibi to give; or if members of a business firm or football team have disputed about what prices they will quote or what tactic they will follow, they nevertheless have an overriding interest in the ultimate consistency of their actions. Once agreement is formally reached, it constitutes the only possible focal point for the necessary subsequent tacit collaboration; no one has a unilateral preference now to do anything but what he is expected to do. In the absence of any other means of enforcement, then, parties might be well advised to try to find agreements that enjoy this property of interdependent expectations, even to the extent of importing in their agreement certain ele-

²¹ Somewhat related is the grant of immunity that strips a reticent witness of protective danger of self-incrimination, and so opens him to the ordinary sanction of contempt proceedings.

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ments whose sole purpose is to create severe jeopardy for non-co-ordination. Tearing the treasure map in half or letting one partner carry the gun and the other the ammunition is a familiar example.

The institution of *hostages* is an ancient technique that deserves to be studied by game theory, as does the practice of drinking wine from the same glass or of holding gang meetings in places so public that neither side could escape if it subjected the other to a massacre. The reported use of only drug addicts as agents or employees in a narcotics ring is a fairly straightforward example of a unilateral hostage.

Perhaps a sufficient interchange of populations between nations that hate each other or an agreement to move the governing agencies of both countries to a single island where they would occupy alternate blocks of the city could be resorted to if both sides became sufficiently desperate to avoid mutual destruction. A principal drawback to the exchange of hostages, on the assumption of rational behavior, is the inherent unknowability of each other's value system adverted to earlier. The king who sends his daughter as a hostage to his enemy's court may be incapable of assuaging his enemy's fears that he really dislikes the girl. We could probably guarantee the Russians against an American surprise attack by having the equivalent of "junior year abroad" at the kindergarten level: if every American five-year-old went to kindergarten in Russia —in American establishments constructed for the purpose, designed solely for "hostage" purposes and not for cultural interchange— and if each year's incoming group arrived before the graduating class left, there would not seem to be the slightest chance that America would ever initiate atomic destruction in Russia. We cannot be quite sure that the Russians would be quite sure of this. Nor can we be quite sure that a reciprocal

program would be as much of a deterrent to the Russian government; unfortunately, even if the Russian government were bound by the fear of harming Russian children, it seems nearly impossible for it to persuade us so. Still, in many surprise-attack situations a unilateral promise is better than none; and the idea of hostages may be worth considering, even when symmetrical exchanges do not seem available.²²

Actually, the hostage idea is logically identical with the notion that a disarmament agreement between the major powers might be more efficacious (and probably more subject to technical control) if it related to defensive weapons and structures. To eschew defense is, in effect, to make hostages of your entire population without bothering to put them physically into the other's possession. Thus we can put our children at the mercy of the Russians and receive similar power over Russian children not only by physically trading them, with enormous discomfort and breach of constitutional rights, but also by simply agreeing to leave them so unprotected that the other can do them as much damage where they are as if he had them in his grasp. Thus the "balance of terror" that is so often adverted to is—if, in fact, it exists and is stable—equivalent to a total exchange of all conceivable hostages. (The analogy requires that the balance be

²² The precise definition of hostages is a little difficult. They seem to be as pertinent to threats as to promises: the four American divisions that were stationed in Europe principally to demonstrate that America could not avoid becoming engaged in a European conflict can probably be viewed as hostages; if they cannot, their wives and children can, and perhaps their wives and children have been a more persuasive commitment of "trip wire" than the troops themselves. As a general rule, invaders may have to avoid the peak tourist season in countries they covet, to avoid provoking the countries that have yielded inadvertent hostages.

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stable, i.e., that neither side be able, by surprise attack, to destroy the other's power to strike back, but just able to inflict a surfeit of civilian agony.)

**Denial of enforcement.** Enforcement of promises is also relevant to the influence of a third party that wishes to make an efficient outcome more difficult for the other two players. A potent means of banning illegal activities has often been the outlawing of them, so that contracts became unenforcible. Failure to enforce gambling contracts or contracts in restraint of trade or contracts for the delivery of liquor during prohibition has always been part of the process of discouraging the activities themselves. Sometimes, of course, prohibition of this sort delivers enormous power into the hands of anyone who can enforce contracts or make enforcible promises.²³ The denial of copyright liquor labels during prohibition meant that only the bigger gangs could guarantee the quality of their liquor and hence assisted them in developing monopoly control of the business. By the same token, laws to protect brands and labels can perhaps be viewed as devices that facilitate business based on unwritten contracts.

## RELINQUISHING THE INITIATIVE

What makes the threat or ordinary commitment a difficult tactic to employ and an interesting one to study is the problem of finding a means to commitment, the available “penalty” to invoke against one’s own non-performance. There is consequently a related set of tactics that consists of maneuvering one’s self into a position in which one

²³ It has been argued that an important function of the racketeer is sometimes to help enforce agreements that are beyond the law. Price-cutting in the Chicago garment trade was punishable by explosion—the fee for the explosion being paid by the price-fixing organization—according to R. L. Duffus (7).

no longer has any effective choice over how he shall behave or respond. The purpose of these tactics is to get rid of an embarrassing initiative, making the outcome depend solely on the other party's choice.

This is the kind of tactic that Secretary of State John Foster Dulles (8) was looking for in the following passage:

> In the future it may thus be feasible to place less reliance upon deterrence of vast retaliatory power. . . . Thus, in contrast to the 1950 decade, it may be that by the 1960 decade the nations which are around the Sino-Soviet perimeter can possess an effective defense against full-scale conventional attack and thus confront any aggressor with the choice between failing or himself initiating nuclear war against the defending country. Thus the tables may be turned, in the sense that instead of those who are non-aggressive having to rely upon all-out nuclear retaliatory power for their protection, would-be aggressors would be unable to count on a successful conventional aggression, but must themselves weigh the consequence of invoking nuclear war.²⁴

The distinction between the type of deterrence he imputes to the 1950's and the type he imputes to the 1960's differs in the matter of who has to make that final decision; and the difference is important because the United States cannot find, or bring itself to trust, a persuasive means of commitment to the threat of massive retaliation against certain types of aggression.

There was a time, shortly after the first atomic bomb was exploded, when there was some journalistic speculation about whether the earth's atmosphere had a limited tolerance to nuclear fission; the idea was bruited

²⁴ Very similar language is used by Dean Acheson (1, pp. 87–88) in discussing the role of a sizable defense force in Europe: by requiring of the enemy a major attack, rather than a small one, it makes him believe that retaliation would ensue, because “he would be making the decision for us. . . . A defense in Europe of this magnitude will pass the decision to risk everything from the defense to the offense.”

[LC]

THE STRATEGY OF CONFLICT

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about that a mighty chain reaction might destroy the earth's atmosphere when some critical number of bombs had already been exploded. Someone proposed that, if this were true and if we could calculate with accuracy that critical level of tolerance, we might neutralize atomic weapons for all time by a deliberate program of openly and dramatically exploding n-1 bombs.

This tactic of shifting responsibility to the other player was nicely accomplished by Lieutenant Colonel (then Major) Stevenson B. Canyon, U.S.A.F., in using his aircraft to protect a surface vessel about to be captured by Communist surface forces in his comic strip. Unwilling and unauthorized to initiate hostilities and knowing that no threat to do so would be credited, he directed his planes to jettison gasoline in a burning ring about the aggressor forces, leaving to them the last clear chance of reversing their engines. He could neither drop gasoline on the enemy ships nor threaten to; so he dropped the initiative instead.

The same tactic is involved in those dramatic forms of "passive resistance" that might be better called "active non-resistance." According to *The New York Times* (29), "Striking railway workers sat down on the tracks at more than 300 stations in Japan today, halting 48 passenger and 144 freight trains."²⁵

A more dramatic instance, also Japanese, was reported in the same paper (28): "A public debate is being held here this week on whether to send a 'suicide sit-down fleet' to the forbidden waters around Christmas Island, the site of the forthcoming British hydrogen bomb experiment. . . . The first object of the expedition would be to prevent the British blast."

## IDENTIFICATION

An important characteristic of any game is how much each side knows about the

other's value system; but a similar information problem arises with respect to sheer identification. The bank employee who would like to rob the bank if he could only find an outside collaborator and the bank robber who would like to rob the bank if only he could find an inside accomplice may find it difficult to collaborate because they are unable to identify each other, there being severe penalties in the event that either should declare his intentions to someone who proved not to have identical interests. The boy who is afraid to ask a girl for a date because she might rebuff him is in a similar position. Similarly, the kidnaper cannot operate properly if he cannot tell the rich from the poor in advance; and the antisegregation minority in the South may never know whether it is large or small because of the penalties on declaration.

Identification, like communication, is not necessarily reciprocal; and the act of self-identification may sometimes be reversible and sometimes not. One may achieve more identification than he bargained for, once he declares his interest in an object. A nice example occurs in Shakespeare's *Measure for Measure*. The acting Duke has a prisoner whom he proposes to kill. He could torture him, but he has no incentive to. The victim has a sister, who arrives to plead for

²⁵ The appropriate countertactic seems to be the following: The engineer sets the throttle for slow forward speed, conspicuously climbs down from his cab and jumps off the moving train, walks through the station and jumps back on his engine when it catches up with him. The weakness of his position while he is driving the train is that he can stop it more quickly than his adversaries can get off the tracks, particularly if they have arranged to crowd themselves so that they could not vacate the track quickly. They can forestall his countertactic by locking themselves to the tracks and throwing away the key—if they can persuasively inform the engineer of this before he has relinquished his own control of the engine.

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his life. The Duke, finding the sister attractive, proposes a dishonorable bargain; the sister declines. The Duke then threatens to torture the brother unless the sister submits. At this point the game has been expanded simply by the establishment of identity and of a line of communication. The Duke's only interest in torturing the brother is in what he may gain by making a threat to do so; once there is somebody available to whom the threat can profitably be communicated, the possibility of torture has value for the Duke—not the torture itself, but the threatening of it. The sister has gotten negative value out of her trip; having identified her interest and made herself available to receive the threatening message, she has been forced to suffer what she would not have had to suffer if she had never made her identity known or if she could have disappeared into the crowd before the threat was made.

A nice identification game was uncovered in a New York suburb a few years ago. Certain motorists carried identity cards which identified them to policemen as members in a club; if the motorist with a membership card was arrested, he simply showed the card to the policeman and paid a bribe. The role of these cards was to identify the motorist as a person who, if the bribe was received, would keep quiet. It identified the motorist as a man whose promise was enforceable. But the card identifies the motorist only *after* he has been arrested; if the police could identify card-carrying motorists by looking at them, they could concentrate their arrests on card-carrying drivers, threatening a ticket unless payment were received. The card is contingent identification, at the option of the motorist. A similar situation —pertinent to the discussion of promises as well as to identification—is described by Sutherland (36, p. 126): "Most coppers are

more or less fair in their dealings with thieves simply because it pays them to be so. They will extend favors even after a pinch which they would not extend to non-professionals whom they lock up. They realize that it is safe to do this and that high officials will not be informed, as might be the case if favors were extended to amateurs."

Identification is also relevant to an important economic fact that tends to be ignored in the conventional economics of production and exchange, namely, the enormous potential for destruction that is available and that is relevant because of the extortionate threats that could be supported by it. The ordinary healthy high-school graduate, of slightly below average intelligence, has to work fairly hard to produce more than $3,000 or $4,000 of value per year; but he could destroy a hundred times that much if he set his mind to it, according to the writer's hasty calculations. Given an institutional arrangement in which he could generously abstain from destruction in return for a mere fraction of the value that he might have destroyed, the boy clearly has a calling as an extortionist rather than as a mechanic or clerk. It is fortunate that extortion usually depends on self-identification and overt communication by the extortionist himself.

The importance of self-identification is attested by the significance attached to the doctrine that an accused person should be permitted to know and to confront his accuser. It is also reflected in secret testimony before a Grand Jury, in cases where identifiable witnesses might be intimidated by potential defendants, and in efforts to keep secret the identity of eyewitnesses to a crime until the criminal is apprehended. (The strategy of law and of law enforcement and criminal deterrence is a rich field for the application of game theory.)