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## DELEGATION

Another "move" that is sometimes available is the delegation of part or all of one's interest, or part or all of one's initiative for decision, to some agent who becomes (or perhaps already is) another player in the game. Insurance schemes permit the sharing of interests; the insurance company has a different incentive structure from the insured party and may be better able to make threats or resist them for that reason. Requiring several signatures on a check accomplishes a similar purpose. The use of a professional collecting agency by a business firm for the collection of debts is a means of achieving unilateral rather than bilateral communication with its debtors and of being therefore unavailable to hear pleas or threats from the debtors. Providing ammunition to South Korean troops or giving them access to prisoner-of-war camps so that they can unilaterally release prisoners is a tactical means of relinquishing an embarrassing power of decision—embarrassing because it subjects one to coercive or deterrent threats or leaves one the capacity to back out of his own threat, hence the incapacity to make the threat persuasive.

The mutual-defense agreement with the Nationalist government of China is probably to be viewed partly as a means of shifting the decision for response to someone whose resolution would be less doubtful; and more recently the proposal to put nuclear weapons in the hands of European governments has been explicitly argued on grounds that it would enhance deterrence by giving the visible power to retaliate to countries that might in certain contingencies be thought less irresolute than the United States.

The use of thugs and sadists for the collection of extortion or the guarding of prisoners, or the conspicuous delegation of authority to a military commander of known

motivation, exemplifies a common means of making credible a response pattern that the original source of decision might have been thought to shrink from or to find profitless, once the threat had failed. (Just as it would be rational for a rational player to destroy his own rationality in certain game situations, either to deter a threat that might be made against him and that would be premised on his rationality or to make credible a threat that he could not otherwise commit himself to, it may also be rational for a player to select irrational partners or agents.)

In the matrix in Figure 7—disregarding

FIG. 7

the numbers in parentheses—if Row has second move, he loses in the lower right-hand corner, Column gaining his own preferred outcome. If a third party without power of decision is scheduled to receive, as a by-product, the payoff in parentheses, Row can win if some means is available for irreversibly surrendering his move to the third player. The payoffs of the latter are such that with second move he wins in the upper left-hand corner, leaving the original Row-player a payoff of 5 as a by-product. (If the third party's rewards had to be financed by Row, whose own payoffs were correspondingly reduced, it would still be worth his while to make an irrevocable assignment of portions of his various payoffs to the third player, together with assignment of the decision; with the figures shown, he would

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THOMAS C. SCHELLING

still carry away a net value of 3 in the up-
per left-hand corner, in contrast to 1 in the
lower right.)

MEDIATION

The role of mediator is another element for analysis in game theory. A mediator, whether imposed on the game by its original rules or adopted by the players to facilitate an efficient outcome, is probably best viewed as an element in the communication arrangements or as a third player with a payoff structure of his own who is given an influential role through his control over communication. But a mediator can do more than simply constrain communications—putting limits on the order of offers, counter-offers, etc.—since he can invent contextual material of his own and make potent suggestions. That is, he can influence the other player's expectations on his own initiative, in a manner that both parties cannot help mutually recognizing. When there is no apparent focal point for agreement, he can create one by his power to make a dramatic suggestion. The bystander who jumps into an intersection and begins to direct traffic at an impromptu traffic jam is conceded the power to discriminate among cars by being able to offer a sufficient increase in efficiency to benefit even the cars most discriminated against; his directions have only the power of suggestion, but co-ordination requires the common acceptance of some source of suggestion. Similarly, the participants of a square dance may all be thoroughly dissatisfied with the particular dances being called, but as long as the caller has the microphone, nobody can dance anything else. The white line down the center of the road is a mediator, and very likely it can err substantially toward one side or the other before the disadvantaged side finds advantage in denying its authority.

Mediators can also be a means by which rational players can put aside some of their rational faculties. A mediator can consummate certain communications while blocking off certain facilities for memory. (In this regard he serves a function that can be reproduced by a computing machine.) He can, for example, compare two parties' offers to each other, declaring whether or not the offers are compatible without revealing the actual offers. He is a scanning device that can suppress part of the information put into it. He makes possible certain limited comparisons that are beyond the mental powers of the participants, since no player can persuasively commit himself to forget something.

The problem of persuasively denying one's self the knowledge that one receives by the left hand, while actively seeking it with the right hand, is nicely illustrated by the efforts of parts of governments to obtain accurate data on incomes for the purpose of statistical programs, while another part of the government is seeking the same data in order to impose taxes or to prosecute evasion. Governments have found it important to seek ways of guaranteeing that the statistical agency will deny the information it receives to the taxing agency, in order to receive the information in the first place. An analogous case of relying on an explicit mediator is that of companies that turn trade secrets over to a statistical bureau that is committed to destroy the individual data after computing the sums and averages that it will make public for the benefit of the contributing companies, or of public opinion services that suppress potentially embarrassing individual data on political or sexual practices, publishing only the aggregates. The use of mediators to forestall identification seems to be a common tactic when a buyer of large resources thinks a painting,

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[LC]

or a right-of-way can be bought cheap if the owner is unaware who it is that is interested.

Mediators may be converted into arbitrators by the irrevocable surrender of authority to him by the players. But arbitration agreements have to be made enforcible by the players' deliberately incurring jeopardy, providing the referee with the power to punish or surrendering to him something complementary to their own value systems. In turn, they must be able to trust him or to extract an enforcible promise from him. But in any case he increases the totality of means for enforcing promises: two people who do not trust each other may find a third person that they both trust, and let him hold the stakes.²⁶

## COMMUNICATION AND ITS DESTRUCTION

Many interesting game tactics and game situations depend on the structure of communication, particularly asymmetries in communication and unilateral options to initiate communication or to destroy it. Threats are no good if they cannot be communicated to the persons for whom they are intended; extortion requires a means of conveying the alternatives to the intended victim. Even the threat, "Stop crying or I'll give you something to cry about," is ineffectual if the child is already crying too loud to hear it. (It sometimes appears that children know this.) A witness cannot be intimidated into giving false testimony if he is in custody that prevents his getting

²⁶ I have been told that in countries where no strong tradition of business morality exists, a few partners or directors for a business may deliberately be chosen from another culture where simple honesty and fairness are considered to be common traits or where a reputation for them is considered of much higher value.

instructions on what to say, even though he might infer the sanction of the threat itself.

When the outcome depends on co-ordination, the timely destruction of communication may be a winning tactic. When a man and his wife are arguing by telephone over where to meet for dinner, the argument is won by the wife if she simply announces where she is going and hangs up. And the status quo is often preserved by a person who evades discussion of alternatives, even to the extent of simply turning off his hearing aid.

As discussed in the earlier part of this paper, mob action often depends on communication in a way that makes it possible for the authorities to obstruct mob action by forbidding groups of three or more to congregate. But mobs can themselves intimidate the authorities if they are able to identify them and to communicate with them. Even a tacit threat of subsequent ostracism or violence may be communicated from a riotous mob to the local police, if the police are known to them and are persons who have to reside among them when the occasion is over. In that case the use of outsiders may forestall the mob's intimidating threats against the authorities, partly by reducing the subsequent occasion for carrying out the threat but partly also through the difficulty of tacit communication between mob and police. Federal troops in Little Rock may have enjoyed some immunity to intimidation just by being outside the tacit communication structure of the local populace and being patently less conversant with the local value system than were the local police. State troops were dramatically successful in quelling the Detroit race riot of 1943, when the local police were ineffectual. The use of Moors, Sikhs; and other foreign-language

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troops against local uprising may owe some of its success to their poor capacity to receive the threats and promises that the enemies or victims might otherwise seek to convey. Even the isolation of officers from enlisted men in military service may tend to make officers less capable of receiving and perceiving threats, hence less capable of being effectively threatened, and thus deterring intimidating threats themselves.

It is important, of course, whether or not the threatener knows that his threat cannot be received; for if he thinks it can, and it cannot, he may make the threat and fail in his objective, being obliged to carry out his threat to the subsequent disadvantage of both himself and the one threatened. So the soldiers in quelling the riot should not only be strangers and not only keep moving sufficiently to avoid "acquaintance" with particular portions of the mob; they should behave with an impassivity to demonstrate that no messages are getting through. They must catch no one's eye; they must not blush at the jeers; they must act as if they cannot tell one rioter from another, even if one has been making himself conspicuous. Figuratively, if not literally, they should wear masks; even the uniform contributes to the suppression of identification and so itself makes reciprocal communication difficult.

**Conveyance of evidence.** "Communication" refers to more than the transmission of messages. To communicate a threat, one has to communicate the commitment that goes with it, and similarly with a promise; and to communicate a commitment requires more than communication of words. One has to communicate *evidence* that the commitment exists; this may mean that one can communicate a threat only if he can make the other person see something with his own eyes or if he can find a device to

authenticate certain allegations. One can send a signed check by mail, but one cannot demonstrate over the telephone that a check bears an authentic signature; one may show that he has a loaded gun but not prove it by simply saying so. From a game-theory point of view, the Paris pneumatique differs from an ordinary telegraph system, and television differs from radio. (One role of a mediator may be to authenticate the statements that the players make to each other; for example, a code system for identification might make it possible for people to transmit funds orally by telephone, the recipient being assured by the bank's code response that it is in fact the bank at the other end of the line assuring him that the payer has been identified by code and that the transaction is complete.) The importance and the difficulty of communicating evidence is exemplified by President Eisenhower's "open-skies" proposal and other suggested devices for dealing with the instability that may be caused by the reciprocal fear of surprise attack. Leo Szilard (37) has even pointed to the paradox that one might wish to confer immunity on foreign spies rather than subject them to prosecution, since they may be the only means by which the enemy can obtain persuasive evidence of the important truth that we are 'making no preparations for embarking on a surprise attack.

It is interesting to observe that political democracy itself depends on a game structure in which the communication of evidence is impossible. What is the secret ballot but a device to rob the voter of his power to sell his vote? It is not alone the secrecy, but the *mandatory* secrecy, that robs him of his power. He not only *may* vote in secret, but he *must* if the system is to work. He must be denied any means of proving which way he voted. And what he is robbed of is not just an asset that he

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might sell; he is stripped of his power to be intimidated. He is made impotent to meet the demands of blackmail. There may be no limit to violence that he can be threatened with if he is truly free to bargain away his vote, since the threatened violence is not carried out anyway if it is frightening enough to persuade him. But when the voter is powerless to prove that he complied with the threat, both he and those who would threaten him know that any punishment would be unrelated to the way he actually voted. And the threat, being useless, goes idle.

An interesting case of tacit and asymmetrical communication is that of a motorist in a busy intersection who knows that a policeman is directing traffic. If the motorist sees, and evidently sees, the policeman's directions and ignores them, he is insubordinate; and the policeman has both an incentive and an obligation to give the man a ticket. If the motorist avoids looking at the policeman, cannot see the directions, and ignores the directions that he does not see, taking a right of way that he does not deserve, he may be considered only stupid by the policeman, who has little incentive and no obligation to give the man a ticket. Alternatively, if it is evident that the driver knew what the instructions were and disobeyed them, it is to the policeman's advantage not to have seen the driver, otherwise he is obliged, for the reputation of the corps, to abandon his pressing business and hail the driver down to give him a ticket. Children are skilled at avoiding the receipt of a warning glance from a parent, knowing that if they perceive it the parent is obliged to punish non-compliance; adults are equally skilled at not requesting the permission they suspect would be denied, knowing that explicit denial is a sterner sanction, obliging the denying authorities to take cognizance of the transgression.²⁷

The efficacy of the communication structure can depend on the kinds of rationality that are imputed to the players. This is illustrated by the game situation known as “having a bear by the tail.” The minimum requirement for an efficient outcome is that the bear be able to incur an enforcible promise and that he be able to transmit credible evidence that he is committed, either by a penalty incurred or by a maneuver that destroys his power not to comply (like extracting his own teeth and claws). But if the bear is of limited rationality, having a capacity for making rational and consistent choices among the alternatives that he perceives but lacking the capacity to solve games-i.e., lacking the capacity to determine introspectively the choices that a partner would make—the communication system must make it possible for him to receive a message from his partner. The partner must then formulate

²⁷ What might be called the “legal status” of communication is nicely developed by Goffman (11, pp. 224, 226): “Tact in regard to face-work often relies for its operation on a tacit agreement to do business through the language of hint—the language of innuendo, ambiguities, well placed pauses, carefully worded jokes, and so on. The rule regarding this unofficial kind of communication is that the sender ought not to act as if he had officially conveyed the message he has hinted at, while the recipients have the right and the obligation to act as if they have not officially received the message contained in the hint. Hinted communication, then, is deniable communication.” He refers to the “unratified” participation that can occur in spoken interaction: “A person may overhear others unbeknown to them; he can overhear them when they know this to be the case and when they choose either to act as if he were not overhearing them or to signal to him informally that they know he is overhearing them.” He points out that the obligation to respond, for example, to an insulting remark that one has inadvertently overheard may depend on whether the overhearing has acquired “ratification.”

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the proposition (choice) for the bear and communicate it to him, in order that the bear may then respond by accepting the promise (now that he sees what the "solution" is) and transmitting authoritative evidence back to his own partner.

### INCORPORATION OF MOVES IN A GAME MATRIX

One is led to suppose that, if a game has potential moves like threats, commitments,

payoffs in all cells that do not correspond to the strategy so selected.²⁸

Specifically, in Figure 8, A, Row would commit himself to ii by subtracting from his own payoffs in the first row sufficiently large quantities—5 in the example shown—to make ii a dominant strategy, i.e., a strategy that he would follow no matter which column the other player selects. The result would be the modified matrix shown in Figure 8, B. (Committing himself to i

FIG. 8

and promises that are susceptible of formal analysis, it must be possible to represent such moves in the traditional form of strategy choices, with the payoff matrix of the original game expanded to allow for the choices among these various moves.

The first point to observe is that a commitment, a promise, or a threat can usually be characterized in a fashion equivalent to the following: to make one of these moves, a player selectively reduces—visibly and irreversibly—some of his own payoffs in the matrix. This is what the move amounts to. We could also say that one openly selects a strategy in advance for responding to the other's choice; but more than selection is required. The player must invoke penalty on his own failure to pursue subsequently the particular strategy of response that he has selected beforehand. And to invoke a penalty on failure to follow a strategy is mathematically equivalent to subtracting the amount of the penalty from one's own

²⁸ Threats, promises, and unconditional commitments have already been illustrated; a more general "reaction function" is illustrated in the accompanying matrix. If Row can attach adequate penalties to his own selection of any cells other than those starred, he leaves Column a simple maximization problem which Column solves by choosing his third strategy. Row has "won" almost his favorite cell; specifically, he has secured for himself the most favorable cell among those that leave Column no lower than his "minimax" value. This is the generalization of the tactic that, for simple two-way or three-way choices, can be identified as a "commitment," "threat," "promise," or combination of them.

<table><thead><tr><th></th><th>I</th><th>II</th><th>III</th><th>IV</th><th>V</th></tr></thead><tbody><tr><th>i</th><td>1</td><td>6<br>11</td><td>10<br>10</td><td>2<br>2</td><td>9<br>10</td></tr><tr><th>ii</th><td>8</td><td>9<br>12</td><td>*4<br>25</td><td>0<br>20</td><td>1<br>3</td></tr><tr><th>iii</th><td>9</td><td>20<br>2</td><td>15<br>16</td><td>*6<br>18</td><td>*1<br>14</td></tr><tr><th>iv</th><td>6</td><td>*2<br>8</td><td>10<br>7</td><td>7<br>5</td><td>4<br>20</td></tr></tbody></table>

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241

with penalty of 5 would yield the matrix in Fig. 8, C.) Can we now build up a larger matrix that represents not only the actual choices of rows and columns in the original game, such as those in Figure 8, A, but also the strategies of *commit, threaten, promise, etc.?* Certainly, once we have specified what moves are available and the order in which they are to be taken. Take the simple game in which Row has the power to commit himself visibly in advance, and Column has first move in the original game, that is, chooses his column before Row makes his final choice of row.

Originally Row, having second move, had four strategies available. He could pick i no matter what; he could pick ii no matter what; he could play i to column I and ii to column II; or he could play ii to column I and i to column II. Including the possibility of commitment, he now has first the choice of committing himself; and to each of these first choices he can attach any one of the four strategies just mentioned for his final move. For example, he can commit himself to ii and play ii no matter what; he can commit himself to ii and play i no matter what; he can commit himself to ii and play

<table><thead><tr><th></th><th></th><th>I</th><th>II</th><th>III</th><th>IV</th><th>V</th><th>VI</th><th>VII</th><th>VIII</th></tr></thead><tbody><tr><td></td><td></td><td>0-I<br>1-I<br>2-I</td><td>0- I<br>1- I<br>2-II</td><td>0- I<br>1-II<br>2- I</td><td>0- I<br>1-II<br>2-II</td><td>0-II<br>1- I<br>2- I</td><td>0-II<br>1- I<br>2-II</td><td>0-II<br>1-II<br>2- I</td><td>0-II<br>1-II<br>2-II</td></tr><tr><td>i</td><td>0; I-i, II-i</td><td>5<br>2</td><td>5<br>2</td><td>5<br>2</td><td>5<br>2</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td></tr><tr><td>ii</td><td>0; I-ii, II-ii</td><td>1<br>0</td><td>1<br>0</td><td>1<br>0</td><td>1<br>0</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td></tr><tr><td>iii</td><td>0; I-i; II-ii</td><td>5<br>2</td><td>5<br>2</td><td>5<br>2</td><td>5<br>2</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td><td>2<br>5</td></tr><tr><td>iv</td><td>0; I-ii, II-i</td><td>1<br>0</td><td>1<br>0</td><td>1<br>0</td><td>1<br>0</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td></tr><tr><td>v</td><td>1; I-i, II-i</td><td>5<br>2</td><td>5<br>2</td><td>0<br>1</td><td>0<br>1</td><td>5<br>2</td><td>5<br>2</td><td>0<br>1</td><td>0<br>1</td><td>0<br>1</td></tr><tr><td>vi</td><td>1; I-ii, II-ii</td><td>1<br>-5</td><td>1<br>-5</td><td>2<br>0</td><td>2<br>0</td><td>1<br>-5</td><td>1<br>-5</td><td>2<br>0</td><td>2<br>0</td><td>2<br>0</td></tr><tr><td>vii</td><td>1; I-i, II-ii</td><td>5<br>2</td><td>5<br>2</td><td>2<br>0</td><td>2<br>0</td><td>5<br>2</td><td>5<br>2</td><td>2<br>0</td><td>2<br>0</td><td>2<br>0</td></tr><tr><td>viii</td><td>1; I-ii, II-i</td><td>1<br>-5</td><td>1<br>-5</td><td>1<br>1</td><td>0<br>1</td><td>0<br>1</td><td>1<br>-5</td><td>1<br>-5</td><td>0<br>1</td><td>0<br>1</td></tr><tr><td>ix</td><td>2; I-i, II-i</td><td>5<br>-3</td><td>0<br>-4</td><td>5<br>-3</td><td>0<br>-4</td><td>5<br>-3</td><td>0<br>-4</td><td>5<br>-3</td><td>0<br>-4</td><td>0<br>-4</td></tr><tr><td>x</td><td>2; I-ii, II-ii</td><td>1<br>0</td><td>*2<br>5</td><td>1<br>0</td><td>*2<br>5</td><td>1<br>0</td><td>*2<br>5</td><td>1<br>0</td><td>*2<br>5</td><td>1<br>0</td></tr><tr><td>xi</td><td>2; I-i, II-ii</td><td>5<br>-3</td><td>2<br>5</td><td>5<br>-3</td><td>2<br>5</td><td>5<br>-3</td><td>2<br>5</td><td>5<br>-3</td><td>2<br>5</td><td>2<br>5</td></tr><tr><td>xii</td><td>2; I-ii, II-i</td><td>1<br>0</td><td>0<br>-4</td><td>1<br>0</td><td>0<br>-4</td><td>1<br>0</td><td>0<br>-4</td><td>1<br>0</td><td>0<br>-4</td><td>0<br>-4</td></tr></tbody></table>

FIG. 9

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THOMAS C. SCHELLING

i to column I, ii to column II; or he can commit himself to ii and play ii to column I, i to column II. Altogether, he has twelve possible strategy combinations.

Column has eight possible strategy combinations: for each of three contingencies he has either of two moves, the moves being I and II, the contingencies being Row's commitment to i, Row's commitment to ii, and Row's non-commitment.

If we put these strategies into matrix form, we get Figure 9. The 12 × 8 matrix of Figure 9 represents the tacit ("non-cooperative") game that corresponds to the players' private decisions on *how to play* the original game. The eight possible strategies available to Column, for example, can be thought of as the eight possible distinct *sets of complete instructions* that he might give an agent who would then play the original game for him—that is, play the game at which he chooses one of two columns, depending on whether and how Row committed himself first. There is no loss to either player in being supposed to play this enlarged game tacitly, since what would have been each player's *adaptations* to the other's prior moves is now fully allowed for in the specification of strategies in the enlarged version of the game; they are strategies of response or adaptation.

This is brought out in the labeling of Figure 9. As before, Column's choices in the original two-move game are labeled I and II; Row's choices, i and ii. Additionally, the symbol "2" will denote Row's commitment to row ii, "1" a commitment to row i, and "0" a decision not to commit himself. In the enlarged game, a single "strategy" for Column is now denoted by three pairs of symbols, such as 0-I, 1-II, 2-I, which would mean, "Choose column I if he does not commit himself, column II if he commits himself to row 1, and column I if he commits himself to row 2." For Row, a

strategy consists of a decision on 0, 1, or 2, plus a pair of symbols denoting how he will react to each of Column's possible choices. For example, 1; I-i, II-i would mean, "Commit to row 1, then choose row 1 no matter what Column does." Knowing the payoffs in the original game, Figure 8, A, the players can identify the payoffs in the enlarged game of Figure 9. We can imagine Row and Column, instead of meeting to play the original game, sending their agents to play for them, each agent fully instructed for all contingencies (i.e., given one particular strategy for the enlarged game). To determine what instructions to give, Row and Column consider the matrix in Figure 9; in effect, they play the tacit game in that matrix, leaving to their agents just the role of messenger.

What is the "solution" of this enlarged tacit game? Or, rather, can we identify an evident solution to the original game? And, if so, how does it show up in the enlarged matrix? The original game clearly has a solution for rational players. (A) If Row is committed to row i, with a penalty of 5 for breaking his commitment, Column can see that row i will be chosen, no matter which column he chooses; Column chooses his preferred cell in the upper row, which is the upper left cell, i,I. And Row knows that, if he commits himself to row i, he gets the payoff in that upper-left cell, which is 2. (B) If, instead, Row commits himself to row ii (subtracts 5 from his payoff in row i), Column chooses II in preference to I; and Row knows he will get 5. Finally, (C) if Row remains uncommitted, Column knows that Row will pick the highest row payoff in the column chosen; thus if Column chooses I, Row takes i, and Column gets 5; if Column takes II, Row takes ii, and Column gets 2. Column prefers I; this leaves Row a payoff of 2; and Row can anticipate it. So Row's best outcome is to commit

THE STRATEGY OF CONFLICT

243

himself to row ii. This is the evident "solution"; it has a payoff of [5²], and it corresponds to the strategy 2; I-ii, II-ii for Row, and to all four strategies containing 2-II for Column. (What Column would have done in contingencies 0 and 1 is of no material consequence, once Row has made his first move.) These are the starred cells in Figure 9, row x. (In effect, Row's first move is a choice of which to play among the three different two-move games, A, B, and C, shown in Figure 8, in which he has second move.)

How do we characterize the cells, or
pairs of strategies, that represent the "solu-
tion" in Figure 9? They constitute a solu-
tion of the kind that is called a *solution in
the complete weak sense* (21, pp. 106-9).
It is arrived at by a process of discarding
"dominated" rows and strategies. A row is
dominated by another row if every payoff
to Row in the dominating row is at least
as good as the corresponding payoff in the
dominated row and at least one payoff is
better. Applying this criterion, the first row
is dominated by the third, and we strike it
out. (The argument might be that Row can
safely eliminate the strategy represented in
the first row, since the third is at least as
good in every contingency and better in
some.) So is the second, so is the fourth;
so are all the rest except the tenth. Neither
the third nor the tenth row dominates the
other, so for the moment we keep them
both. Comparing columns, no single col-
umn dominates another; but, having elimi-
nated all rows but the third and tenth
(arguing, perhaps, that Row would not
choose them anyway), Column can make
his comparison between only the third and
tenth cells in the columns. Now it is ap-
parent that the second column dominates
the first, the third, the fifth, and the seventh.
After striking out those columns that are
dominated in the reduced set of rows, we

can look again at rows iii and x. Originally,
neither dominated the other; but, with the
first, third, fifth, and seventh columns gone,
the tenth row dominates the third. Striking
out the third row, we are left with a single
row, row x, intersected by four columns.
The payoffs are the same in the four inter-
sections, indicating that it is inconsequential
which of those four strategies Column plays,
as long as Row plays the tenth row. (That
is, once Row has committed himself to the
second row, it makes no difference what
instructions Column gives his agent regard-
ing the two contingencies that did not
arise.)

This, then, is the way that a solution to
the original sequential-move game shows
up in the static ("moveless," or simultane-
ous-tacit-choice) game. It is a solution ar-
rived at by discarding dominated strategies,
with the criterion for domination reflecting
only the undiscarded strategies at each
stage. This seems to be the general form
of solution in the enlarged tacit game that
corresponds to a sequential-move game
when the latter has a determinate solution.
The discarding of rows and columns can
actually be identified with the process of
first calculating the rational last move for
all possible sets of prior moves, then, know-
ing what last move would follow each next-
to-last move, calculating the best next-to-
last move for all possible sets of prior
moves and so on back to the best first move
of the game.

While it is instructive and intellectually
satisfying to see how such tactics as threats,
commitments, and promises can be ab-
sorbed into an enlarged, abstract "super-
game" (game in "normal form"), it should
be emphasized that we cannot learn any-
thing about those tactics by studying games
that are already in normal form. The ob-
jects of our study, namely, these tactics
together with the communication and en-

244

THOMAS C. SCHELLING

forcement structures that they depend on, and the timing of moves, have all disappeared by the time the game is in normal form. What we want is a theory that systematizes the study of the various universal ingredients that make up the move-structure of games; too abstract a model will miss them.²⁹

The matrix representation of a sequential game does help emphasize, however, that the formal "determinateness" of games that are resolved by tactical moves does not detract from their essential game-of-strategy character. A threat "wins" and determines an outcome only because it induces the other player to choose in one's favor. The other player retains his original freedom of choice; and his choice still depends on his anticipation of the threatener's final choice. The threatener's first choice—to threaten or not—thus depends on what he expects the threatened player to expect the threatener to do. The reciprocal-expectation character of the game remains; the threat, like the unconditional commitment or like the

²⁹ Incidentally, casting a particular game into supergame matrix form is generally not a feasible technique of analysis; the number of rows and columns (i.e., the number of sequential-move strategies) becomes astronomically large, even for quite simple games. To illustrate, consider a 3×3 matrix, with Column to choose first; add a prior opportunity for Row to commit himself to any partially or fully specified strategy of response; finally, to study the "defense" against threats, allow Column a still earlier opportunity to commit his choice of column. That is, Column may first commit himself unconditionally if he pleases, Row may then commit himself conditionally in whatever way he pleases, then Column chooses a column and finally Row chooses a row. Let us not complicate the game by limiting sizes of penalties or by inserting any uncertainty or imperfect communication system. This "simple" game, which is not terribly difficult to analyze in its extensive form, turns out to have more than a "googol" (1 followed by a hundred zeros) of columns.

broader concept of reaction function when many choices of action are available, works by constraining another player's expectations through the manipulation of one's own incentives.

## THE PARADOX OF STRATEGIC ADVANTAGE

It is, of course, a corollary principle that if the payoff matrix to begin with had already shown values for one of the players reduced in the same pattern as that in which he would reduce it deliberately at the winning move, he simply wins without needing to make the move overtly. This is the point that, in an earlier article by the author (30), was referred to as an abstract analogy of the principle that, in bargaining, weakness may be strength. There is probably no single principle of game theory that epitomizes so strikingly the mixed-motive game as this principle that a worsening of some or even all of the potential outcomes for a particular player and an improvement in none of them may be distinctly—even dramatically—advantageous for the player so disadvantaged. It explains why a sufficiently severe and certain penalty on the payment of blackmail can protect the potential victim, how the burning of bridges behind one's self may dishearten an enemy and induce his retirement, or why a lady might, in an earlier era, defy the search party by haughtily placing the sought object in her bosom.³⁰

It was reported unofficially during the Korean War that when the Treasury Department blocked Communist Chinese financial assets, it also knowingly blocked some non-Communist assets as a means of

³⁰ It also explains why a "promise" to abstain from a choice that would damage the other player may not be welcomed by him. A promise that permits him safely to make a particular choice may assure us that he would make it,

THE STRATEGY OF CONFLICT

245

immunizing the owners against extortionate threats against their relatives still in China. Quite likely, for owners located in the United States, the very penalties on transfer of funds to Communist China enhanced their capacity to resist extortion. Deliberately putting one's own assets in a form that made evasion of the law more difficult, or lobbying for more severe penalties on illegal transfer of one's own funds, or even getting one's self temporarily identified as a Communist sympathizer so that his funds would be blocked might have been an indicated tactic for potential victims, to discourage the extortionate threat in advance.

A similar principle is reflected in Article 26 of the Japanese peace treaty, which gives the United States certain claims if subsequent Japanese territorial concessions to

so that we can count on it and make some prior choice that is to his disadvantage. By the same token, adding values selectively to the other's payoffs can absolutely worsen his position—if we have a means of making the addition. In the accompanying matrix, assuming Row has first move, Row can “win”—he can gain 7 at Column's expense—if he unilaterally guarantees to compensate Column in the event of an outcome at i,II, the compensation coming out of his own winnings. If he promises to pay 2 to Column in such an event, he gets 8; Column gets 3; otherwise, without the promised compensation, Row cannot choose i, and the outcome is at ii,I with payoffs of 1 and 10, respectively. Column obviously prefers that Row be unable to commit himself to confer the “benefit.” (If the blackmailer cannot scale down his demands to where what he demands, plus the fine for paying blackmail, are less than the damage he threatens, he may offer to pay his victim's fine. This guarantees what his victim's response to the threat will be; so the threat is made, to the disadvantage of the victim.)

<table><tr><td>2</td><td></td><td>1</td></tr><tr><td>0</td><td>10</td><td></td></tr><tr><td>10</td><td></td><td>0</td></tr><tr><td>1</td><td>2</td><td></td></tr></table>

becomes

<table><tr><td>2</td><td></td><td>3</td></tr><tr><td>0</td><td>8</td><td></td></tr><tr><td>10</td><td></td><td>0</td></tr><tr><td>1</td><td>2</td><td></td></tr></table>

other powers are more favorable. When the Japanese were reported to be under pressure from the Russians for additional territorial concessions in 1956, Secretary of State John Foster Dulles (9) pointedly described that article of the treaty in his press conference and said that he had recently “reminded the Japanese of the existence of that clause.” The evident intention was to strengthen Japanese resistance; and it may be supposed that by “reminding” the Russians of the same clause through the medium of his press conference, Dulles helped to provide the Japanese with the familiar bargaining claim, “If I did it for you, I’d have to do it for everyone else.” It was, in terms used earlier, a “commitment” secured by the penalty of a forfeit to the United States. (Paradoxically, the United States could not give the Japanese the benefit of this bargaining gimmick unless the United States were patently motivated to take advantage of its claim if the tactic failed.)

## “STRATEGIC MOVES”

If the essence of a game of strategy is the dependence of each person's proper choice of action on what he expects the other to do, it may be useful to define a “strategic move” as follows: A strategic move is one that influences the other person's choice, in a manner favorable to one's self, by affecting the other person's expectations on how one's self will behave. One constrains the partner's choice by constraining one's own behavior. The object is to set up for one's self and communicate persuasively to the other player a mode of behavior (including conditional responses to the other's behavior) that leaves the other a simple maximization problem whose solution for him is the optimum for one's self, and to destroy the other's ability to do the same.

There is probably no contrast more striking

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THOMAS C. SCHELLING

ing, in the comparison of the mixed-motive and the pure-conflict (zero-sum) game, than the significance of having one's own strategy found out and appreciated by the opponent. Hardly anything captures the spirit of the zero-sum game quite so much as the importance of "not being found out" and of employing a mode of decision that is proof against deductive anticipation by the other player.³¹ Hardly anything epitomizes strategic behavior in the mixed-motive game so much as the advantage of being able to adopt a mode of behavior that the other party will take for granted.

It can, of course, be an advantage in the zero-sum game to have the opponent believe firmly in a particular mode of play for one's self, but only if that belief is in error. In the mixed-motive game, one is interested in conveying the *truth* about his own behavior—if, indeed, he has succeeded in constraining his own behavior along lines that, when anticipated, win.

Another paradox of mixed-motive games is that genuine ignorance can be an advantage to a player if it is recognized and taken into account by an opponent. This paradox, which can arise either in the coordination problem or in the immunity from a threat, has no counterpart in zero-sum games.

A related contrast between zero-sum and mixed-motive games is that the former, characterized by secrecy and purely unilateral decision processes, can, with reason, be viewed as having rational solutions that are necessarily symmetrical, while the mixed-motive game, if it has strategic moves, is inherently asymmetrical. The

³¹ Concerning this point, Von Neumann and Morgenstern say (38, p. 147): "We have placed considerations concerning the danger of having one's strategy found out by the opponent into an absolutely central position."

richness of the game in its strategic potentialities depends on the fact that all moves are not foredoomed to neutralization by a symmetrical move structure stamped on a game by definition and known to the players as inevitable. (This point concerning symmetry is argued at length in an earlier paper by the author [32].)

### III. Co-ordination of Expectations in the "Pure"-Bargaining Game

There remains the tantalizing case of the "pure"-bargaining game—the game in which, to express it in abstract form, the two or more participants must overtly agree on the division of a sum of gains, or forego the gains altogether—with no obstacles to communication, no secrets, and no "moves" but sheer negotiation. The gains available for distribution may or may not be completely divisible; they may or may not be of similar utility to the two parties; but, among the array of potential efficient outcomes, more for one means less for the other, and they both know it. This kind of game is of special analytical importance because some element of it is present in any bargaining game that does not resolve itself in a sequence of purely tactical moves, i.e., moves (of the kind discussed in Part II of this paper) that make the outcome formally determinate; and this element of "pure" bargaining is in fact the most tantalizing part of non-zero-sum game theory.³²

³² It should be emphasized that a solution to the "pure"-bargaining problem cannot necessarily be superimposed on, say, the Von Neumann-Morgenstern "solution," as though it applied to the residual "range of indeterminacy" left by another theory. Consider the real estate example mentioned in footnote 16 on p. 222 and suppose that B does not yet exist. Something determines a price between 10 and 25. Suppose that the "something," in a particular case, would have led to a price of 12. B appears, approaches C,

THE STRATEGY OF CONFLICT

247

In a two-person model, this game can be represented by a set of points in an X, Y plane, one of which is to be chosen by agreement between players X and Y, to yield rewards denoted by the X, Y co-ordinates, plus a zero point that obtains by default in the absence of an agreed choice. We now have a game in which any outcome is a point from which at least one party would be willing to retreat for the sake of agreement, and the other knows it. For a party to insist on any such point (other than his least favorite among the efficient points) is to take a pure "bargaining position," since one always *would* take less rather than reach no agreement at all and since one always *could* recede if retreat proved necessary to agreement. The only reason for accepting any point short of one's most favored point is the expectation that the other will not concede more; yet there is no logical reason for anybody to expect

and threatens to offer A 13. What is it worth to C to prevent this?

Something must have convinced A that he could not get more than 12 (even, under the strict terms of the theory, knowing that C's reservation price was 25). B can destroy that conviction; he can disturb expectations; he can force C to contradict his earlier behavior, change his role, break his pattern of bargaining, and begin anew after the shock. Who knows what price will be settled on next time-what price the "something" of our unspecified theory will determine with B in there just messing around. The damage to C that B can cause, just by participating in the bargaining process, gives him the basis for an extortionate threat. And there is no a priori reason to suppose that C's judgment of what it is worth to keep B from stirring things up is limited by how badly B might want the house, especially when B's interest is concentrated on getting money, not the house, anyway. Thus the limits set by the original "solution," which left a residue of indeterminacy, may be compromised by the theory that one has to cultivate for dealing with that residue.

anything except what he expects to be expected to expect. Agreement is reached when the two parties' expectations converge on a particular point in such fashion that neither expects the other to concede anything further, knowing that whether or not the other will concede depends simply on what the other expects to be expected to expect. Somehow the infinitely reflexive expectations have to converge on a particular outcome that both parties have come to consider inevitable.

What causes their expectations to converge on some resting place in this fashion? What causes some particular outcome to acquire this aura of finality? What makes each recognize that they both recognize that bargaining has reached an end? And how do we discover what characteristics of a particular outcome make it eligible for this sort of election? How do we discover what kinds of tactics can lead to this point of common consent?

The author (31) argued in an earlier paper that what brings expectations into convergence and brings the negotiations to a close is the "intrinsic magnetism" of particular outcomes, especially those that enjoy prominence, uniqueness, simplicity, precedent, or some rationale that makes them qualitatively differentiable from the continuum of possible alternatives. But what is "intrinsic magnetism"? In rationalizing this idea (which seems supported by empirical evidence, some of which was mentioned in that article) it was proposed that there is often a close analogy between the outcome of an *explicit* bargain and the outcome that would seem most plausible if agreement had had to be reached by a *tacit* process. In tacit bargaining the parties have to let themselves be co-ordinated toward a common choice, much as they would follow the lead of an arbitrator in search of

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THOMAS C. SCHELLING

the agreement that they both want. The conflict of interest was dominated by the need for co-ordination.

The argument was then advanced that a similar psychic phenomenon may occur even when communication is present. Explicit bargaining, like tacit bargaining, requires some "co-ordination" of the participants' expectation of the outcome. It was argued that if in a given game context the participants are intellectually capable of recognizing some unique outcome that can command their agreement through its power to co-ordinate their choices, they will probably be conscious of the focal properties of such an outcome even when they are not dependent on it for their co-ordination. That is, the kinds of properties that would make a particular outcome a strong candidate for election in case of tacit bargaining are likely to be the kinds of properties that have a strong appeal for attention when an explicit agreement is being sought.

Furthermore, the recognition of such a particular outcome by one party implies his recognition that the other can recognize it, too, and his recognition that the other recognizes that the first recognizes it, since in the tacit case the only virtue of such a solution is that each side can recognize that they both will recognize it as inevitable. Neither can disguise or suppress his anticipation because the whole basis for it is the recognition of it as a shared expectation. One's consciousness of the focal properties of such an outcome in the case of explicit bargaining is a consciousness that both are conscious of it. There is, so to speak, involuntary tacit consent that both recognize the attraction of this particular point. What was vaguely referred to as "intrinsic magnetism" is therefore not a mystical concept but can be rationalized as an intellectual

phenomenon experimentally identifiable in games that demand co-ordination of strategies. The co-ordination of expectations in explicit bargaining is at least analogous to the co-ordination of strategies when communication is cut; and it may be the same intellectual or psychic phenomenon.³³

But, although the analogy with tacit bargaining may exercise this influence on the participants, it is by no means unique in doing so; in many cases the tacit-bargaining analogue would not itself have a very powerful solution. This is consequently an influence, not the sole influence, on the mutual choice in a pure-bargaining game. What other influences are there that can focus expectations ultimately on some particular outcome and that are analyzable within the framework of game theory rather than dependent on the whole field of psychology?

The writer suggests a generalization of the phenomenon represented in the tacit-bargaining analogy. It is that in these completely "indeterminate" situations, in which the logical structure of the game provides no rationale for the convergence of expectations anywhere, any mutually perceived analogue may exercise a power of suggestion that can focus expectations and thereby bring the players under its discipline. And this analogue may simply be an al-

³³ It should be added that the "intrinsic magnetism" of particular outcomes in a bargaining situation or in a pure co-ordination problem gets some support and clarification from the very substantial body of experimental evidence provided by the Gestalt psychologists. Their work on the perception of physical forms is especially pertinent. For example, incomplete shapes were shown to people whose vision was damaged in part of the eye, and they often saw the shapes as complete rather than as partial. But the particular shapes that they "completed" for themselves followed certain principles of simplicity; and unfamiliar "simple" figures were completed

THE STRATEGY OF CONFLICT

249

ternative game—that is, an alternative bargaining situation—that differs in having some prescribed set of moves or communication channels that would make it a determine game. In other words, if there is a variant of the game in question that would have a determinate solution, that solution of the variant game provides a focal point for the convergence of expectations in the indeterminate game in question.

where very familiar, but less simple, figures were not. Koffka (19, pp. 140–42, 146–47, 109–10) refers to “spontaneous organization in simple shapes.” We are surrounded by skewed rectangles; but what we “see” about us is rectangles, not departures from perfect rectangles, because “the true rectangle is a better organized figure than the slightly inaccurate one would be.” Adverting to the minimum-maximum properties of stationary processes, Koffka suggests that psychological processes will have these properties: “For we can at least select psychological organizations which occur under simple conditions and can then predict that they must possess regularity, symmetry, simplicity. This conclusion is based on the principle of isomorphism, according to which characteristics of the physiological processes are also characteristic aspects of the corresponding conscious processes.” And, “Thus we have gained a general, though admittedly somewhat vague, principle to guide us in our investigation of psychophysical organization. . . . The principle . . . can briefly be formulated like this: psychological organization will always be as ‘good’ as the prevailing conditions allow. In this definition the term ‘good’ is undefined. It embraces such properties as regularity, symmetry, simplicity and others which we shall meet in the course of our discussion.”

If individual perception and “organization” of forms follow these constraints, the process of “mutual perception” and “mutual organization of forms” involved in the convergence of expectations must depend on similar restraints at least as rigorous. And, since the non-zero-sum game requires some ultimate joint “organization of form,” so to speak, a normative theory of strategy (not just a descriptive psychology) must take these restraints into account.

## AN ILLUSTRATIVE GAME

An example—which here depends entirely on the reader’s intuitive appreciation of it but should be susceptible of experimental verification—will make the point clear. Suppose two players, X and Y, are to pick jointly a point within or on the boundary of the segment of a circle OPQ shown in Figure 10; if they can agree on such a point, they receive sums of money corresponding to the X and Y co-ordinates, and if they reach no agreement they get zero apiece. Here is the pure-bargaining game. Now let us give specific form to the way in which

FIG. 10

they will *express* these choices. X and Y are to choose, respectively, the two components of a vector from the origin, namely, the direction and the length. X, so to speak, will point the steering wheel, and Y will work the accelerator; or X will aim the gun horizontally, and Y will control its elevation. Mathematically speaking, we have introduced nothing of substance into the game, since *explicit agreement is required* or they both get nothing. The logical structure of the game is unaffected by how we require the two participants to translate the point they choose into a co-ordinate system or otherwise to identify it for us. But the details we introduce may have connotative significance and may influence the players'

250

THOMAS C. SCHELLING

expectations of the outcome and hence the
outcome itself.

Note that the particular topical content
that we have introduced into the game is
asymmetrical; the roles of the two players-
or rather the roles that seem to be implied
for the two players—and the manner in
which they must divide between them the
communication of their choice to us differ
in a way that could have logical signif-
icance. X makes a distributive choice; in
giving the direction of the vector, he in-
dicates in what proportions he and his part-
ner will divide whatever sum they ultimate-
ly get. Y is left responsibility for the effi-
ciency of the solution; he, in effect, appears
to decide how near to the boundary the
point will be, that is, how large a joint sum
they will be dividing between them.

The reason we say that this difference in
their *apparent* roles *could* have logical sig-
nificance is that there are very similar
games in which they *would* have logical
significance and would make the game quite
determinate. Let us look at some alternative
games. One variant would have them
choose separately without communicating;
in this tacit game Y seems to be in a com-
pletely subordinate position. He is respon-
sible for a choice that affects him and
his partner together; Y alone is responsible
for how the gain is divided. Whatever X
does, Y can only benefit by choosing the
boundary curve; X is free to secure the
proportion denoted by Q, hence the amount
denoted by Q.

Suppose, instead, they take turns mov-
ing, without prior agreement. If X can move
first, he points the vector toward Q; again
Y has lost and can only lay the point down
on the boundary at Q. Now suppose Y gets
to move first: he loses again; whatever
length he chooses, X will distribute it in the
ratio denoted by Q, and Y can gain only by
choosing the boundary curve. If, in still

another variant, X can commit himself to
a choice, he again wins as though he had
first choice. But if, alternatively, Y is able
to commit himself, it does him no good;
the commitment is only a means of acquir-
ing first choice, which, as we saw, does
him no good. Y could win only if he (not
X) could commit himself to a conditional
choice—out on the boundary if X chooses
favorably for Y, or all the way back at
zero if X chooses unfavorably—i.e., if he
could threaten mutual destruction in order
to win. The threat is the only tactic in
which even an asymmetry in Y's favor can
do him any good. (Even the incentive to
cheat on an agreement would always be in
X's favor, never in Y's.)

Now let us go back to the original game,
in which there is none of this superstructure
about order of moves, etc. Suppose X de-
clares a direction for the vector: he aims it
at Q. This is his offer, and Y says, "No." X
says, "That's it," and writes it out on the
sheet of paper on which he is to transmit
his part of the joint choice. He simply says
that he has no intention of changing it,
that there is nothing Y can do but accept
it, and that if Y chooses anything but the
boundary point at Q he will only hurt him-
self as well as X. X sits in a chair on the
other side of the room and says he does not
plan to get up until Y has written his own
part of the bargain and the game is over.

What can Y do? First, note that Y has no
comparable tactic. Y cannot write down a
number that favors himself and complacent-
ly cross the room and light a cigar. He can-
not even pretend that he has made his
"final" choice, unless he has already written
100 per cent as his choice of length. Thus
in some mechanical sense X has a more
"stable" bargaining position than Y. X can
create a status quo by declaring what he
proposes to choose. Declaring does not con-
stitute a commitment—our rules enforce no

CONFLICT RESOLUTION

VOLUME II NUMBER 3

THE STRATEGY OF CONFLICT

251

commitment. Y can still threaten non-agreement, to force X to get up and change the direction of his choice more toward P or, for that matter, all the way to P. But the roles of the two players are different. X has a complacent role; he can announce and sit still. Y has to make threats and demands. If X could leave the room, he could tell Y he did not plan to come back, having made his choice. Y could not leave the room and say that he was not coming back, until he had made the choice that satisfied X. (This tactic would be a means of commitment, and so is against the rules.)

Now it should be emphasized that there is nothing in the logic of the situation that gives X any advantage over Y. It remains true that, until the two of them reach explicit agreement, they both stand to get nothing; and Y's ability to withhold agreement until X offers a satisfactory outcome is not in any logical way diminished by comparison with what it would be if their mode of expressing their choice had never been prescribed. It just looks as though X had what would be described as a strong psychological advantage. This advantage bears no relation to the mathematical structure of the game and has no meaning within any mathematical formulation of the problem. Nor is it a "tactical" advantage, of the kind involved when one party can make a threat or a commitment or take advantage of discriminatory communication channels. It is, in a strict sense, psychological. It is created by some topical content that is mathematically inessential to the game. It arises in the power of suggestion that resides in certain procedures, details, or connotative superstructure. Essentially, it resides in the fact that X and Y both perceive that in some way X has an undeniable psychological advantage.³⁴

The fact that the advantage rests on intuition is not an objection. We are dealing

with a process that involves intuition; and the argument here is that intuition in game situations may rest on something that is objectively analyzable. It may be the power

³⁴ In particular, it should be noted that the outcome of the game proposed here would violate any axiom of symmetry. The abstract version of the game is completely symmetrical; the two players have identical capacities to withhold action until explicit agreement is reached. The mechanical asymmetry would be a logical part of the abstract game only if there were limits on communication or a prescribed order of moves or if players could commit themselves to their choices before reaching joint agreement. In another paper the author (32) has argued that the "pure" (moveless) explicit bargaining game of the type analyzed in this section may actually not "exist." The point of the argument is that a well-defined game must have a more explicit legal structure than the game informally characterized above on p. 249; in particular, there must be an explicit formula or rule for termination, and there must be an operational definition of what constitutes "agreement" during the game or at its closure. The problem of making up such a game is that, when attention is given to the legal details, explicit-communication games often appear to give way, by default, to a tacit version at some "last moment" unless agreement already exists; and if, then, that tacit version has an efficient solution, it becomes a minimax strategy for a player to force the game into its tacit stage rather than reach prior agreement. If the tacit game has no compelling solution, explicit bargaining acquires the tactical significance of moves designed to create focal points for co-ordination in the final stage, each player interested in creating focal points favorable to himself. The reader can experiment with these possibilities by considering precisely how the game discussed in the text might be given rules sufficiently explicit to permit actual play, i.e., rules for termination (by time limit, roll of dice, etc.) and for establishing whether a legal contract exists at termination or sooner. From this point of view it is not quite fair to say that the nominal asymmetry in the game discussed in the text is not part of the logical structure: we have not given the game a closed logical structure yet. (We have left it in the form in which game theory traditionally leaves it.)

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THOMAS C. SCHELLING

of suggestion of analogous games: the ability of the participants to recognize—or, rather, their inability not to recognize—that various details of their situation and the roles they play point toward particular outcomes, because in situations only slightly different they would *determine* those outcomes. Intuition that is governed by analogy, when the analogues are objectively analyzable, is analyzable.

Of course, we must avoid assuming that everything the analyst can perceive is perceived by the participants in a game. The proposition advanced here relates only to the appreciation by actual participants that certain outcomes would be “obvious” in powerfully similar situations—conceivably, even situations that impress their similarity so strongly on the players that the latter do not consciously recognize the difference.

For this reason, game analogues that have sophisticated mathematical solutions (except when the same solution can also be reached by an alternative, less sophisticated route) would not have this power of focusing expectations and influencing the outcome. Or, rather, they would have this power only if the players perceived each other to be mathematicians. Here is the empirical interpretation of such “solutions” as those of Braithwaite, Nash, Harsanyi, and others. It is that the mathematical properties of a game, like the aesthetic properties, the historical properties, the legal and moral properties, the cultural properties, and all the connotative details, can serve to focus the expectations of certain participants on certain solutions. If X and Y in the above game were mathematical game theorists, they might mutually perceive and be powerfully affected by potential solutions that had compelling mathematical properties.³⁵ Each would transcend, and know that the other would transcend, such adventitious details as how they were to express their

choice. Their analogue would not necessarily be the same one that influences non-mathematician players of the same game.

Thus mathematical solutions are one species of a genus of influences that have the power to focus expectations; but they work through the same psychic mechanism—this power of suggestion that is able to bring expectations into convergence—as the other species. When husband and wife, separated in a department store, gaily traiapse off to the Lost and Found by a tacit and jocular mutual appreciation that it is the “obvious” place to meet, two mathematicians in the same situation—each aware that both are aware that both are mathematicians—might look for a geometrically unique point rather than one that depended on a play on words.³⁶

## The nature of convergent expectations

³⁵ In many cases, though, these properties would be a uniqueness or symmetry that would have non-mathematical definitions and non-mathematical appeal, too, or would happen to coincide with qualitatively distinguishable points that could be rationalized in an equally compelling non-mathematical way.

³⁶ The main point here is independent of whether, under the “rules” of game theory, a rational player must be presumed to know as much mathematics as he ever has need for. We are dealing here with the players’ shared appreciations, preoccupations, obsessions, and sensitivities to suggestion, not with the resources that they can draw on when necessary. If the phenomenon of “rational agreement” is fundamentally psychic—convergence of expectations—there is no presumption that mathematical game theory is essential to the process of reaching agreement, hence no basis for presuming that mathematics is a main source of inspiration in the convergence process. The writer (32) has tried to argue that *no efficient outcome* of a bargaining game, however it divides the gains between the players, can constitute refutation of the rationality hypothesis as long as rationality is not given a definition implicit in a particular theory.

[LC]

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THE STRATEGY OF CONFLICT

tions. One may or may not agree with any particular hypothesis as to how a bargainer's expectations are formed, either in the bargaining process or before it and either by the bargaining itself or by other forces. But it does seem clear that the outcome of a bargaining process is to be described most immediately, most straightforwardly, and most empirically in terms of some phenomenon of stabilized convergent expectations. Whether one agrees explicitly to a bargain or agrees tacitly or accepts it by default, he must, if he has his wits about him, expect that he could do no better and recognize that the other party must reciprocate the feeling. Thus the *fact* of an outcome, which is simply a co-ordinated choice, should be analytically characterized by the notion of converging expectations.

The intuitive formulation, or even a careful formulation in psychological terms, of what it is that a rational player "expects" in relation to another rational player in the "pure" bargaining game, poses a problem in sheer scientific description. Both players, being rational, must recognize that the only kind of "rational" expectation that they can have is a fully shared expectation of an *outcome*. It is not quite accurate—as a description of the psychological phenomenon—to say that one expects the second to concede something or to accept something; the second's readiness to concede or to accept is only an expression of what he expects the first to accept or to concede, which, in turn, is what he expects the first to expect the second to expect the first to expect, and so on. To avoid an "ad infinitum" in the descriptive process, we have to say that both sense a shared expectation of an *outcome*; one's expectation is a belief that both identify the *same* outcome as being indicated by the situation, hence as virtually inevitable. Both players, in effect, accept a common authority—the power of the game to dictate

its own solution through their intellectual capacity to perceive it—and what they “expect” is that they both perceive the same solution.

Viewed in this way, the intellectual process of arriving at "rational expectations" in the full-communication "pure"-bargaining game is virtually identical with the intellectual process of arriving at a co-ordinated choice in the tacit game. The actual solutions might be different because the game contexts might be different, with different suggestive details; but the intellectual nature of the two solutions seems virtually identical, since both depend on an agreement that is reached by tacit consent. This is true because the explicit agreement that is reached in the full-communication game corresponds to a priori expectations that were reached (or in theory could have been reached) jointly but independently by the two players before the bargaining started. And it is a tacit "agreement" in the sense that both can hold confident rational expectations only if both are aware that both accept the indicated solution in advance as *the* outcome that they both know they both expect.

**A hypothetical experiment.** As an illustration of what the author has in mind, the following hypothetical experiment can be considered. (Hopefully, some such experiment could be carried out.) It is offered here as a conceptual analogue or, conceivably, an empirical test of the psychic phenomenon involved in bargaining.

The first stage in the experiment is to invent a machine, perhaps on the principle of the lie detector, that will record or measure a person's "recognition" or the focus of his attention or his alertness or his excitement. What we want is a machine that measures, as the player scans an array of possible outcomes in some orderly fashion, the extent to which particular outcomes

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THOMAS C. SCHELLING

catch his attention or generate excitement
in the course of actual bargaining.

Given the machine, set up a bargaining
game. For simplicity, make it one in which
there are certain gains to be shared when
agreement is reached on the shares. Give
the game enough "topical content" to pro-
vide some room for argument, casuistry,
alternative rationales, etc.; that is, provide
more than a bare mathematical range with
a conspicuous mid-point.

Now have the two players connected to
their machines in such a way that each can
see the meter on his own machine, each
can see the meter on the other's machine,
and each is aware that both are aware that
both can see both meters. In other words,
they mutually perceive that they both can
see each other's reactions to particular out-
comes as they come within view of the
scanning device. We employ a mechanical
scanning device, which moves about in the
range of possible outcomes, pointing to,
lighting up, or focusing on one possible
outcome after another. It follows perhaps
some regular course, perhaps a random
course. Let this machine scan; let the play-
ers watch it scan, watch their own and
each other's meters, and watch each other's
faces if they wish to.

Finally, we go through with the game;
and there may be several variants. An in-
teresting possibility would be to exclude
explicit bargaining and simply let the scan-
ning proceed, back and forth or round and
round among the array of alternative out-
comes. We watch to see whether the re-
corded reactions of the two players tend
eventually to converge on a single outcome,
in the sense that their involuntary, physi-
cally identifiable reactions are at some kind
of maximum for the same particular out-
come among all those to which the scanning
device elicits their reactions. (For control
purposes, we might once have subjected

each player to a scanning session in which
the other player was absent, to get some
notion of each player's reactions independ-
ently of any interaction between the play-
ers.) If convergence does occur, we have
certainly identified a significant phenome-
non, whether or not we can allege that this
is *the* psychic bargaining process. We shall
have demonstrated (a) that players do re-
act to the content of the bargaining situa-
tion and (b) that their reactions are subject
to a mutual interaction that results from
the fact that each can see the other's reac-
tion and each knows that his own visible
reaction is yielding information about his
own expectations. (The writer conjectures
that, like Lot's wife, players will often be
unable to keep their attention from being
drawn to particular outcomes, even unfavor-
able outcomes, and that a conscious effort
to ignore a "focal point" may often enhance
the focal power.)<sup>37</sup>

Another variant would be to let the
players bargain explicitly during the scan-
ning and metering, with the scanning de-
vice inexorably eliciting their physical re-
actions in the course of the discussion in a

37 The following observation, quoted by K. Koffka (19), may be hard to believe but is certainly to the point: "When an expert . . . follows a football game attentively he will also notice that the goalkeeper, standing before the comparatively large goal, is more often hit than can be accounted for by the mere adventitious kicking of the contestants, even when one takes account of the fact that the goalkeeper whenever he can will try to intercept the ball. The goalkeeper furnishes a prominent point in space which attracts the eyes of the opposing kickers. If the motor activity takes place while the kicker's eye is fixed on the goalkeeper, then the ball will generally land near him. But when the kicker learns to reconstruct his field, to change the phenomenal 'centre of gravity' from the goalkeeper to another point in space, the new centre of gravity will have the same attraction as the goalkeeper had before."

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255

manner visible to both of them. (We could even, in this latter case, let a player adduce the evidence of the visible reaction meters if he wished to as a bargaining tactic, pointing out to his partner, for example, that the latter "obviously" cannot expect to hold out for, say, the $60 he is verbally demanding when it is clear from his blood pressure that his mind is settled on $40.)

This experiment would rest on three hypotheses. First, that an individual player would have physically identifiable "reactions" upon contemplating different alternatives among the range of possible game outcomes and that these reactions would be conspicuously different among the different alternatives. Second, that these reactions, when the player knows that they are naked to his partner's eye, would behave in a manner suggestive of bargaining; i.e., that the reactions of the two players, when visible to both of them, would interact in a kind of "bargaining process." Third, that this measured phenomenon, which we liken to a bargaining process, is part of, or is involved in, or is related to, *the* bargaining process as defined in the ordinary way. (An experiment of the sort described might prove especially interesting for the case of more than two persons.)

The experiment has not been carried out and is not adduced as evidence. It has been described here in order to give an operational representation of the theoretical system that the author has in mind in referring to the "convergence" of expectations and to suggest that the convergence that ultimately occurs in a bargaining process may depend on the dynamics of the process itself and not solely on the a priori data of the game.

**Some dynamic characteristics of focal-point solutions.** The dependence of a "focal-point" solution on some characteristic that distinguishes it qualitatively from the

surrounding alternatives has important dynamic considerations. For example, it often makes small concessions less likely than large ones; it often means that the focal point is more persuasive as an *exact* expected outcome than as an approximation. If a bargainer has persistently been unsuccessfully demanding 50 per cent, compromise at 47 per cent is unlikely; the small concession may be a sign of collapse. Qualitative principles are hard to compromise, and focal points generally depend on qualitative principles. One cannot expect to satisfy an aggressor by letting him have a few square miles on this side of a boundary; he knows that we both know that we both expect our side to retreat until we find some persuasive new boundary that can be rationalized.

In fact, a focal point for agreement often owes its focal character to the fact that small concessions would be impossible, that small encroachments would lead to more and larger ones. One draws a line at some conspicuous boundary or rests his case on some conspicuous principle that is supported mainly by the rhetorical question, "If not here, where?" The more it is clear that concession is collapse, the more convincing the focal point is. The same point is illustrated in the game that we play against ourselves when we try to give up cigarettes or liquor. "Just one little drink," is a notoriously unstable compromise offer; and more people give up cigarettes altogether than manage to reach a stable compromise at a small daily quota. Once the virgin principle is gone, there is no confidence in any resting point, and expectations converge on complete collapse. The very recognition of this keeps attention focused on the point of complete abstinence.

Sometimes the focal point itself is inherently unstable. In that case it serves not as an outcome but as a sign of where to

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look for the outcome. This is often true of a "test vote" in a legislative body or a "test issue" that arises in the relations between the players in some continuing game. Often it is a challenge or a dare or an act of defiance that, by its nature, must either elicit a submissive response from the other party or be submissively withdrawn. It is a small piece of the game that comes to symbolize the game itself, setting a pattern of expectations that extends beyond the substance of the point involved. Sometimes it is so intended and constitutes a deliberate tactic; in other cases the act or the issue develops an unintended symbolic significance, making compromise impossible.

Diplomatic recognition of the Communist regime in China, loyalty oaths at universities, a strike settlement in a key industry, surrender of the floor to an interrupter at a cocktail party, or the vote on some particular motion at a political convention may all have this kind of significance. Sometimes, it is true, the outcome on this particular issue simply yields evidence of how other issues would be decided, as when a test vote indicates exactly how large the opposition to a measure is; but often the particular issue is not representative of the rest of the game, it just acquires tacit recognition as a clue to all that will follow, so that each side is the prisoner or beneficiary of the mutual expectations that are created.

Often this phenomenon can be identified as an actual signal in a co-ordination game. The members of an unorganized coalition can often recognize the potentialities of concerted action without being sure that "agreement" exists to act in concert. One wants to know how everyone else is going to act and whether everyone else will do what he knows he ought to. A test vote in a legislature or some particular simultaneous action among the group, like a mass protest, is often a means of "ratifying" the

existence of the coalition and of demonstrating that everybody expects everybody else to act in concert. But even in a two-person game, as typified by the dare, the phenomenon of psychological dominance or submissiveness may prove to be psychologically identical with the resolution of a bargaining game.

This process, by which particular moves in a game or offers and concessions achieve symbolic importance as indicators where expectations should converge in the rest of the game, seems to be an area in which experimental psychology can contribute to game theory.

#### IV. Game Theory and Experimental Research

The foregoing discussion suggests several conclusions about the methodology appropriate to a study of bargaining games. One is that the mathematical structure of the payoff function should not be permitted to dominate the analysis. A second one, somewhat more general, is that there is a danger in too much abstractness: we change the character of the game when we drastically alter the amount of contextual detail that it contains or when we eliminate such complicating factors as the players' uncertainties about each other's value systems. It is often contextual detail that can guide the players to the discovery of a stable or, at least, mutually non-destructive outcome. In terms of an earlier example, the ability of Holmes and Moriarty to get off at the same station may depend on the presence of something in the problem other than its formal structure. It may be something on the train or something in the station, something in their common background, or something that they hear over the loudspeaker when the train stops; and, though it may be difficult to derive scientific generalizations about what it is that serves their need for co-

CONFLICT RESOLUTION

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NUMBER 3

THE STRATEGY OF CONFLICT

257

ordination, we have to recognize that the kinds of thing that determine the outcome are what a highly abstract analysis may treat as irrelevant detail.

A third conclusion, which is particularly applicable whenever the facilities for communication are short of perfect, where there is inherent uncertainty about each other's value systems or choices of strategies, and especially when an outcome must be reached by a sequence of moves or maneuvers, is that some essential part of the study of mixed-motive games is necessarily empirical. This is not to say just that it is an empirical question how people do actually perform in mixed-motive games, especially games too complicated for intellectual mastery. It is a stronger statement: that the principles relevant to successful play, the strategic principles, the propositions of a normative theory, cannot be derived by purely analytical means from a priori considerations.

In a zero-sum game the analyst is really dealing with only a single center of consciousness, a single source of decision. True, there are two players, each with his own consciousness; but minimax strategy converts the situation into one involving two essentially unilateral decisions. No spark of recognition needs to jump between the two players; no meeting of minds is required; no hints have to be conveyed; no impressions, images, or understandings have to be compared. No social perception is involved. But in the mixed-motive game, two or more centers of consciousness are dependent on each other in an essential way. Something has to be communicated; at least some spark of recognition must pass between the players. There is generally a necessity for some social activity, however rudimentary or tacit it may be; and both players are dependent to some degree on the success of their social perception and

interaction. Even two completely isolated individuals, who play with each other in absolute silence and without even knowing each other's identity, must tacitly reach some meeting of minds.

There is, consequently, no way that an analyst can reproduce the whole decision process either introspectively or by an axiomatic method. There is no way to build a model for the interaction of two or more decision units, with the behavior and expectations of those decision units being derived by purely formal deduction. An analyst can deduce the decisions of a single rational mind if he knows the criteria that govern the decisions; but he cannot infer by purely formal analysis what can pass between two centers of consciousness. It takes at least two people to test that. (Two analysts can do it, but only by using themselves as subjects in an experiment.) Taking a hint is fundamentally different from deciphering a formal communication or solving a mathematical problem; it involves discovering a message that has been planted within a context by someone who thinks he shares with the recipient certain impressions or associations. One cannot, without empirical evidence, deduce what understandings can be perceived in a non-zero-sum game of maneuver any more than one can prove, by purely formal deduction, that a particular joke is bound to be funny.

To illustrate, consider the question whether two people, looking at the same ink blot, can identify the same picture or suggestion in it if each is trying and knows that the other is trying to concert on the same picture or suggestion? The answer to this question can be found only by trying. But if they can, they can do something that no purely formal game theory can take into account; they can do better than a purely deductive game theory would predict. And if they can do better—if they can rise above

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the limitations of a purely formal game theory—even a normative, prescriptive, strategic theory cannot be based on purely formal analysis. We cannot build either a descriptive theory or a prescriptive theory on the assumption that there are certain intellectual processes that rational players are *not* capable of, of the kind involved in “taking a hint”; it is an empirical question whether rational players, either jointly or individually, can actually do better than a purely formal game theory predicts and should consequently ignore the strategic principles produced by such a theory.³⁸

Again it should be emphasized that the reason why this kind of consideration does not arise in the zero-sum game is that any such social interaction could not be to the advantage of both players simultaneously and that at least one of the rational players would have both motive and ability to destroy all social communications. But in a non-zero-sum game that involves any initial uncertainty over which among the possible outcomes are in fact efficient and any need for co-ordinated mutual accommodation to get to an efficient outcome, a rational player cannot absent himself in self-defense from the social process; he cannot turn off his hearing aid to avoid being constrained by what he hears, if complete radio silence makes efficient collaboration impossible. Nor can he rationally fail to open a letter, once it is delivered, since the other party will have assumed that he will open it and have acted accordingly.

At this point a question arises whether the game-theory trail ramifies indefinitely over the whole domain of social psychology or leads into a more limited area particularly congenial to game theory. Are there some general propositions about co-operative behavior in mixed-motive games that can be discovered by experiment or observation and that yield a widely applicable insight into

the universe of bargaining situations? Although success is not assured, there are certainly some promising areas for research; and, even if we cannot discover general propositions, we may at least disprove em-

³⁸ A good laboratory example of the communication-perception part of game strategy is the experiment reported by M. M. Flood (10), who presented his players with a 2 × 2 non-zero-sum matrix for 100 consecutive tacit plays. The special property of the matrix is that the players can win only by co-operating on a particular cell on each play, but to distribute the winnings for the 100-play sequence they must co-operate on some pattern of alternation among two or more cells that discriminate differently between the two players. And the only means of negotiating over the distribution to be sought and concerting on a pattern of alternating play that achieves it is through the choices they actually make as the play proceeds. This “communication” stage—and any later stage when one player may depart from the tacitly agreed pattern to cheat a little and have to be punished by a reprisal pattern—is jointly expensive to them, since an unco-ordinated choice is a lost chance to make some money.

The question of how to communicate a proposal effectively and how to interpret the other player's proposal implicit in his pattern of play is evidently dependent on some mutual perception of a shared sense of pattern—a jointly recognized ability to complete a pattern of which a fragment has been displayed—not unlike the process involved in the experiments of the Gestalt psychologists mentioned in an earlier footnote. And, while a purely formal theory of communication may derive certain minimum standards of “efficiency” in communication that rational players ought to achieve, it is an empirical question whether players can do better than that. How well one can take a hint and what kinds of hints are most successful are empirical questions of social perception, probably amenable to experimental study. (The same problem arises if two men at an auction recognize that they are jointly losing money by bidding against each other and try, without giving any overt evidence of collusion, to concert on some pattern of reciprocal and alternating abstention from bidding that both saves them money jointly and distributes the savings and the opportunities between them.)

THE STRATEGY OF CONFLICT

259

pirically some that are widely held. It does
appear that game theory is badly under-
developed from the experimental side.

Consider a game like the one described
earlier, involving the movement of counters
over a map, or the modified chess game
that was made non-zero-sum. These can be
taken to represent games in "limited war";
both players can gain by successfully avoid-
ing mutually destructive strategies. Here is
a game in which the ability of the two
players to avoid mutual destruction may well
depend on what *means* for successful co-
ordination of intentions are provided by
the incidental details of the game, by such
things as a configuration of the map or
board, the suggested names of the pieces,
the tradition or precedent that goes with
the game, and the scenario or connota-
tive background that is instilled into the
players before the game begins. It is a suf-
ficiently complicated game to require per-
ceptive play by both sides and the suc-
cessful conveyance of intentions. If we sup-
pose for a moment that the technical prob-
lem of constructing a playable game of that
type has been mastered, it is worth while
to consider what line of questions we might
try to investigate or what hypotheses we
might test.

One such question would be this: by
and large, does it appear that the players
are any more successful in reaching an
efficient solution, i.e., a mutually non-de-
structive solution, when (a) full or nearly
full communication is allowed, (b) no com-
munication or virtually none is allowed,
other than what can be conveyed by the
moves themselves, or (c) communication is
asymmetrical, with one party more able to
send messages than he is to receive them?
There is no guaranty that a single, uni-
versally applicable answer would emerge;
nevertheless, some quite general valid prop-
ositions about the role of communication

might well be discovered. The enormous
significance of this question is attested by
some of the current controversies about
whether the possibility of keeping war lim-
ited is greater if there is good communica-
tion between both sides, or if there are
unilateral declarations ahead of time by one
side or the other, or if there is virtually no
overt communication between the bellig-
erents.³⁹

Another set of questions, also pertinent
to problems of limited war, international or
other, would be whether a stable, efficient
outcome is more likely when the connota-
tions of the game—the names and interpre-
tations that are overtly attached to the
moves and pieces and objects on the board—
are familiar and recognizable or when they
are quite novel, unfamiliar, and unlikely
to inspire similar notions in the two players.
Is it—to speak of the game in a particular
extensive form—more likely that rational
players can keep a war limited in Southeast
Asia, using conventional and atomic weap-
ons, or in a battle against an unknown
adversary on the surface of the moon, using
strange bacterial weapons? These are im-
portant questions; they are at the very cen-

³⁹ To preclude any possible misunderstanding: the writer is not suggesting that limited war can be simulated in the laboratory or that experimental results regarding the limiting process can be directly transferred to the outside world. Experiments of the kind described would come under the heading of "basic research." And it would be concerned mainly with the perceptual and communicative side of the problem, not the motivational-except to the extent that motivations affect social perception. The probability that the results of such research would find ready application, however, is enhanced by the observation that much current theorizing on, for instance, the role of communication in limited war or the types of limitations most likely to be observed seems itself to be based only on what might be described as implicit experimental games played introspectively.

260

THOMAS C. SCHELLING

ter of game theory; and they are questions
that cannot possibly be given a confident an-
swer without empirical evidence. And there
is no arguing that rational players have the
intellectual capacity to rise above these de-
tails of the game and ignore them; the im-
portance of the details is that they can be
supremely helpful to both players and that
rational players know that they may be
dependent on using these details as props in
the course of their mutual accommodation.

Is a stable, efficient outcome more likely
between two players of similar tempera-
ment and cultural background or between
two quite different players? Is a stable, ef-
ficient solution more likely with two prac-
ticed players, two novices, or one novice
and a practiced player; and in the latter
pair, who has the advantage?

In a game of this sort, how crucial are
the opening moves? If stable patterns of
behavior, i.e., "rules of the game," are
not discovered early, will they be discov-
ered at all? Is mutually successful play
more likely if the general philosophy of
each player is to begin with "tight" rules or
highly "limited" weapons and resources,
loosening them a little only as the occasion
demands it, or if each player sets himself
wider limits at the outset in order to avoid
having to establish a practice of loosening
rules as he goes?

How much influence on a game of this
sort can a "mediator" have, and what kinds
of mediating roles are most effective? Does
it help or hinder the other two players if
the mediator has a stake of his own in the
outcome? To what extent can a mediator
discriminate in favor of one of the two
players and still increase the likelihood of a
stable, efficient outcome?

It would be interesting in a game of this
sort to have the players score both them-
selves and their partners from time to time
on such matters as who is playing the more

aggressively or the more co-operatively,
and what "rules" each thinks are in force
and thinks the other thinks are in force; of
who is "winning" in a bilateral sense (it
being recalled that the substantial ignorance
of each other's value system makes this al-
ways a matter of interpretation); of when
the game has reached a "critical" turning
point, or when an "innovation" in tactics
has been introduced, or when a particular
move by the other side is to be interpreted
as "retaliation" or a new initiative.

Because a "law of reprisal" is essentially
casuistic in nature; because the mutually
recognized restraints in any form of "lim-
ited war" are essentially based on some-
thing psychologically and sociologically akin
to tradition; and because the received body
of casuistry and tradition is often wholly
inadequate to the game at hand (say, grad-
uated atomic reprisal on the U.S.S.R. and
America while limited atomic war obtains
in Europe, or the bombing of grammar
schools in an area without recent experience
in racial violence, or the introduction of
new forms of non-price competition in a
particular industry), it seems likely that
the empirical part of game theory will
include experimental work like that of
Muzafer Sherif (35) reported in "The De-
velopment of Social Norms." He finds that
when no norms exist for a laboratory
judgment, they are created by the subjects;
and when norms are created for two parties
in the same process, each player's develop-
ing norm influences the other's. There is a
process of genuine learning with respect to
values; each side adapts its own system of
values to the other's, in forming its own.
When the supply of available "objective"
criteria is incapable of yielding a complete
set of rules, i.e., when the game is "inde-
terminate," norms of some sort must be
developed, mutually perceived, and ac-
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THE STRATEGY OF CONFLICT

261

have to be legitimized.⁴⁰ In an almost un-
consciously co-operative way, adversaries
must reach a mutually recognized definition
of what constitutes an innovation, a chal-
lenging or assertive move, or a co-operative
gesture, and they must develop some com-
mon norm regarding the kind of retaliation
that fits the crime when a breach of the
rules occurs.⁴¹

A “scenario” might, for example, identify
one of the players as “aggressor”; it might
give the outcomes of previous plays of the

⁴⁰ A splendid example of the creation of norms in practice—and one that suggests that the process is susceptible of analysis—was the acceptance during the 1957 disarmament discussions of the notion that any inspection zone ultimately agreed on had to be selected from among the array of possible pie-shaped zones with apex at the North Pole.

⁴¹ One may hope, as a game theorist, that a clear line can be drawn between the experimental psychology pertinent to game theory and the rest of social psychology; this is still supposed to be a theory of *strategy*, not the entire domain of conflict behavior. But it is not clear just where the line can be drawn in advance. “Hostility,” for example, might seem to be an emotional or temperamental quality best kept out of game theory; but if a player’s hostility in the game is a significant constraint on his ability to perceive the other player’s meaning, it becomes part of the “communication structure.” An experiment by Morton Deutsch (6, pp. 113–14) is pertinent. He let pairs of players play non-zero-sum games (in matrix form) tacitly for a sequence of two plays, the game providing both a “co-operative” and an “unco-operative” choice. Those who played unco-operatively against a co-operative partner had an opportunity, on the second play, to respond to the implicit offer of co-operation. But, “when their expectation of the other person’s choice was not confirmed, they tended to interpret his choice as being a function of in-difference or a basic lack of understanding as to how the game ‘should’ be played. . . . In this group, knowledge of the other person’s choice, because of the meaning attributed to it, tended to reinforce the previous negative sentiments re-garding the intentions of the other person.”

same game by other players; it might give
a background story that would tend to
identify some particular division of the
terrain as corresponding to an original "sta-
tus quo"; or it might seem to attach a kind
of moral claim of one of the players to par-
ticular parts of the board. These background
data would have no influence on the logical
or mathematical structure of the game; they
would be intended to have no force except
power of suggestion. Again, one might set
up the board so that on the first play it cor-
responds to the way it stood in the middle
of the same game as played earlier by two
other players, and see whether the out-
come can be effected by informing the
players of what the starting lineup was in
that earlier game. If players tend to develop
"norms" based on the static configuration
of the game as they appreciate it at the
outset, it may be possible to distort those
norms by providing, in a completely "non-
authoritative" way, a background story that
suggestively indicates some other hypotheti-
cal starting point.⁴²

It should also be interesting to see
whether each player can really discern
when the other is "testing" his determina-
tion, "daring" him, and so forth; and it
might be possible to study the process by
which particular encounters become in-
vested with symbolic importance, such that
each player recognizes that he is establish-
ing a role and reputation in the way he
conducts himself at a particular point in
the game.

Another dimension of the game that
seems susceptible of analysis is the signif-
icance of the *incrementalism* that is in-
volved in the moves and value systems.
Take, for example, a game that involves
moving pieces over a board or troops over

⁴² The income tax games described in the author's earlier paper in this *Journal* indicate the force of this power of suggestion (31, pp. 7-8, 9).

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some terrain. If players move in turn, each
moving one piece one square at a time, the
game proceeds at a slow tempo by small
increments; the situation on the board may
change character in the course of play, but
it does so by a succession of small changes
that can be observed, appreciated, and
adapted to, with plenty of time for the mis-
takes of individual players or mutual mis-
takes that destroy value for both of them
to be observed, adapted to, and avoided in
subsequent play. If there is communication,
there is time for the players to bargain
verbally and to avoid moves that involve
mutual destruction. But suppose that, in-
stead, the pieces can be moved several at
a time in any direction and any distance
and that the rules make the outcome of any
hostile clash enormously destructive for one
or both sides. Now the game is not so in-
cremental; things can happen abruptly.
There may be a temptation toward surprise
attack. While one can see what the situa-
tion is at a particular moment, he cannot
project it more than a move or two ahead.
There seems to be less chance to develop a
modus vivendi, or tradition of trust, or
dominant and submissive roles for the two
players, because the pace of the game
brings things to a head before much experi-
ence has been gained or much of an under-
standing reached. But does a more incre-
mental game make successful collaboration
easier, or does it just invite a riskier mode
of play? Or does this depend on what kinds
of people the players are and on what sug-
gestions we plant in the game itself? Is the
critical factor the incrementalism of the
moves in the game or incrementalism in the
value systems of the players (i.e., of the
scoring system)? Or can these be made
commensurate with each other, so that in-
crementalism can be introduced into a game
in one dimension to offset the lack of it in
another? The relevance of these questions

is attested by the controversy over the role
of nuclear weapons in limited war, the sig-
nificance of the temptation to surprise at-
tack in a situation that depends on mutual
deterrence, and various proposals to reduce
the tempo of modern war and to isolate it
geographically, together with disagreement
over whether there can be such a thing as
limited war on the continent of western
Europe. Incrementalism may be compara-
tively amenable to formal analysis, once the
necessary empirical benchmarks have been
identified by experiment or observation.⁴³

These questions have concerned two-per-
son games, except for the possible role of
the mediator. Similar games could be
played by three or more participants, each
on his own account; and the author con-
jectures that—at least among “successful”
players—many of the empirical results
would appear in sharper relief with the
larger number of players. More generally,
the kind of co-ordination involved in the
formation of mobs and coalitions may lend
itself to experimental study. In contrast to
the more sanitary, symmetrical schemes that
have sometimes been used to study the for-
mation of coalitions in game theory, it might
prove more interesting to introduce delib-
erately certain asymmetries, precedents,
orders of moves, imperfect communication
structures, and various connotative details,
in order to study the crystallization of
groups. Certainly the influence exerted on
the formation of coalitions by various kinds
of asymmetrical and otherwise imperfect

⁴³ Kissinger (18, p. 225) says: "It is not only that limited war must find means to prevent the most extreme violence; it must also seek to slow down the tempo of modern war lest the rapidity with which operations succeed each other prevent the establishment of a relation between political and military objectives. If this relationship is lost, any war is likely to grow by imperceptible stages into one all-out effort."

THE STRATEGY OF CONFLICT

263

communication systems often lends itself to systematic experimental study.⁴⁴

REFERENCES

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7. DUFFUS, R. L. "The Function of the Racket-

⁴⁴ Alex Bavelas (2, p. 493) has described an experiment in pure co-ordination in which each of five separated players must pass geometric pieces among themselves until they reach a dis-tribution of the pieces that permits the formation of five separate squares. The pieces are so cut that many "wrong" squares can be formed, i.e., squares that use a combination of pieces that makes it impossible for four more squares to be formed with the remaining pieces. He is inter-ested in what happens when these deceptive "successes" occur. "For an individual who has completed a square it is understandably difficult to tear it apart. The ease with which he can take a course of action 'away from the goal' should depend to some extent upon his perception of the total situation. In this regard the pattern of communication should have well-defined effects. ... Preliminary runs ... have revealed ... that the binding forces against restructuring are very great, and that, with any considerable amount of communication restriction, a solution is improbable."

eer," *New Republic*, March 27, 1929, pp. 166-68.

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[LC]

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