This is a series of closely interrelated essays in a field that is variously described as "theory of bargaining," "theory of conflict," or "theory of strategy." Strictly speaking, the subject falls within the theory of games, but within the part of game theory in which the least satisfactory progress has been made, the situations in which there is common interest as well as conflict between adversaries: negotiations, war and threats of war, criminal deterrence, tacit bargaining, extortion. The philosophy of the book is that in the strategy of conflict there are enlightening similarities between, say, maneuvering in limited war and jockeying in a traffic jam, between deterring the Russians and deterring one's own children, or between the modern balance of terror and the ancient institution of hostages.

XREF: Bridges game theory to everyday situations like traffic jams and parenting, echoing how behavioral economics applies formal models to ordinary life.

Motivation for the purer theory came almost exclusively from preoccupation with (and fascination with) "applied" problems; and the clarification of theoretical ideas was absolutely dependent on an identification of live examples.

XREF: Echoes the science-vs-engineering dynamic and the gap-tooth model of innovation common in histories of computing and applied math. SEED: Could make a piece on how pure theory thrives when anchored to concrete problems — the antithesis of research-for-its-own-sake.

In fact, the richness of the subject arises from the fact that, in international affairs, there is mutual dependence as well as opposition. Pure conflict, in which the interests of two antagonists are completely opposed, is a special case; it would arise in a war of complete extermination, otherwise not even in war. For this reason, “winning” in a conflict does not have a strictly competitive meaning; it is not winning relative to one’s adversary. It means gaining relative to one’s own value system; and this may be done by bargaining, by mutual accommodation, and by the avoidance of mutually damaging behavior.

DEFINE: Redefines 'winning' in conflict from beating an adversary to making relative gains against one's own value system.

Thus, strategy—in the sense in which I am using it here—is not concerned with the efficient application of force but with the exploitation of potential force. It is concerned not just with enemies who dislike each other but with partners who distrust or disagree with each other. It is concerned not just with the division of gains and losses between two claimants but with the possibility that particular outcomes are worse (better) for both claimants than certain other outcomes. In the terminology of game theory, most interesting international conflicts are not “constant-sum games” but “variable-sum games”: the sum of the gains of the participants involved is not fixed so that more for one inexorably means less for the other. There is a common interest in reaching outcomes that are mutually advantageous.

DEFINE: Defines the key game-theory distinction between constant-sum and variable-sum games, where variable-sum games involve mutual gains.

To study the strategy of conflict is to take the view that most conflict situations are essentially bargaining situations. They are situations in which the ability of one participant to gain his ends is dependent to an important degree on the choices or decisions that the other participant will make.

DEFINE: Establishes Schelling's core framework: conflict as a bargaining situation where outcomes depend on mutual decisions, not unilateral action.

We have observed that the rationality of the adversary is pertinent to the efficacy of a threat, and that madmen, like small children, can often not be controlled by threats. We have recognized that the efficacy of the threat may depend on what alternatives are available to the potential enemy, who, if he is not to react like a trapped lion, must be left some tolerable recourse.

XREF: Connects to game theory concepts of deterrence and the importance of keeping options open for the opponent to avoid escalation.

We have learned that a threat has to be credible to be efficacious, and that its credibility may depend on the costs and risks associated with fulfillment for the party making the threat. We have developed the idea of making a threat credible by getting ourselves committed to its fulfillment, through the stretching of a “trip wire” across the enemy’s path of advance, or by making fulfillment a matter of national honor and prestige—as in the case, say, of the Formosa Resolution.

But here a useful distinction can be made between the application of force and the threat of force. Deterrence is concerned with the exploitation of potential force. It is concerned with persuading a potential enemy that he should in his own interest avoid certain courses of activity.

Deterrence is concerned with the exploitation of potential force. It is concerned with persuading a potential enemy that he should in his own interest avoid certain courses of activity. There is an important difference between the intellectual skills required for carrying out a military mission and for using potential military capability to pursue a nation’s objectives. A theory of deterrence would be, in effect, a theory of the skillful nonuse of military forces, and for this purpose deterrence requires something broader than military skills.

XREF: Connects to Schelling's work on strategy and the credibility of threats, where the art lies in what forces are NOT used.

A deterrent threat meets this definition nicely; it works only because of what the other player expects us to do in response to his choice of moves, and we can afford to make the threat only because we expect it to have an influence on his choice. But in international strategy the promise of game theory is so far unfulfilled. Game theory has been extremely helpful in the formulation of problems and the clarification of concepts, but its greatest successes have been in other fields. It has, on the whole, been pitched at a level of abstraction where it has made little contact with the elements of a problem like deterrence.

DEFINE: Clarifies what a deterrent threat is and why game theory fits it conceptually yet fails practically.

Some aspects of deterrence stand out vividly in child discipline: the importance of rationality and self-discipline on the part of the person to be deterred, of his ability to comprehend the threat if he hears it and to hear it through the din and noise, of the threatener’s determination to fulfill the threat if need be—and, more important, of the threatened party’s conviction that the threat will be carried out. Clearer perhaps in child discipline than in criminal deterrence is the important possibility that the threatened punishment will hurt the threatener as much as it will the one threatened, perhaps more. There is an analogy between a parent’s threat to a child and the threat that a wealthy paternalistic nation makes to the weak and disorganized government of a poor nation in, say, extending foreign aid and demanding “sound” economic policies or cooperative military policies in return.

XREF: Connects to the reader's understanding of Schelling's broader deterrence theory and game-theoretic models of commitment and credibility. SEED: The parent-child/foreign aid analogy is a striking framing that could power an essay on how credibility operates in asymmetric power relationships. Builds on: "Deterrence is concerned with the exploitation of potential force."

The deterrence concept requires that there be both conflict and common interest between the parties involved; it is as inapplicable to a situation of pure and complete antagonism of interest as it is to the case of pure and complete common interest. Between these extremes, deterring an ally and deterring an enemy differ only by degrees, and in fact we may have to develop a more coherent theory before we can even say in a meaningful way whether we have more in common with Russia or with Greece, relative to the conflicts between us.

DEFINE: Schelling clarifies that deterrence is only meaningful in mixed-motive situations, not in pure antagonism or pure harmony.

Collision being about as mutual as anything can be, and often the only thing that one can threaten, the maneuvers by which one conveys a threat of mutual damage to another driver aggressing on one’s right of way are an instructive example of the kind of threat that is conveyed not by words but by actions, and of the threat in which the pledge to fulfill is made not by verbal announcement but by losing the power to do otherwise.

DEFINE: Distinguishes credible threats that are communicated through action—committing so fully that one loses the ability to back down—from verbal threats.

What is the need for trust, or enforcement of promises? Specifically, in addition to threatening damage, need one also guarantee to withhold the damage if compliance is forthcoming; or does this depend on the configuration of "payoffs" involved? What "legal system," communication system, or information structure is needed to make the necessary promises enforceable?

QUESTION: This opens an interesting line: under what payoff structures does a bare threat suffice, and when is a promise to withhold damage additionally required? Worth tracing through game-theoretic examples.

But where trust and good faith do not exist and cannot be made to by our acting as though they did, we may wish to solicit advice from the underworld, or from ancient despotisms, on how to make agreements work when trust and good faith are lacking and there is no legal recourse for breach of contract. The ancients exchanged hostages, drank wine from the same glass to demonstrate the absence of poison, met in public places to inhibit the massacre of one by the other, and even deliberately exchanged spies to facilitate transmittal of authentic information. It seems likely that a well-developed theory of strategy could throw light on the efficacy of some of those old devices, suggest the circumstances to which they apply, and discover modern equivalents that, though offensive to our taste, may be desperately needed in the regulation of conflict.

SEED: Could become an essay on how pre-legal enforcement mechanisms (hostages, shared wine, public meetings) map onto modern deterrence and verification practices in international relations.

It is not true, as illustrated in the example of extortion, that in the face of a threat it is invariably an advantage to be rational, particularly if the fact of being rational or irrational cannot be concealed. It is not invariably an advantage, in the face of a threat, to have a communication system in good order, to have complete information, or to be in full command of one's own actions or of one's own assets. Mossadeq and my small children have already been referred to; but the same tactic is illustrated by the burning of bridges behind oneself to persuade an adversary that one cannot be induced to retreat. An old English law that made it a serious crime to pay tribute to coastal pirates does not necessarily appear either cruel or anomalous in the light of a theory of strategy.

It is interesting that political democracy itself relies on a particular communication system in which the transmittal of authentic evidence is precluded: the mandatory secret ballot is a scheme to deny the voter any means of proving which way he voted. Being stripped of his power to prove how he voted, he is stripped of his power to be intimidated. Powerless to prove whether or not he complied with a threat, he knows—and so do those who would threaten him—that any punishment would be unrelated to the way he actually voted.

XREF: Connects to conspiracy theory literature on the function of secrecy as a defense against coercion and intimidation networks.

We tend to identify peace, stability, and the quiescence of conflict with notions like trust, good faith, and mutual respect. To the extent that this point of view actually encourages trust and respect it is good. But where trust and good faith do not exist and cannot be made to by our acting as though they did, we may wish to solicit advice from the underworld, or from ancient despotisms, on how to make agreements work when trust and good faith are lacking and there is no legal recourse for breach of contract.

XREF: Connects to classic principal-agent and contract theory, as well as game-theoretic treatments of cooperation without trust (e.g., Schelling, Axelrod, hostage mechanisms in history).

The essence of these tactics is some voluntary but irreversible sacrifice of freedom of choice. They rest on the paradox that the power to constrain an adversary may depend on the power to bind oneself; that, in bargaining, weakness is often strength, freedom may be freedom to capitulate, and to burn bridges behind one may suffice to undo an opponent.

XREF: This connects to commitment strategies in game theory and Robert Cialdini's work on commitment and consistency in persuasion.

But, if the terms imply that it is an advantage to be more intelligent or more skilled in debate, or to have more financial resources, more physical strength, more military potency, or more ability to withstand losses, then the term does a disservice. These qualities are by no means universal advantages in bargaining situations; they often have a contrary value.

Builds on: "Binding oneself can paradoxically strengthen one's bargaining position"

Incidentally, some of the more contractual kinds of commitments are not as effective as they at first seem. In the example of the self-inflicted penalty through the bet, it remains possible for the seller to seek out the third party and offer a modest sum in consideration of the latter's releasing the buyer from the bet, threatening to sell the house for $16,000 if the release is not forthcoming. The effect of the bet—as of most such contractual commitments—is to shift the locus and personnel of the negotiation, in the hope that the third party will be less available for negotiation or less subject to an incentive to concede. To put it differently, a contractual commitment is usually the assumption of a contingent "transfer cost," not a "real cost"; and if all interested parties can be brought into the negotiation the range of indeterminacy remains as it was.

Consider a culture in which "cross my heart" is universally recognized as absolutely binding. Any offer accompanied by this invocation is a final offer, and is so recognized. If each party knows the other's true reservation price, the object is to be first with a firm offer. Complete responsibility for the outcome then rests with the other, who can take it or leave it as he chooses (and who chooses to take it). Bargaining is all over; the commitment (that is, the first offer) wins.

DEFINE: Illustrates the concept of a binding commitment that preempts negotiation.

An asymmetry in communication may well favor the one who is (and is known to be) unavailable for the receipt of messages, for he is the one who cannot be deterred from his own commitment by receipt of the other's.

Suppose only part of the population belongs to the cult in which "cross my heart" is (or is believed to be) absolutely binding. If everyone knows (and is known to know) everyone else's affiliation, those belonging to this particular cult have the advantage. They can commit themselves, the others cannot. If the buyer says "$16,000, cross my heart" his offer is final; if the seller says "$19,000" he is (and is known to be) only "bargaining."

XREF: Connects to Schelling's work on commitment devices and signaling theory (e.g., Thomas Schelling, Michael Spence's signaling models). DEFINE: Illustrates how asymmetric commitment capacity functions as a strategic advantage, clarifying the mechanism of credible commitment.

These examples have certain characteristics in common. First, they clearly depend not only on incurring a commitment but on communicating it persuasively to the other party. Second, it is by no means easy to establish the commitment, nor is it entirely clear to either of the parties concerned just how strong the commitment is. Third, similar activity may be available to the parties on both sides. Fourth, the possibility of commitment, though perhaps available to both sides, is by no means equally available; the ability of a democratic government to get itself tied by public opinion may be different from the ability of a totalitarian government to incur such a commitment. Fifth, they all run the risk of establishing an immovable position that goes beyond the ability of the other to concede, and thereby provoke the likelihood of stalemate or breakdown.

XREF: Connects to Schelling's general theory of commitment and signaling in strategic interaction.

Second, an "agent" may be brought in as a principal in his own right, with an incentive structure of his own that differs from his principal's. This device is involved in automobile insurance; the private citizen, in settling out of court, cannot threaten suit as effectively as the insurance company since the latter is more conspicuously obliged to carry out such threats to maintain its own reputation for subsequent accidents.

A potent means of commitment, and sometimes the only means, is the pledge of one's reputation. If national representatives can arrange to be charged with appeasement for every small concession, they place concession visibly beyond their own reach. If a union with other plants to deal with can arrange to make any retreat dramatically visible, it places its bargaining reputation in jeopardy and thereby becomes visibly incapable of serious compromise. (The same convenient jeopardy is the basis for the universally exploited defense, "If I did it for you I'd have to do it for everyone else.") But to commit in this fashion publicity is required. Both the initial offer and the final outcome would have to be known; and if secrecy surrounds either point, or if the outcome is inherently not observable, the device is unavailable.

The second party is simultaneously the "third party" to whom one's bargaining reputation can be pledged. This situation occurs in the threat of local resistance to local aggression. The party threatening achieves its commitment, and hence the credibility of its threat, not by referring to what it would gain from carrying out the threat in this particular instance but by pointing to the long-run value of a fulfilled threat in enhancing the credibility of future threats.

DEFINE: Clarifies how a third party (the reputation audience) undergirds the credibility of threats without direct payoff.

The protection against extortion depends on refusal, unavailability, or inability, to negotiate. But if the object of the extortionate threat can be brought onto the agenda with the other topic, the latent threat becomes effective.

When there are two objects to negotiate, the decision to negotiate them simultaneously or in separate forums or at separate times is by no means neutral to the outcome, particularly when there is a latent extortionate threat that can be exploited only if it can be attached to some more ordinary, legitimate, bargaining situation. The protection against extortion depends on refusal, unavailability, or inability, to negotiate. But if the object of the extortionate threat can be brought onto the agenda with the other topic, the latent threat becomes effective.

XREF: Connects to negotiation theory and the strategic use of agenda-setting, as in Thomas Schelling's work on commitment and bargaining.

But if the bargaining representatives of the threatened party are confined to the cheese-automobile agenda, and have no instructions that permit them even to take cognizance of other commodities, or if there are ground rules that forbid mention of other tariffs while cheese and automobiles remain unsettled, this extortionate weapon must await another opportunity. If the threat that would be brought to the conference table is one that cannot stand publicity, publicity itself may prevent its effective communication.

If there are two projects, each with a cost of three, and each with a value of two to A and a value of four to B, and each is inherently a "one-man" project in its execution, and if compensation is institutionally impossible, B will be forced to pay the entire cost of each as long as the two projects are kept separate. He cannot usefully threaten nonperformance, since A has no incentive to carry out either project by himself. But if B can link the projects together, offering to carry out one while A carries out the other, and can effectively threaten to abandon both unless A carries out one of them, A is left an option with a gain of four and a cost of three, which he takes, and B cuts his cost in half.

XREF: Classic theory of the firm / transaction cost economics territory — Coase, and the hold-up problem familiar from Williamson.

Rules that permit a president to veto an appropriation bill only in its entirety, or that require each amendment to be voted before the original act is voted on, or a priority system accorded to different kinds of motions, substantially alter the incentives that are brought to bear on each action. One who might be pressured into choosing second best is relieved of his vulnerability if he can vote earlier to eliminate that possibility, thereby leaving only first and third choices about which his preference is known to be so strong that no threat will be made.

XREF: Connects to public choice theory and constitutional design literature, where institutional rules are seen as shaping strategic behavior.

The distinctive feature of this threat is that the threatener has no incentive to carry it out either before the event or after. He does have an incentive to bind himself to fulfill the threat, if he thinks the threat may be successful, because the threat and not its fulfillment gains the end; and fulfillment is not required if the threat succeeds. The more certain the contingent fulfillment is, the less likely is actual fulfillment. But the threat's efficacy depends on the credulity of the other party, and the threat is ineffectual unless the threatener can rearrange or display his own incentives so as to demonstrate that he would, ex post, have an incentive to carry it out.

DEFINE: Distinguishes a threat from an empty bluff: the threat works if credible, and credibility requires binding oneself to the contingent action. Builds on: "Threats fail against adversaries who ignore rationality or lack alternatives."

In case a threat is made and fails to deter, there is a second stage prior to fulfillment in which both parties have an interest in undoing the commitment. The purpose of the threat is gone, its deterrence value is zero, and only the commitment exists to motivate fulfillment. … Furthermore, “undoing” does not mean neglecting a commitment regardless of reputation; “undoing,” if the commitment of reputation was real, means disconnecting the threat from one’s reputation, perhaps one’s own reputation with the threatened person himself. It is therefore a subtle and tenuous situation in which, though both have an interest in undoing the commitment, they may be quite unable to collaborate in undoing it.

XREF: Connects to game theory, Schelling's commitment strategies, and deterrence literature.

In order to make a threat precise, so that its terms are visible both to the threatened party and to any third parties whose reaction to the whole affair is of value to the adversaries, it may be necessary to introduce some arbitrary elements. The threat must involve overt acts rather than intentions; it must be attached to the visible deeds, not invisible ones; it may have to attach itself to certain ancillary actions that are of no consequence in themselves to the threatening party. It may, for example, have to put a penalty on the carrying of weapons rather than their use; on suspicious behavior rather than observed misdemeanors; on proximity to a crime rather than the crime itself. And, finally, the act of punishment must be one whose effect or influence is clearly discernible.

In order that one be able to pledge his reputation behind a threat, there must be continuity between the present and subsequent issues that will arise. This need for continuity suggests a means of making the original threat more effective; if it can be decomposed into a series of consecutive smaller threats, there is an opportunity to demonstrate on the first few transgressions that the threat will be carried out on the rest. Even the first few become more plausible, since there is a more obvious incentive to fulfill them as a “lesson.”

In foreign aid programs the overt act of terminating assistance may be so obviously painful to both sides as not to be taken seriously by the recipient, but if each small misuse of funds is to be accompanied by a small reduction in assistance, never so large as to leave the recipient helpless nor to provoke a diplomatic breach, the willingness to carry it out will receive more credulity; or if it does not at first, a few lessons may be persuasive without too much damage.

Similar to decomposing a threat into a series is starting a threat with a punitive act that grows in severity with the passage of time. Where a threat of death by violence might not be credited, cutting off the food supply might bring submission. … Furthermore, the threatener may be embarrassed by his adversary’s collapse but not by his discomfort; and the device may therefore transform a dangerous once-for-all threat into a less costly continuous one. Tenants are less easily removed by threat of forcible eviction than by simply shutting off the utilities.

DEFINE: This explains the 'continuous growing threat' device as a distinct coercive tactic in Schelling's taxonomy. XREF: Connects to Schelling's broader theory of coercive bargaining and deterrence curves.

Among the legal privileges of corporations, two that are mentioned in textbooks are the right to sue and the “right” to be sued. Who wants to be sued! But the right to be sued is the power to make a promise: to borrow money, to enter a contract, to do business with someone who might be damaged. If suit does arise, the “right” seems a liability in retrospect; beforehand it was a prerequisite to doing business.

DEFINE: Recasts a legal 'liability' as a functional prerequisite for contracts and business.

In brief, the right to be sued is the power to accept a commitment. In the commitments discussed up to this point, it was essential that one’s adversary (or “partner,” however we wish to describe him) not have the power to release one from the commitment; the commitment was, in effect, to some third party, real or fictitious. The promise is a commitment to the second party in the bargain and is required whenever the final action of one or of each is outside the other’s control. It is required whenever an agreement leaves any incentive to cheat.

This need for promises is more than incidental; it has an institutional importance of its own. It is not always easy to make a convincing, self-binding, promise. Both the kidnapper who would like to release his prisoner, and the prisoner, may search desperately for a way to commit the latter against informing on his captor, without finding one. … If the law will not enforce price agreements; or if the union is unable to obligate itself to a no-strike pledge; or if a contractor has no assets to pay damages if he loses a suit, and the law will not imprison debtors; or if there is no “audience” to which one can pledge his reputation; it may not be possible to strike a bargain, or at least the same bargain that would otherwise be struck.

XREF: Relates to Thomas Schelling's work on commitment devices and the economics of credible commitments.

Bargaining may have to concern itself with an “incentive” system as well as the division of gains. Oligopolists may lobby for a “fair-trade” law; or exchange shares of stocks. An agreement to stay out of each other’s market may require an agreement to redesign the products to be unsuitable in each other’s area. Two countries that wish to agree not to make military use of an island may have to destroy the usefulness of the island itself. (In effect, a “third-party commitment” has to be assumed when an effective “second-party commitment” cannot be devised.)

XREF: This connects to commitment and threat-credibility themes in bargaining theory, likely from Schelling's work on strategy.

Fulfillment is not always observable. If one sells his vote in a secret election, or a government agrees to recommend an act to its parliament, or an employee agrees not to steal from inventory, or a teacher agrees to keep his political opinions out of class, or a country agrees to stimulate exports “as much as possible,” there is no reliable way to observe or measure compliance. The observable outcome is subject to a number of influences, only one of which is covered by the agreement. The bargain may therefore have to be expressed in terms of something observable, even though what is observable is not the intended object of the bargain.

DEFINE: Clarifies the concept that bargains may need to be framed around observable proxies when the actual obligation is unobservable. XREF: Connects to principal-agent theory and the economics of incomplete contracts and moral hazard.

The tactic of decomposition applies to promises as well as to threats. What makes many agreements enforceable is only the recognition of future opportunities for agreement that will be eliminated if mutual trust is not created and maintained, and whose value outweighs the momentary gain from cheating in the present instance. Each party must be confident that the other will not jeopardize future opportunities by destroying trust at the outset. This confidence does not always exist; and one of the purposes of piecemeal bargains is to cultivate the necessary mutual expectations.

XREF: This echoes the shadow of the future concept from repeated-game theory and Schelling's trust-building mechanism.

Even if the future will bring no recurrence, it may be possible to create the equivalence of continuity by dividing the bargaining issue into consecutive parts. If each party agrees to send a million dollars to the Red Cross on condition the other does, each may be tempted to cheat if the other contributes first, and each one’s anticipation of the other’s cheating will inhibit agreement. But if the contribution is divided into consecutive small contributions, each can try the other’s good faith for a small price. Furthermore, since each can keep the other on short tether to the finish, no one ever need risk more than one small contribution at a time. Finally, this change in the incentive structure itself takes most of the risk out of the initial contribution; the value of established trust is made obviously visible to both.

Figure 4 is the same as Fig. 3 except that αB has been moved leftward. Here, in the absence of communication, North wins at αβ regardless of whether he or East moves first or moves are simultaneous. If, however, East can communicate a conditional commitment, he can force North to choose A and an outcome of Aβ. But this commitment is something more than either a promise or a threat; it is both a promise and a threat. He must threaten αB if North chooses α; and he must promise “not AB” if North chooses A. The threat alone will not induce North to avoid α; αB is better than AB for North, and AB is what he gets with A if East is free to choose B. East must commit himself to do, for either α or A, the opposite of what he would do if he were not committed: abstention from AB or immolation at αB.

DEFINE: Schelling clarifies that a single-strategy commitment (threat only or promise only) is insufficient; a mixed commitment combining both is needed to shift the opponent's choice.

But suppose the two games are simultaneously up for decision, and the same two parties are involved in both. If either party can commit himself to a threat he may improve his position. East, for example, could threaten to choose β rather than B in game 6, unless North chose A rather than α in game 5; alternatively, North could threaten α in game 6 unless East chose β in game 5. Assuming the intervals large enough in game 6, and the threat persuasively committed and communicated, the threatener gains in game 5 at no cost in game 6. Because his threat succeeds he does not carry it out; so he gets AB in 6 as well as his preferred choice in game 5. To express this result differently, game 6 supplies what was ruled out earlier, namely, the threat of an act “outside the game.” From the point of view of game 5, game 6 is an extraneous act, and East might as well threaten to burn North’s house down if he does not choose A in 5. But such purely extortionate threats are not always easy to make; they often require an occasion, an object, and a means of communication, and additionally often suffer from illegality, immorality, or resistance out of sheer stubbornness. The joining of two negotiations on the same agenda may thus succeed where a purely gratuitous threat would be impracticable.

XREF: Connects to game theory and commitment strategies, linking to Schelling's broader work on strategic moves and credible commitment.

Incidentally, dropping AB vertically in Fig. 2 to below the level αβ would illustrate an important principle, namely, that moving one point in a manner “unfavorable” to North may actually improve the outcome for him. The threat that kept him from winning in Fig. 2 depends on the comparative attractiveness of AB over αB for North; if AB is made worse for him than αβ he becomes immune to the threat, which then is not made, and he wins at αB. This is an abstract example of the principle that, in bargaining, weakness may be strength.

XREF: Connects to game theory concepts like commitment devices and the 'madman theory' in diplomacy, where appearing irrational or constrained can improve outcomes. Builds on: "Advantages in bargaining are not universal and can backfire."

When a man loses his wife in a department store without any prior understanding on where to meet if they get separated, the chances are good that they will find each other. It is likely that each will think of some obvious place to meet, so obvious that each will be sure that the other is sure that it is “obvious” to both of them. One does not simply predict where the other will go, since the other will go where he predicts the first to go, which is wherever the first predicts the second to predict the first to go, and so ad infinitum. Not “What would I do if I were she?” but “What would I do if I were she wondering what she would do if she were I wondering what I would do if I were she. ..?” What is necessary is to coordinate predictions, to read the same message in the common situation, to identify the one course of action that their expectations of each other can converge on. They must “mutually recognize” some unique signal that coordinates their expectations of each other.

XREF: Recalls Thomas Schelling's focal point theory in game theory.

These problems are artificial, but they illustrate the point. People can often concert their intentions or expectations with others if each knows that the other is trying to do the same. Most situations—perhaps every situation for people who are practiced at this kind of game—provide some clue for coordinating behavior, some focal point for each person's expectation of what the other expects him to expect to be expected to do. Finding the key, or rather finding a key—any key that is mutually recognized as the key becomes the key—may depend on imagination more than on logic; it may depend on analogy, precedent, accidental arrangement, symmetry, aesthetic or geometric configuration, casuistic reasoning, and who the parties are and what they know about each other. Whimsy may send the man and his wife to the “lost and found”; or logic may lead each to reflect and to expect the other to reflect on where they would have agreed to meet if they had had a prior agreement to cover the contingency.

DEFINE: Introduces the key concept of focal points as the mechanism by which parties coordinate without explicit agreement. XREF: This is Schelling's foundational focal point concept, foundational to game theory and coordination problems in economics and social science.

A prime characteristic of most of these “solutions” to the problems, that is, of the clues or coordinators or focal points, is some kind of prominence or conspicuousness. But it is a prominence that depends on time and place and who the people are. Ordinary folk lost on a plane circular area may naturally go to the center to meet each other; but only one versed in mathematics would “naturally” expect to meet his partner at the center of gravity of an irregularly shaped area. Equally essential is some kind of uniqueness; the man and his wife cannot meet at the “lost and found” if the store has several. The writer's experiments with alternative maps indicated clearly that a map with many houses and a single crossroads sends people to the crossroads, while one with many crossroads and a single house sends most of them to the house. Partly this may reflect only that uniqueness conveys prominence; but it may be more important that uniqueness avoids ambiguousness.

A conflict of interest enters our problem if the parachutists dislike walking. With communication, which is not allowed in our problem, they would have argued or bargained over where to meet, each favoring a spot close to himself or a resting place particularly to his liking. In the absence of communication, their overriding interest is to concert ideas; and if a particular spot commands attention as the “obvious” place to meet, the winner of the bargain is simply the one who happens to be closer to it. Even if the one who is farthest from the focal point knows that he is, he cannot withhold his acquiescence and argue for a fairer division of the walking; the “proposal” for the bargain that is provided by the map itself—if, in fact, it provides one—is the only extant offer; and without communication, there is no counterproposal that can be made. The conflict gets reconciled—or perhaps we should say ignored—as a by-product of the dominant need for coordination.

Better still may be to have the power to send but not to receive messages: if one can announce his position and state that his transmitter works but not his receiver, saying that he will wait where he is until the other arrives, the latter has no choice. He can make no effective counteroffer, since no counteroffer could be heard.

XREF: This is Schelling's classic doctrine of commitment and the strategic value of disabling one's own options — connects to burning bridges, unilateral disarmament strategies, and bargaining theory.

Beggars cannot be choosers about the source of their signal, or about its attractiveness compared with others that they can only wish were as conspicuous.

XREF: The beggars metaphor echoes positions-dependence in trade and negotiation dynamics, where the weaker party accepts unfavorable terms.

Most bargaining situations ultimately involve some range of possible outcomes within which each party would rather make a concession than fail to reach agreement at all. In such a situation any potential outcome is one from which at least one of the parties, and probably both, would have been willing to retreat for the sake of agreement, and very often the other party knows it. Any potential outcome is therefore one that either party could have improved by insisting; yet he may have no basis for insisting, since the other knows or suspects that he would rather concede than do without agreement. Each party's strategy is guided mainly by what he expects the other to accept or insist on; yet each knows that the other is guided by reciprocal thoughts. The final outcome must be a point from which neither expects the other to retreat; yet the main ingredient of this expectation is what one thinks the other expects the first to expect, and so on. Somehow, out of this fluid and indeterminate situation that seemingly provides no logical reason for anybody to expect anything except what he expects to be expected to expect, a decision is reached. These infinitely reflexive expectations must somehow converge on a single point, at which each expects the other not to expect to be expected to retreat.

If one is about to make a concession, he needs to control his adversary's expectations; he needs a recognizable limit to his own retreat. If one is to make a finite concession that is not to be interpreted as capitulation, he needs an obvious place to stop. A mediator's suggestion may provide it; or any other element that qualitatively distinguishes the new position from surrounding positions. If one has been demanding 60 per cent and recedes to 50 per cent, he can get his heels in; if he recedes to 49 per cent, the other will assume that he has hit the skids and will keep sliding.

If some troops have retreated to the river in our map, they will expect to be expected to make a stand. This is the one spot to which they can retreat without necessarily being expected to retreat further, while, if they yield any further, there is no place left where they can be expected to make a determined stand. Similarly, the advancing party can expect to force the other to retreat to the river without having his advance interpreted as an insatiable demand for unlimited retreat. There is stability at the river—and perhaps nowhere else.

But, if this general line of reasoning is valid, any analysis of explicit bargaining must pay attention to what we might call the “communication” that is inherent in the bargaining situations, the signals that the participants read in the inanimate details of the case. And it means that tacit and explicit bargaining are not thoroughly separate concepts but that the various gradations from tacit bargaining up through types of incompleteness or faulty or limited communication to full communication all show some dependence on the need to coordinate expectations. Hence all show some degree of dependence of the participants themselves on their common inability to keep their eyes off certain outcomes.

This is not necessarily an argument for expecting explicit outcomes as a rule to lean toward exactly those that would have emerged if communication had been impossible; the focal points may certainly be different when speech is allowed, except in some of the artificial cases we have used in our illustrations. But what may be the main principle in tacit bargaining apparently may be at least one of the important principles in the analysis of explicit bargaining. And, since even much so-called “explicit” bargaining includes maneuver, indirect communication, jockeying for position, or speaking to be overheard, or is confused by a multitude of participants and divergent interests, the need for convergent expectations and the role of signals that have the power to coordinate expectations may be powerful.

DEFINE: Extends Schelling's focal point concept beyond tacit communication into the explicit bargaining realm, reframing how coordination signals work.

Perhaps many kinds of social stability and the formation of interest groups reflect the same dependence on such coordinators as the terrain and the circumstances can provide: the band wagon at political conventions that often converts the slightest sign of plurality into an overwhelming majority; the power of constitutional legitimacy to command popular support in times of anarchy or political vacuum; the legendary power of an old gang leader to bring order into the underworld, simply because obedience depends on the expectation that others will be obedient in punishing disobedience.

What useful insight does this line of analysis provide into the practical problems of tacit bargaining that usually confront us, particularly the problems of strategic maneuver and limited war? It certainly suggests that it is possible to find limits to war—real war, jurisdictional war, or whatever—without overt negotiation. But it gives us no new strong sense of probability. War was limited in Korea, and gas was not used in World War II; on the possibility of limited war these two facts are more persuasive than all the suggestions contained in the foregoing discussion. If the analysis provides anything, then, it is not a judgment of the probability of successfully reaching tacit agreement but a better understanding of where to look for the terms of agreement.

XREF: Connects to Schelling's broader work on tacit coordination and focal points, as well as game theory's coordination problems.

If there are important conclusions to be drawn, they are probably these: (1) tacit agreements or agreements arrived at through partial or haphazard negotiation require terms that are qualitatively distinguishable from the alternatives and cannot simply be a matter of degree; (2) when agreement must be reached with incomplete communication, the participants must be ready to allow the situation itself to exercise substantial constraint over the outcome; specifically, a solution that discriminates against one party or the other or even involves “unnecessary” nuisance to both of them may be the only one on which their expectations can be coordinated.

Gas was not used in World War II. The agreement, though not without antecedents, was largely a tacit one. It is interesting to speculate on whether any alternative agreement concerning poison gas could have been arrived at without formal communication (or even, for that matter, with communication). “Some gas” raises complicated questions of how much, where, under what circumstances: “no gas” is simple and unambiguous. Gas only on military personnel; gas used only by defending forces; gas only when carried by vehicle or projectile; no gas without warning—a variety of limits is conceivable; some may make sense, and many might have been more impartial to the outcome of the war. But there is a simplicity to “no gas” that makes it almost uniquely a focus for agreement when each side can only conjecture at what rules the other side would propose and when failure at coordination on the first try may spoil the chances for acquiescence in any limits at all.

XREF: Connects to Schelling's broader work on focal points and tacit bargaining — the idea that simple rules make natural coordination points.

In Korea, weapons were limited by the qualitative distinction between atomic and all other; it would surely have been much more difficult to stabilize a tacit acceptance of any limit on size of atomic weapons or selection of targets. {37} No definition of size or target is so obvious and natural that it goes without saying, except for “no size, on any target.” American assistance to the French forces in Indochina was persuasively limited to material, not people; and it was appreciated that an enlargement to include, say, air participation could be recognized as limited to air, while it would not be possible to establish a limited amount of air or ground participation. One’s intentions to abstain from ground intervention can be conveyed by the complete withholding of ground forces; one cannot nearly so easily commit some forces and communicate a persuasive limit to the amount that one intends to commit.

QUESTION: If no natural qualitative threshold exists in a domain, how do tacit limits ever form there?

In sum, the problem of limiting warfare involves not a continuous range of possibilities from most favorable to least favorable for either side; it is a lumpy, discrete world that is better able to recognize qualitative than quantitative differences, that is embarrassed by the multiplicity of choices, and that forces both sides to accept some dictation from the elements themselves. The writer suggests that the same is true of restrained competition in every field in which it occurs.

XREF: Connects to game theory and Thomas Schelling's work on limited war and bargaining, where discrete rather than continuous choice sets shape strategic outcomes.

While the main burden of this paper has been that tacit bargaining is possible and is susceptible of systematic analysis, there is no assurance that it will succeed in any particular case or that, when it succeeds, it will yield to either party a particularly favorable outcome compared with alternatives that might have been available if full communication had been allowed. There is no assurance that the next war, if it comes, will find mutually observed limits in time and of a sort to afford protection, unless explicit negotiation can take place. There is reason, therefore, to consider what steps can be taken before the time for tacit bargaining occurs, to enhance the likelihood of a successful outcome.

But all such efforts may suffer from the unwillingness of an adversary to engage in any preparatory steps. Not only may an adversary balk at giving signs of eagerness to come to agreement; it is even possible that one side in a potential war may have a tactical interest in keeping that war unrestrained and aggravating the likelihood of mutual destruction in case it comes. Why? Because of the strategy of threats, bluffs, and deterrents. The willingness to start a war or take steps that may lead to war, whether aggression or retaliation to aggression, may depend on the confidence with which a nation’s leaders think a war could be kept within limits. To be specific, the willingness of America to retaliate against local aggression with atomic attack depends—and the Russians know that it depends—on how likely we consider it that such retaliation could itself remain limited. That is, it depends on how likely it is in our judgment that we and the Russians, when we both desperately need to recognize limits within which either of us is willing to lose the war without enlarging those limits, will find such limits and come to mutually recognized acquiescence in them. If, then, Russian refusal to engage in any activity that might lead to the possibility of limited war deters our own resolution to act, they might risk forgoing such limits for the sake of reducing the threat of American action. One parachutist in our example may know that the other will be careless with the plane if he is sure they can meet and save themselves; so if the first abstains from discussing the contingency, the other will have to ride quietly for fear of precipitating a fatal separation in the terrain below.

XREF: Game-theory deterrence logic parallel to Schelling's own work on threats and commitments.

When some signal is desperately needed by both parties and both parties know it, even a poor signal and a discriminatory one may command recognition, in default of any other. Once the contingency is upon them, their interests, which originally diverged in the play of threats and deterrents, substantially coincide in the desperate need for a focus of agreement.

XREF: Relates to Schelling's focal point theory — the notion that shared context can coordinate divergent parties toward common ground.

Recall the man who proposed the letter R in one of the bargaining games: as long as the partner heard—and it is obvious that he heard—the letter R is the only extant proposal, and, being unchallenged, it may coordinate in default of any counterproposal nearly as well as if it had been explicitly accepted. (Even denial of it by the other party might not manage to dislodge its claim to prominence but rather simply prove his awareness of it, as long as no rival claim was made that created ambiguousness.)

DEFINE: Explains how prominence in bargaining works: a salient proposal coordinates behavior not by explicit acceptance but by being the only unchallenged alternative on the table.

Whether this consideration or just the usual inhibitions on serious negotiation make prior discussion impossible, there is still a useful idea that emerges from one of our earlier games. It is that negotiation or communication for the purpose of coordinating expectations need not be reciprocal: unilateral negotiation may provide the coordination that will save both parties. Furthermore, even an unwilling member cannot necessarily make himself unavailable for the receipt of messages.

XREF: Relates to Schelling's coordination game theory and focal points from earlier in the work. DEFINE: Clarifies 'unilateral negotiation' as one-way message-sending that still achieves coordination.

On the strategy of pure conflict—the zero-sum games—game theory has yielded important insight and. advice. But on the strategy of action where conflict is mixed with mutual dependence—the nonzero-sum games involved in wars and threats of war, strikes, negotiations, criminal deterrence, class war, race war, price war, and blackmail; maneuvering in a bureaucracy or in a traffic jam; and the coercion of one’s own children—traditional game theory has not yielded comparable insight or advice. These are the “games” in which, though the element of conflict provides the dramatic interest, mutual dependence is part of the logical structure and demands some kind of collaboration or mutual accommodation—tacit, if not explicit—even if only in the avoidance of mutual disaster.

These are also games in which, though secrecy may play a strategic role, there is some essential need for the signaling of intentions and the meeting of minds. Finally, they are games in which what one player can do to avert mutual damage affects what another player will do to avert it, so that it is not always an advantage to possess initiative, knowledge, or freedom of choice.

DEFINE: Distinguishes coordination games, which require signaling intentions and meeting minds, from secrecy-heavy strategic games.

Suggestions and inferences, threats and promises, are of no consequence in the accepted theory of zero-sum games. They are of no consequence because they imply a relation between the two players that, unless perfectly innocuous, must be to the disadvantage of one player; and he can destroy it by adopting a minimax strategy, based, if necessary, on a randomizing mechanism.

Whenever the communication structure does not permit players to divide the task ahead of time according to an explicit plan, it may not be easy to coordinate behavior in the course of the game. Players have to understand each other, to discover patterns of individual behavior that make each player’s actions predictable to the other; they have to test each other for a shared sense of pattern or regularity and to exploit clichés, conventions, and impromptu codes for signaling their intentions and responding to each other’s signals. They must communicate by hint and by suggestive behavior.

The third payoff would show Holmes and Moriarty both being rewarded if they succeed in getting off at the same station, but Holmes gaining more if both he and Moriarty get off at one particular station, Moriarty gaining more if both get off at some other particular station, both losing unless they get off at the same station. This is the usual nonzero-sum game, or “imperfect-correlation-of-preferences” game. This is the mixture of conflict and mutual dependence that epitomizes bargaining situations.

The values of any particular outcome of the game, for the two players, would be represented by the two coordinates of a point. All possible outcomes of a pure-conflict game would be represented by some or all of the points on a negatively inclined line, those of a pure common-interest game by some or all of the points on a positively inclined line. In the mixed game, or bargaining situation, at least one pair of points would denote a negative slope and at least one pair a positive slope.

The role of "incidents" can thus be seen as a coordinating role; it is a substitute for overt leadership and communication. Without something like an incident, it may be difficult to get action at all, since immunity requires that all know when to act together. Similarly, the city that provides no "obvious" central point or dramatic site may be one in which mobs find it difficult to congregate spontaneously; there is no place so "obvious" that it is evident to everyone that it is obvious to everyone else.

Quota systems in housing developments, schools, and so forth, can be viewed as efforts to substitute an explicit game with communication and enforcement for a tacit game that has an undesirably extreme "solution."

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

XREF: Links to Thomas Schelling's focal point concept from The Strategy of Conflict, and David Lewis' convention theory; the collective-expectation regress mirrors these established frameworks.

The individual character of one of these units seems to be largely a matter of convergent expectations—everyone's expectation of what everyone expects of everyone—with the new arrivals' expectations being molded in time to help mold the expectations of subsequent arrivals. There is a sense of "social contract," the particular terms of which are sensed and accepted by each incoming generation. I am told that this persistence of a tradition in a social entity is one of the reasons why the legal identity of an army division or regiment—its name and number and history—is often deliberately preserved when its strength has fallen to where abolition might seem indicated: the tradition that goes with the legal identity of the group is an asset worth preserving for a future buildup.

DEFINE: Defines the character of a social unit as a matter of convergent expectations forming a social contract.

It should be emphasized that coordination is not a matter of guessing what the "average man" will do. One is not, in tacit coordination, trying to guess what another will do in an objective situation; one is trying to guess what the other will guess one's self to guess the other to guess, and so on ad infinitum.

DEFINE: Clarifies coordination as an iterative guessing process, not prediction of an average person's behavior.

In the pure-coordination game, the player's objective is to make contact with the other player through some imaginative process of introspection, of searching for shared clues; in the minimax strategy of a zero-sum game—most strikingly so with randomized choice—one's whole objective is to avoid any meeting of minds, even an inadvertent one.

XREF: Relates to coordination games and game theory concepts also explored in Schelling's and game-theoretic literature.

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

DEFINE: Tightly distinguishes normative (strategy of correct play) from predictive (generalizations about actual choice) game theory, and argues a normative theory must incorporate symbolic details.

One player is "located" in Cincinnati, the other in San Francisco; they have identical maps of the United States and are to divide the country between them. Each is to draw a line dividing the United States into two parts; the line may be straight or curved, related or unrelated to physical or political landmarks. If the two of them divide the map differently, neither gets anything; but if they draw identical division lines on their maps, they are both rewarded. The reward for each player depends on what is contained in his piece after the division, that is, the piece that contains the city in which he is located.

DEFINE: This describes a pure coordination game where players must independently make identical decisions to succeed, with no communication.

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

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

Furthermore, if there are moves available to the players, so that it is an advantage to get on with the maneuver even while negotiating, and particularly if some maneuvers become visible to the other player only after a time lag, there is no reason to suppose that an instantaneous moratorium on maneuver will reign from the outset; in that case, the game progresses while the talk is going on. … One may say and say that a gun is loaded without being able to prove it until he actually shoots; one may say and say that he considers an area strategically important and not be believed until he incurs expense or risk in its protection. Thus moves can reveal information about a player's value system or about the choices of action available to him; moves can commit him to certain actions when speech often cannot; and moves can often progress at a speed that is determined unilaterally, not dependent on formalities of agreement at a conference.

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

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

XREF: Connects to game theory and strategic communication concepts — could link to signaling theory or how deliberate randomness cuts off inference.

The fundamental psychic and intellectual process is that of participating in the creation of traditions; and the ingredients out of which traditions can be created, or the materials in which potential traditions can be perceived and jointly recognized, are not at all coincident with the mathematical contents of the game.

DEFINE: Clarifies that traditions are the core unit of chess culture, distinct from and irreducible to the mathematical/game-theoretic content.

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

XREF: Connects to game theory and signalling theory — the idea that a threat or retaliation is only credible if the other party can recognize the pattern being signaled.

It is not being argued that players just do respond to the non-mathematical properties of the game but that they ought to take them into account, hence that even a normative theory—a theory of the strategy of games—must recognize that rational players may jointly take advantage of them. And even when one rational player realizes that the configuration of these details discriminates against him, he may also rationally recognize that he has no recourse—that the other player will rationally expect him to submit to the discipline of the suggestions that emanate from the game's concrete details and will take actions that, on pain of mutual damage, assume he will co-operate.

DEFINE: Defines a normative game theory that extends beyond mathematical payoffs to include the concrete, non-mathematical properties of a game, which rational players are obligated to consider.

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.

XREF: Connects to bargaining theory and Schelling's focal points, where discrete qualitative distinctions outweigh quantitative concessions.

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.

XREF: Relates to Schelling's focal point theory and commitment devices in behavioral economics; parallels the idea of 'just one drink' as an unstable equilibrium.

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

XREF: Relates to signaling theory and game-theoretic concepts of focal points from Schelling's work on conflict and coordination.

We must avoid assuming that everything the analyst can perceive is perceived by the participants in a game, or that whatever exerts power of suggestion on the analyst does so on the participant in a game. In particular, game characteristics that are relevant to sophisticated mathematical solutions (except when the same solution can also be reached by an alternative, less sophisticated route) might not have this power of focusing expectations and influencing the outcome. They might have it only if the players perceived each other to be mathematicians.

XREF: Connects to Schelling's focal point theory and game theory's bounded rationality — the idea that mathematical elegance matters less than what players actually notice and expect.

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.

How can we know how badly the Russians would dislike an all-out war in which both sides were annihilated? We cannot; and the reason we cannot is not solely that the Russians are necessarily unwilling that we should know. On the contrary, circumstances may arise in which they are desperate that we should know the truth. But how can they make us know it? How can they make us believe that what they tell us is true? How can the prisoner being tortured for secrets that he really does not know persuade his captors that he does not know them?

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

Von Neumann and Morgenstern illustrated their solution concept for the nonzero-sum game with the example of a seller, A, prepared to sell his house for any price above 10, and two buyers, B and C, prepared to pay up to 15 and 25, respectively. {58} (My numbers.) The novel part of the solution was that C might pay B a share of his saving if, through B’s staying out of the market, C got the house for less than 15. They proposed—and this limitation was inherent in their concept of solution—that the most B might receive from C was 15 – 10 = 5. What is interesting about the information requirement of this solution is not that B’s reservation price of 15 is something that he might try to misrepresent, but that in the ordinary world he could not convincingly communicate the truth if he wanted to.

XREF: Connects to game theory concepts of cooperative solutions and side payments that generalize to mechanisms like auction theory and market design.

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

XREF: Connects to negotiation and game theory literature on empathy gaps and perspective-taking, and to the broader strategic literature on asymmetric opponents.

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

XREF: Connects to the classic economics principle that costly signals and revealed preferences carry more information than cheap talk.

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

By abstracting from communication and enforcement systems and by treating perfect symmetry between players as the general case rather than a special one, game theory may have overshot the level at which the most fruitful work could be done and may have defined away some of the essential ingredients of typical nonzero-sum games. Preoccupied with the solution to the nonzero-sum game, game theory has not done justice to some typical game situations or game models and to the “moves” that are peculiar to nonzero-sum games of strategy.

To illustrate, an instructive model is that of twenty men held up for robbery or ransom by a single man who has a gun and six bullets. They can overwhelm him if they are willing to lose six of themselves, if they have a means of deciding which six to lose. They can defeat him without loss if they can visibly commit themselves to a threat to do so, if they can simultaneously commit themselves to a promise to abstain from capital punishment, once they have caught him. He can deter their threat if he can visibly commit himself to shoot in disregard of any subsequent threat they might make, or if he can show that he could not believe their promise. … If the twenty cannot find a way to divide the risk, there may be no one to go first to carry out the threat, hence no way to make the threat persuasive; and if he can announce a formula for shooting, such as that those who move first get shot first, he can deter them unless they find a way to move together without a “first.” If fourteen of the twenty can overpower the remaining six and force them to advance, they can demonstrate that they could overwhelm the man; if so, the threat succeeds and the gunman surrenders, and even the six “expendables” gain through their own inability to avoid jeopardy. If the twenty could overwhelm the man but have no way of letting him escape, a promise of immunity may be necessary; but if they cannot deny their capacity to identify him and testify against him later, it may be necessary to let him take a hostage.

XREF: This echoes Schelling's coordination problems and free-rider logic familiar from game theory and collective action theory.

If the institutional environment makes it possible for a potential buyer to make a single “final” offer subject to extreme penalty in the event he should amend the offer—to commit himself—there remains but a single, well-determined decision for the seller: to sell at the price proposed or to forego the sale. The possibility of commitment converts an indeterminate bargaining situation into a two-move game; one player assumes a commitment, and the other makes a final decision. The game has become determinate.

The distinctive character of a threat is that one asserts that he will do, in a contingency, what he would manifestly prefer not to do if the contingency occurred, the contingency being governed by the second party’s behavior. Like the ordinary commitment, the threat is a surrender of choice, a renunciation of alternatives, that makes one worse off than he need be in the event the tactic fails; the threat and the commitment are both motivated by the possibility that a rational second player can be constrained by his knowledge that the first player has altered his own incentive structure. Like an ordinary commitment, a threat can constrain the other player only insofar as it carries to the other player at least some appearance of obligation; if I threaten to blow us both to bits unless you close the window, you know that I won’t unless I have somehow managed to leave myself no choice in the matter.

The threat differs from the ordinary commitment, however, in that it makes one’s course of action conditional on what the other player does. While the commitment fixes one’s course of action, the threat fixes a course of reaction, of response to the other player. The commitment is a means of gaining first move in a game in which first move carries an advantage; the threat is a commitment to a strategy for second move.

DEFINE: Distinguishes threat from commitment by temporal position: commitment is first-move strategy, threat is second-move strategy conditional on opponent action.

The fact that some kind of commitment, or at least appearance of commitment, must lie behind the threat and be successfully communicated to the threatened party is in contradiction to another notion that often appears in game theory. This is the notion that a threat is desirable, or admissible, or plausible, only if the reaction threatened would cause worse damage to the threatened party than to the party making the threat. This is the view of Luce and Raiffa, who characterize threats by the phrase, "This will hurt you more than it hurts me," explicitly making threats depend on interpersonal utility comparisons.

DEFINE: Distinguishes commitment-based credibility from the Luce/Raiffa view that threats hinge on interpersonal utility comparison. XREF: Relates to Schelling's broader work on commitment and credible threats in strategic interaction.

But if we allow Row to make a threat, he declares that he will choose strategy ii unless Column chooses II; that is, he gives Column a choice of ii,I or i,II by committing himself to that conditional choice. If Column went ahead and chose I, of course, Row would prefer to choose i; and they both know it. The tactic succeeds only if Column believes that Row must choose ii in the event of I.

DEFINE: Distinguishes a genuine strategic threat from an empty one depends on credibility of the conditional commitment.

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

Builds on: "Threats condition action on rivals' moves; commitments fix first moves."

"Commitment" is to be interpreted broadly here. It includes maneuvers that leave one in such a position that the option of non-fulfilment no longer exists (as when one intimidates the other car by driving too fast to stop in time), maneuvers that shift the final decision beyond recall to another party whose incentive structure would provide an ex post motive for fulfilment (as when the authority to punish is deliberately given to sadists, or when one shifts his claims and liabilities to an insurance company), and maneuvers that simply "worsen" one's own payoff in the contingency of non-fulfilment so that even the horror of a mutually damaging fulfilment becomes more attractive (as when one arranges for himself to appear a public coward if he fails to fulfil, or when he puts a plate-glass window in front of his wares or stations women and children on the particular bit of territory that he has threatened somewhat implausibly to defend at great cost).

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

DEFINE: Distinguishes a promise from a commitment by emphasizing second-party enforcement control, while noting timing and irreversibility complicate the definition.

Trust is often achieved simply by the continuity of the relation between parties and the recognition by each that what he might gain by cheating in a given instance is outweighed by the value of the tradition of trust that makes possible a long sequence of future agreement. By the same token, “trust” may be achieved for a single discontinuous instance, if it can be divided into a succession of increments.

XREF: This echoes classic game-theoretic reasoning about iterated prisoner's dilemma and reputation models found in economics literature.

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

QUESTION: How do these self-enforcing coordination agreements relate to Schelling's broader theory of focal points and tacit bargaining?

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

XREF: Connects to game theory and commitment problems — hostage institutions and shared-ritual practices both solve credibility dilemmas.

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

DEFINE: Reframes the 'balance of terror' as a total exchange of hostages, achieved not by physical trading but by mutual exposure to destruction.

What makes the threat or ordinary commitment a difficult tactic to employ and an interesting one to study is the problem of finding a means to commitment, the available “penalty” to invoke against one’s own nonperformance. There is consequently a related set of tactics that consists of maneuvering one’s self into a position in which one no longer has any effective choice over how he shall behave or respond. The purpose of these tactics is to get rid of an embarrassing initiative, making the outcome depend solely on the other party’s choice.

This is the kind of tactic that Secretary of State John Foster Dulles was looking for in the following passage: … In the future it may thus be feasible to place less reliance upon deterrence of vast retaliatory power....Thus, in contrast to the 1950 decade, it may be that by the 1960 decade the nations which are around the Sino-Soviet perimeter can possess an effective defense against full-scale conventional attack and thus confront any aggressor with the choice between failing or himself initiating nuclear war against the defending country.

DEFINE: Clarifies the 'tactic' Dulles sought — a shift away from massive retaliation toward perimeter defense against conventional attack.

Angelo’s only interest in torturing the brother is in what he may gain by making a threat to do so; once there is somebody available to whom the threat can profitably be communicated, the possibility of torture has value for Angelo—not the torture itself, but the threatening of it. The sister has gotten negative value out of her trip; having identified her interest and made herself available to receive the threatening message, she has been forced to suffer what she would not have had to suffer if she had never made her identity known or if she could have disappeared into the crowd before the threat was made.

DEFINE: Illuminates how the value of a threat lies in its communication to an identifiable recipient, not in the act it threatens.

A nice identification game was uncovered in a New York suburb a few years ago. Certain motorists carried identity cards which identified them to policemen as members in a club; if the motorist with a membership card was arrested, he simply showed the card to the policeman and paid a bribe. The role of these cards was to identify the motorist as a person who, if the bribe was received, would keep quiet. It identified the motorist as a man whose promise was enforcible. But the card identifies the motorist only after he has been arrested; if the police could identify card-carrying motorists by looking at them, they could concentrate their arrests on card-carrying drivers, threatening a ticket unless payment were received. The card is contingent identification, at the option of the motorist.

XREF: Connects to signaling theory and reputation economics patterns, like Akerlof's lemons or costly signaling in biology. SEED: Could angle an article on how privileges and trust signals work — the design insight that withholding identification until needed preserves asymmetry.

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 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.)

XREF: Connects to Schelling-style commitment and credibility problems in strategic bargaining and deterrence theory — deliberately delegating to actors unable to back down.

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 coordination requires the common acceptance of some source of suggestion.

DEFINE: Explains how coordination and suggestion function in traffic contexts, illustrating a broader principle about power and consensus.

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, and so forth—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.

XREF: Connects to coordination games and focal point theory in Schelling's framework; mediator as a mechanism for resolving coordination failures.

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.

XREF: Connects to game theory and mechanism design — the mediator as a trusted third party who can process information without full disclosure, relevant to trustless computation and MPC (secure multi-party computation) literature.

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.

When the outcome depends on coordination, 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.

FICTION: A compelling dramatic beat: a character wins a conflict not by out-arguing but by unilaterally ending the conversation, forcing coordination on their terms.

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.

DEFINE: Distinguishes communicating a commitment (which needs evidence/authentication) from merely communicating words.

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 might sell; he is stripped of his power to be intimidated. He is made impotent to meet the demands of blackmail.

XREF: Connects to game theory and mechanism design, where information asymmetry is deliberately engineered in political institutions.

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 noncompliance; 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.

DEFINE: Characterizes tacit, asymmetrical communication where non-perception strategically avoids triggering reciprocal obligations.

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—that is, 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 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.

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).

DEFINE: Defines the two-part requirement for credible commitment: an enforceable promise plus credible evidence of the commitment through penalty or destroying one's own ability to defect.

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. {81} 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 payoffs in all cells that do not correspond to the strategy so selected.

(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 himself to row ii. This is the evident “solution”; it has a payoff of [5 2], 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.)

DEFINE: Illustrates the concept of commitment as a strategic device that changes opponents' expectations and thus outcomes. Pays off: "Commitment transforms indeterminate bargaining into a determinate game."

This, then, is the way that a solution to the original sequential-move game shows up in the static ("moveless," or simultaneous-tacit-choice) game. It is a solution arrived 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, knowing 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.

DEFINE: Explains backward induction as iterative elimination of dominated strategies in the enlarged simultaneous-tacit-choice game.

While it is instructive and intellectually satisfying to see how such tactics as threats, commitments, and promises can be absorbed into an enlarged, abstract "supergame" (game in "normal form"), it should be emphasized that we cannot learn anything about those tactics by studying games that are already in normal form. The objects of our study, namely, these tactics together with the communication and enforcements 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.

XREF: Echoes critiques that abstraction erases the very phenomena under study — comparable to how formalized models in other fields strip away processual detail.

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 while facing an enemy 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.

XREF: Connects to Schelling's commitment and credibility work, and the concept of 'burning bridges' as a negotiation tactic. QUESTION: When does self-imposed disadvantage backfire rather than help? There must be boundary conditions on this principle.

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 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.

XREF: Ties to understanding how sanctions and financial controls produce unintended protective side effects alongside their intended coercive purpose.

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 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 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."

XREF: Connects to the broader diplomatic/territorial concessions dynamics being discussed in the surrounding text.

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.

DEFINE: Defines strategic moves in game theory as self-constraint that shapes an opponent's expectations rather than direct action.

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. And, similarly, in a zero-sum game between rational players with full information it can never be an advantage to move first (to play the "minorant game" in the language of von Neumann and Morgenstern); in the mixed game it certainly can.

XREF: Connects to game theory concepts of von Neumann and Morgenstern, and the broader idea that asymmetries of information can be strategically useful rather than purely disadvantageous.

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.

DEFINE: Schelling distinguishes the empirical question of how people perform from the stronger claim that normative strategic principles themselves cannot be derived analytically.

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 mini-max 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 nonzero-sum game of maneuver any more than one can prove, by purely formal deduction, that a particular joke is bound to be funny.

But in a nonzero-sum game that involves any initial uncertainty over which among the possible outcomes are in fact efficient and any need for coordinated 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.

Because a “law of reprisal” is essentially casuistic in nature; because the mutually recognized restraints in any form of “limited war” are essentially based on something 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, graduated 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. 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 developing 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, that is, when the game is “indeterminate,” norms of some sort must be developed, mutually perceived, and accepted; patterns of action and response have to be legitimized.

XREF: Connects to game theory and Schelling-style coordination norms; similar to how emergent conventions solve coordination problems.

A “scenario” might, for example, identify one of the players as “aggressor”; it might give the outcomes of previous plays of the 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 “status quo”; or it might seem to attach a kind of moral claim of one of the players to particular 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 corresponds to the way it stood in the middle of the same game as played earlier by two other players, and see whether the outcome can be affected 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 hypothetical starting point.

SEED: Suggests an essay experiment on how suggestive framing — even explicitly non-authoritative — shapes people's sense of fairness and baseline norms in negotiation or game theory.

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

QUESTION: Notes an empirical question about how players recognize symbolic investment in encounters — a research direction worth pursuing.

Another dimension of the game that seems susceptible of analysis is the significance of the incrementalism that is involved in the moves and value systems. Take, for example, a game that involves moving pieces over a board or troops over 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 mistakes of individual players or mutual mistakes 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, instead, 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 incremental; things can happen abruptly. There may be a temptation toward surprise attack.

XREF: Connects to game theory and Schelling's work on brinkmanship, mutual destruction, and the destabilizing effect of first-strike advantages — likely from a chapter on conflict and strategy. Builds on: "Strategic moves win by constraining one's own behavior."

It has been a device to make indivisible objects divisible, or incommensurate objects homogeneous. Their "expected values" are divisible by lottery when the objects themselves are not. We flip coins to see who gets the object, and play "double or nothing" when we cannot make change. We can divide the obligation of citizenship equally by selecting draftees through a lottery, when we want a fraction of the eligibles for a long period of service rather than all of them for a short one.

XREF: Connects to broader work on lotteries and fairness in allocation — likely from a book on lottery logic (possibly by Aaron Brown or similar). Dovetails with any discussion of chance as an equalizing mechanism.

This particular mixed commitment can be called a combination of a fractional threat with a fractional promise. Row, in effect, “threatens” a relatively high probability of i in the event that II is chosen and “promises” it if I is chosen.

DEFINE: Defines the concept of a 'mixed commitment' as composed of a fractional threat plus a fractional promise across two alternative choices.

As a rule, one must threaten that he will act, not that he may act, if the threat fails. To say that one may act is to say that one may not, and to say this is to confess that one has kept the power of decision—that one is not committed. To say only that one may carry out the threat, not that one certainly will, is to invite the opponent to guess whether one will prefer to punish himself and his opponent or to pass up the occasion. Furthermore, if one says that he may—not that he will—and the opponent fails to heed the threat, and the threatener chooses not to carry it out, he only confirms his opponent's belief that when he has a clear choice to act or to abstain he will choose to abstain

The key to these threats is that, though one may or may not carry them out if the threatened party fails to comply, the final decision is not altogether under the threateneds control. The threat is not quite of the form "I may or may not, according as I choose," but, has an element of, "I may or may not, and even I can't be altogether sure." … Where does the uncertain element in the decision come from? It must come from somewhere outside of the threatener's control. Whether we call it "chance," accident, third-party influence, imperfection in the machinery of decision, or just processes that we do not entirely understand, it is an ingredient in the situation that neither we nor the party we threaten can entirely control.

The threat—if we call this contingent-behavior mechanism a "threat"—has some interesting features. It may exist whether we realize it or not. Even those who have doubted whether our massive-retaliation threat was a potent deterrent to minor aggression during the last several years, but are perplexed that the Russians have not engaged in more mischief than they have, can note that the threat we voiced was backed by an additional implicit threat that we might be triggered by Soviet actions in spite of ourselves. Furthermore, even if we prefer not to incur even a small probability of inadvertent war, and would not use this mechanism deliberately, the "threat" in question may be a byproduct of other actions that we have a powerful incentive to take. We may get this threat whether we like it or not when we (and the Russians) take precautions commensurate with a crisis; knowing this, the Russians may have to take the risk into account.

XREF: Connects to Schelling's broader work on commitment, brinkmanship, and the rationality of threats — the irrational/semi-automatic threat is a recurring theme in his strategic thought. Builds on: "Mixed commitments combine partial threats with partial promises to guide choice."

Finally, the threat is not discredited even if the Russians accomplish their purpose without triggering war. If the Russians estimate that the chance of inadvertent war during a particular month rises from very small to not-so-small if they create a crisis, and they go ahead anyway, and no major war occurs, they still have little reason to suppose that their original estimate was wrong, and little reason to suppose that repetition would be less risky, any more than a person who survives a single play of Russian roulette should decide it isn't dangerous after all.

QUESTION: This probabilistic reasoning about repeated risk-taking is striking — worth reflecting on how it maps to other domains beyond nuclear brinkmanship.

Here again is a threat that all-out war may occur, not that it certainly will occur, if the other party engages in certain actions. Again, whether it does or does not occur is not a matter altogether controlled by the threatener. Just how all-out war would occur—just where the fault, initiative, or misunderstanding may occur—is not sure. Whatever it is that makes limited war between great powers a risky thing, the risk is a genuine one that neither side can altogether dispel if it wants to. The final decision, or the critical action that initiates an irreversible process, is not something that should necessarily be expected to be taken altogether deliberately. "Chance" helps to decide whether general war occurs or not, with odds that are a matter of judgment based on the nature of the limited war and the context in which it occurs.

DEFINE: Clarifies the concept of 'threat' as a conditional (war may occur if) rather than a certainty, and distinguishes deliberate decision from chance-driven escalation.

But if we can threaten him with a one-in-twenty chance of all-out war in the event he proceeds, and he does proceed, we can hold our breath and have nineteen-to-one odds of getting off without general war. Of course, if we scale down the risk to us, we scale it down to him too; it may degrade the threat to put too much safety in it. But in cases where there is danger that we completely misjudge the enemy's commitment to an action, or completely misjudge his ability to control his own agents, allies, or commanders, the more moderate risk may deter anything that is still within his control.

XREF: Connects to Schelling's broader work on deterrence and the credibility of threats.

What we have is a graduated series of trip wires, each attached to a chance mechanism, with the daily probability of detonation increasing as the enemy moves from wire to wire.

The critical feature of the analogy, it should be emphasized, is that whether or not the trip wire detonates general war is—at least to some extent—outside our control, and the Russians know it.

One can argue that the Chinese or Russians were deterred by the prospect of major war, not just by the prospect of losing a limited war or winning one at excessive cost. Even if they were convinced that we would exercise every skill and caution to keep a war limited, and they were prepared to exercise skill and caution themselves, they may simply have felt that the process that leads to bigger and bigger wars is not one that they or we fully understand or can foresee, and that the risk, though numerically small, was appreciable.

XREF: Connects to deterrence theory and escalation dynamics, relevant to nuclear strategy literature like Schelling's work on shared expectations and credible threats. Builds on: "Limited war between great powers carries genuine, uncontrollable escalation risk."

If one of the functions of limited war, then, is to pose the deliberate risk of all-out war, in order to intimidate the enemy and to make pursuit of his limited objectives intolerably risky to him, the usual precepts for behavior in limited war need revision.

XREF: This inverts conventional deterrence logic — the deliberate courting of disaster as a strategy parallels 'threat that leaves something to chance' in Schelling's work.

The supreme objective may not be to assure that it stays limited, but rather to keep the risk of all-out war within moderate limits above zero. At least this may be the strategy for the side that is in danger of "losing" a limited war. The less likely it is that the enemy's aggressive advances can be contained by limited and local resistance, the more reason there may be to fall back upon the deliberate creation of mutual risk.

To share such an increase in risk with an enemy may provide him an overpowering incentive to lay off. Preferably one creates the shared risk by irreversible maneuvers or commitments, so that only the enemy's withdrawal can tranquilize the situation; otherwise it may turn out to be a contest of nerves.

DEFINE: Explains the strategic mechanism of 'brinkmanship' — deliberately creating shared, irreversible risk to make an adversary back down.

As in limited war, there again may be two parts to the "cost" imposed on an enemy. One is a direct cost: casualties, destruction, humiliation, or whatever it may be. The other is the created risk of all-out war. Nobody quite knows what happens if one country explodes a nuclear weapon in an enemy country.

Builds on: "Deterrence rests on fear of uncontrolled escalation, not just defeat."

There is typically a difference between a threat intended to make an adversary do something (or cease doing something) and a threat intended to keep him from starting something. The distinction is in the timing, in who has to make the first move, in whose initiative is put to the test. To deter by threat an enemy’s advance it may be enough to burn the bridges behind me as I face the enemy; to compel by threat an enemy’s retreat I have to be committed to move forward, and this requires setting fire to the grass behind me with the wind blowing toward the enemy. I can block your car in the road by placing my car in your way; my deterrent threat is passive, the decision to collide is up to you. If you, however, find me in your way and threaten to collide unless I move, you enjoy no such advantage; the decision to collide is still yours, and I enjoy deterrence. You have to arrange to have to collide unless I move, and that is a degree more complicated.

DEFINE: Draws the deterrence/compellence distinction: deterrence is passive (the adversary decides to collide), compellence requires the threatener to commit to moving, a more complicated arrangement.

Given the fact that, even if the enemy complies, there is some risk that the box detonates war before he has a chance to collect himself and do our bidding, there is an advantage in making it less than certain that the box will explode on any given day. In ordinary deterrence—where nothing happens unless the enemy acts contrary to our demand—to threaten too much may be superfluous but not self-defeating; in the present case—where the threat starts fulfilling itself at a specified rate over time as soon as we commit ourselves to it—too big a threat can defeat its purpose. In this situation the small-probability threat is not just a possible substitute for the large certain threat; it is a superior and necessary alternative.

DEFINE: The Doomsday Machine model distinguishes ordinary deterrence from bounded deterrence, where the threat self-fulfills over time and certainty becomes self-defeating.

What is the country doing? The principal thing the country is doing—in addition to damaging or humiliating Russia—is incurring a painful risk that both it and Russia (and the rest of the world) will be engaged in all-out war in the near future, a war that neither it nor Russia wants. The country is saying in effect, “If you do not get out of Hungary, we may cause an all-out war to occur.” By when must the Russians get out? The sooner they get out, the sooner the risk of war (from this cause) will be terminated or reduced. The country applying the pressure is not saying, “Get out or we shall deliberately start a war.” The decision is not up to them, and does not depend on their displaying the manifest resolution for a final act. The Russians may suppose that the country concerned will do everything it can to prevent total war; but they also have to recognize that with these things flying around, exploding now and then, and with themselves responding in whatever way they feel obliged to, it is not altogether clear that the country concerned, and the Russians, know how to keep total war from occurring.

QUESTION: How does this Cold War framing about the Hungarian crisis map to contemporary escalation dynamics? The mechanism of risk-creation as coercion is worth examining.

The brink is not, in this view, the sharp edge of a cliff where one can stand firmly, look down, and decide whether or not to plunge. The brink is a curved slope that one can stand on with some risk of slipping, the slope gets steeper and the risk of slipping greater as one moves toward the chasm. But the slope and the risk of slipping are rather irregular; neither the person standing there nor onlookers can be quite sure just how great the risk is, or how much it increases when one takes a few more steps downward. One does not, in brinkmanship, frighten the adversary who is roped to him by getting so close to the edge that if one decides to jump one can do so before anyone can stop him. Brinkmanship involves getting onto the slope where one may fall in spite of his own best efforts to save himself, dragging his adversary with him.

DEFINE: Redefines brinkmanship by contrast with a cliff-edge metaphor; key insight is the slope where one may fall despite best efforts, dragging the adversary along. Builds on: "Limited war can deliberately risk total war to deter escalation."

Underlying this threat that one “may” retaliate or precipitate war—the decision being somewhat beyond his control—is the notion that some of the most momentous decisions of government are taken by a process that is not entirely predictable, not fully “under control,” not altogether deliberate. It implies that a nation can get even into a major war somewhat inadvertently, by a decision process that might be called “imperfect” in the sense that the response to particular contingencies cannot exactly be foretold by any advance calculations, that the response to a particular contingency may depend on certain random or haphazard processes, or that there will be faulty information, faulty communication, misunderstanding, misuse of authority, panic, or human or mechanical failure.

XREF: Connects to Kahneman's work on decision-making, and to the 'fog of war' concept from Clausewitz.

For this reason there is no such thing as a “firm” plan, intention, or policy of a government to cover every contingency—even all important foreseeable contingencies. How the considerations add up, what interests are brought to bear, and how the collective decision procedure works in future crises is simply not fully determinable in advance.

The usual idea that a trip wire either does work or does not work, that the Russians either expect it to work or expect it not to work, is mistaking two simple extremes for a more complicated range of probabilities.

If I go downstairs to investigate a noise at night, with a gun in my hand, and find myself face to face with a burglar who has a gun in his hand, there is danger of an outcome that neither of us desires. Even if he prefers just to leave quietly, and I wish him to, there is danger that he may think I want to shoot, and shoot first. Worse, there is danger that he may think that I think he wants to shoot. Or he may think that I think he thinks I want to shoot. And so on. “Self-defense” is ambiguous, when one is only trying to preclude being shot in self-defense.

DEFINE: Illustrates why 'self-defense' is ambiguous in reciprocal-fear situations where either party might preemptively shoot. Builds on: "Nuclear strikes impose uncertain risk beyond direct destruction."

But, if the gains from even successful surprise are less desired than no war at all, there is no “fundamental” basis for an attack by either side. Nevertheless, it looks as though a modest temptation on each side to sneak in a first blow—a temptation too small by itself to motivate an attack—might become compounded through a process of interacting expectations, with additional motive for attack being produced by successive cycles of “He thinks we think he thinks we think...he thinks we think he’ll attack; so he thinks we shall; so he will; so we must.”

XREF: This echoes game-theoretic escalation dynamics and the security dilemma familiar from deterrence theory.

It is interesting that this problem, though it arises most dramatically in situations that would usually be characterized as conflict, like that between the Russians and us or between the burglar and me, is logically equivalent to the problem of two or more partners who lack confidence in each other. If each is under some temptation to abscond with the joint assets; if each has a little suspicion that the other may be contemplating the same thing; if each realizes that the other may suspect too, and may suspect himself the object of suspicion; we have a pay-off matrix identical with that of a surprise attack problem. If the heat is on some members of the mob, the rest of the mob may be tempted to rub them out to keep them from squealing, and those in danger may be tempted to squeal in self-defense. So the game structure of “preclusive self-defense” is the same as that of “partnership confidence.”

XREF: This is classic game theory — the prisoner's dilemma logic applied to trust and preemptive defense. Connects to Schelling-style coordination problems.

These two parameters, Pc and Pr, are assumed to be plainly visible to the two players; there is nothing secret or conjectural about them. This assumption might seem to beg the question we are trying to answer, but it does not. These two exogenous likelihoods of attack do not by themselves indicate what the probability is that the players will in fact attack. They are only one element. The problem is to see whether, given these basic sources of uncertainty, the interaction of the two players’ expectations generate additional motive to attack.

DEFINE: Distinguishes exogenous likelihood parameters from the endogenous probability of attack that emerges from interactive expectations.

Even a player whose own probability of “irrational” attack is known to be zero must consider that the second may attack not only irrationally but also out of fear that the first, fearing the second’s attack, may try to strike first to forestall it. Thus it does seem as though we might get a compounding of motives.

XREF: This is the classic security dilemma / first-strike problem from Schelling's The Strategy of Conflict and game theory — fear itself becomes the mechanism that drives escalation. Builds on: "Mutual suspicion can compound small incentives into preemptive attack."

But we do not. We do not get any regular kind of “multiplier” effect out of this. The probabilities of attack by the two sides do not interact to yield a higher probability, except when they yield certainty. That is, the outcome of this game, starting with finite probabilities of “irrational” attack on both sides, is not an enlargement of those probabilities by the fear of surprise attack; it is either joint attack or no attack. That is, it is a pair of decisions, not a pair of probabilities about behavior.

XREF: Counterintuitive game-theoretic result linking to deterrence theory and nuclear brinkmanship literature. Builds on: "Mutual suspicion can compound small incentives into preemptive attack."

The limits to the values of our two parameters, Pr and Pc, beyond which they make the situation unstable and provoke joint attack, are—letting h stand for the value obtained by unilateral surprise attack, –h the value obtained by being attacked while not attacking, 0 the value obtained by simultaneous attack, and 1.0 the value of joint no-attack, for each player—

DEFINE: Defines the payoff structure Pr and Pc and the threshold h beyond which unilateral surprise attack destabilizes the joint no-attack equilibrium.

The same seems to be true now if we go back to that burglar downstairs. If he behaves “rationally” as defined in our behavior hypothesis above, he must consider the likelihood that I will shoot him out of sheer preference; and he must consider that I may shoot him if I think there is a strong likelihood that he will shoot me out of sheer preference. But, if we both know what these two basic (exogenous) “likelihoods” are, we need not go any further. Either these basic probabilities are sufficient to make at least one of us shoot to forestall surprise, and hence to make both shoot, so that the second and higher degree fears are superfluous, or else they are insufficient by themselves to make either of us shoot in self-defense, and we know it and have nothing to fear beyond the exogenous likelihoods themselves.

But a different situation obtains if I shoot not by calculation but by nervousness. Suppose that my nervousness depends on how frightened I am, and my fright depends on how likely I think it that he may shoot me; and suppose he acts the same way. Then when I consider the exogenous probability that he may shoot me out of sheer preference, it makes me nervous; this nervousness enhances the likelihood that I may shoot him even though I prefer not to. He sees my nervousness and gets nervous himself; that scares me more, and I am even more likely to shoot. He sees this increment in my nervousness, and matches it with one of his own, scaring me further; and the probability that I will shoot goes up again. Now we can denote each person's nervousness as a function of the other's, and the likelihood of shooting as a function of nervousness, and have a simple pair of simultaneous differential equations that seem to yield precisely the kind of phenomenon we started off to study.

XREF: Connects to escalation dynamics in game theory, arms races, and conflict spirals—the classic prisoner's dilemma and security dilemma literature.

But now we can have a model of a rational decider who responds to an estimate of the probability of being attacked not by an overt decision to act or abstain, but by adjusting the likelihood that he may mistakenly attack. One’s response to an increase in the probability of being attacked is to shift the criteria for decision that are used in the warning system in the direction of lesser likelihood of a failure to respond, and hence in the direction of greater likelihood of a false alarm that provokes one’s own “retaliation.” If each player’s response to an increased danger of surprise attack is to enhance his own proclivity toward inadvertent attack, the probability of each player’s attack is now a rising function of the other’s.

XREF: A strategic analogue of a positive feedback loop or arms race dynamic — echoes escalation spirals and prisoner's dilemma logics.

We can express each player’s optimum value of B as a function of the other’s, solve the two equations, and deduce the stability conditions for the equilibrium. We can also compute “multipliers” relating each player’s changes of B and R to shifts in the f function or to changes in the A parameters.

Since f′ is presumed positive, the denominator must be positive if Vr is to be maximized with R < 1; but the condition that the denominator be positive is precisely the condition that Pc must meet in order that player R still prefer joint no-attack. Thus, if both players have optimal adjustments with R < 1, those optimal values of R and B are also perforce consistent with joint preference at no-attack.

But, in general, this becomes a wide-open bargaining problem. It is even wider open than the present formulation suggests, since the players may not only manipulate values of R and B but of course can now threaten direct attack, or operate on the institutional arrangements that determine the values of the A’s.

There is an enforcement difficulty with any agreement on reduction of values of R and B in the mutual interest; it is that each other’s values of R and B may not be observable. They depend—at least to an important extent—on the criteria that will govern future decisions, not solely on the observable, physical mechanics of an alert system. They depend on how long one will wait to be “sure,” and on what risks one will accept in an emergency. Furthermore, failure to keep the agreement, if it leads to anything, leads to war itself; so recriminations and damage suits are out of the question if our model represents all-out war rather than a border scrape or a minor transgression of one partner against another.

DEFINE: R and B seem to denote response and brinkmanship parameters; the enforcement difficulty rests on their non-observability.

We are interested not only in assuring ourselves with our own eyes that he is not preparing an attack against us; we are interested as well in assuring him through his own eyes that we are preparing no deliberate attack against him.

As General Leslie R. Groves remarked in a speech, "If Russia knows we won't attack first, the Kremlin will be very much less apt to attack us....Our reluctance to strike first is a military disadvantage to us; but it is also, paradoxically, a factor in preventing a world conflict today."

XREF: Connects to nuclear deterrence theory and the rational-actor paradox of mutually assured destruction. Builds on: "Superpowers need credibility of non-aggression, not just defensive assurance."

"Self-defense" becomes peculiarly compounded if we have to worry about his striking us to keep us from striking him to keep him from striking us....The surprise-attack problem, when viewed as a problem of reciprocal suspicion and aggravated "self-defense," suggests that there are not only secrets we prefer not to keep, but military capabilities we might prefer not to have.

DEFINE: Illuminates the 'security dilemma' concept via the self-defense paradox. XREF: Echoes Schelling's stability theory and Cold War deterrence thinking on the dynamics of reciprocal fear.

But, if either side can obliterate the other, what does it matter who strikes first? The answer, of course, is that we are not particularly concerned with outliving the Russians by a day; we are worried about whether a surprise attack might have such prospects of destroying the power to retaliate as to be undeterred itself by the threat of retaliation. It is not our existing capacity to destroy Russia that deters a Russian attack against us, but our capacity to retaliate after being attacked ourselves. We must assume that a Russian first-strike, if it came, would be aimed at the very power that we rely upon for retaliation.

DEFINE: Distinguishes simple first-strike capacity from credible second-strike deterrence, the core logic of mutual assured destruction.

There is a difference between a balance of terror in which either side can obliterate the other and one in which both sides can do it no matter who strikes first. It is not the "balance"—the sheer equality or symmetry in the situation—that constitutes mutual deterrence; it is the stability of the balance. The balance is stable only when neither, in striking first, can destroy the other's ability to strike back.

DEFINE: Clarifies the distinction between a symmetric balance of terror and a stable one, where stability requires both sides retain second-strike capability.

The "equalizer" of the Old West made it possible for either man to kill the other; it did not assure that both would be killed. The tense consequences of this weapon system can be seen on TV almost any night. The advantage of shooting first aggravates any incentive to shoot. As the survivor might put it, "He was about to kill me in self-defense, so I had to kill him in self-defense." Or, "He, thinking I was about to kill him in self-defense, was about to kill me in self-defense, so I had to kill him in self-defense." But if both were assured of living long enough to shoot back with unimpaired aim, there would be no advantage in jumping the gun and little reason to fear that the other would try it.

DEFINE: This introduces the 'equalizer' concept — how equal-payload weapons create first-strike incentive rather than deterrence, a key game-theory mechanism. Builds on: "Deterrence depends on surviving retaliation capacity, not raw destruction."

We have to try to detect submarines, because we cannot afford to let the Russians find a technique that we do not know, and because we have to learn all we can about detection to make our submarines less detectable; but like a person who has entered into an agreement with a partner that he cannot trust, we may search like the devil for a loophole, knowing that our partner is searching just as hard, while hoping that no loophole is to found.

If our problem is to guarantee to an enemy that we have the ability to strike a punitive blow after being struck ourselves—and to assure him that we know that he knows it so that we are under no temptation to doubt the potency of our own deterrence and strike first—we should find virtue in technological discoveries that enhance the anti-population potency of our retaliatory weapons. If it is logical to take measures to guarantee that a larger proportion of our retaliatory forces could survive a first strike on them, the same logic should make us welcome an increase in the potency of those that do survive. As Bernard Brodie has said, “When we consider the special requirements of deterrence, with its emphasis on the punitive aspect of retaliation, we may find a need even for super-dirty bombs. Since the emphasis must be on making certain that the enemy will fear even the smallest number of bombs that might be sent in retaliation, one wants these bombs to be, and thus to appear before the event, as horrendous as possible.”

The novelty of this reasoning disappears as soon as we recognize that the “balance of terror,” if it is stable, is simply a massive and modern version of an ancient institution: the exchange of hostages. In older times, one committed himself to a promise by delivering his hostages physically into the hands of his distrustful “partner”; today’s military technology makes it possible to have the lives of a potential enemy’s women and children within one’s grasp while he keeps those women and children thousands of miles away. As long as each side has the manifest power to destroy a nation and its population in response to an attack by the other, the “balance of terror” amounts to a tacit understanding backed by a total exchange of all conceivable hostages. We may not, of course, want to exchange quite that many hostages in support of this particular understanding with this particular enemy. But in a lawless world that provides no recourse to damage suits for breach of this unwritten contract, hostages may be the only device by which mutually distrustful and antagonistic partners can strike a bargain.

XREF: Frames nuclear deterrence as a variant of hostage exchange, linking international relations to older legal and customary practices. Pays off: "Hostage-taking structurally parallels wine-sharing and public gang meetings"

This line of reasoning is not simply an enormous rationalization for an arms race. It does indeed suggest that “disarmament” in the literal sense, aimed indiscriminately at weapons of all kinds—or even selectively aimed at the most horrifying weapons of mass destruction—could produce instability rather than stability, and might have to be completely successful in order not to be disastrous. Nevertheless, there is an important area of arms limitations that is not only compatible with the foregoing analysis but is suggested by it.

QUESTION: Which specific categories of arms limitation remain compatible with this stability logic? Worth examining the proposed boundaries of feasible disarmament. Builds on: "Publicly declared no-first-use policy paradoxically deters rival attack."

It suggests making a distinction between the kinds of weapons that are peculiarly suitable to the exploitation of a first strike and weapons that are peculiarly suitable to the retaliatory role. At one extreme is the “pure” strike-back type of weapon: the relatively inaccurate vehicle with a super-dirty bomb that can kill just about everything in the enemy’s country except a well-protected or well-hidden retaliatory force, and that itself is so well-protected or well-hidden as to be invulnerable to any weapons that the other side might possess. Ideally, this weapon would suffer no disadvantage in waiting to strike second and gain no advantage in striking first. At the opposite extreme is a weapon that is itself so vulnerable that it could not survive to strike second, or a weapon so specialized for finding and destroying the enemy’s retaliatory forces before they are launched that it would lose most of its usefulness if it were held until the other side has already started. These “strike-first” weapons not only give their possessor a powerful incentive to strike first, and an incentive to jump the gun in the event of ambiguous warning rather than to wait and make absolutely sure; they are a tacit declaration to the enemy that one expects to strike first. They consequently invite the enemy to strike a little before that and to act with haste in the event he thinks that we think it’s time to act quickly.

DEFINE: Schelling introduces the strike-back vs. strike-first weapon distinction as a conceptual framework for nuclear deterrence. XREF: This is the classic deterrence theory logic that also applies to cybersecurity, market strategy, and competitive posturing in other domains.

In other ways an inspection scheme on the scale required for protection against surprise attack might yield excessive information about the disposition of the other's forces and make them more vulnerable. … Imagine the state of tension that could occur if either side's strategic-force personnel began to suffer a severe epidemic that threatened to immobilize them temporarily before the eyes of the other side's inspection. Much better—if we and they are occasionally to land in a very unalert position for reasons that are impossible to prevent—that neither of us should be in a position to know too much about the other's occasional disabilities.

XREF: Relates to Schelling's deterrence and mutual vulnerability ideas, and to more general thinking on the strategic value of ignorance and opacity.

Finally, while there may be arrangements that have a high probability of providing warning of the enemy's preparation for an attack, the value of the system depends on what we can do if we do get warning. We can send off our own anticipatory strike, hoping to get in first; but this is an unattractive course if the warning is ambiguous. A false alarm then leads to war. And a true one precludes any last-minute deterrence.

SEED: The false-alarm-to-war dilemma is a strong essay angle about the hidden costs of intelligence capacity.

The solution is in two parts. First, one must try to design a super-alert response that has good endurance and little fatigue, recognizing that this means compromising its peak effectiveness. Second, and most pertinent to the present subject, one may have to engage in a kind of crash disarmament negotiation with the enemy during the period that one has in fact taken measures to insure his own invulnerability of retaliation. … It would mean negotiating not just under the ordinary pressure of knowing that sneak attack is a long-term danger, but doing it with clear notice that if measures to make successful first-strike impossible have not been devised, agreed upon, and taken by a quick deadline, war by mutual consent has become inevitable.

DEFINE: Describes Schelling's concept of crash disarmament negotiation, where mutual security requires bargaining against a deadline before first-strike capability forces war.

There are two quite distinct criteria for judging the efficacy of an inspection system, or for designing the system itself. One is how well the system gets at the truth in spite of efforts to conceal it; the other is how well it helps one to reveal the truth convincingly when it is in his interest to do so. The difference is like that between a scheme for discovering the guilty and a scheme for permitting the innocent to establish innocence.

DEFINE: Distinguishes two distinct purposes of inspection systems: uncovering hidden truth versus presenting truth convincingly, using the guilt-versus-innocence framing.

But, in most of the cases that one can imagine, it is insufficient simply to assert that one is not engaging in a strategic strike or that one is not in a menacing posture. There has to be some way of authenticating certain facts, the facts presumably involving the disposition of forces. We would have to prove not only that we were not intending to exploit our position, but that our actual position was one that could not be exploited to doublecross the enemy if he should take us at our word and restrain his own forces.

DEFINE: Clarifies the concept of authentication — that trustworthy posture in strategic contexts demands verifiable proof of one's actual position, not just assertion of intent.

The requirements for a successful outcome can analytically be divided into two parts. First there has to be discovered some “solution”—some pattern of action that reverses the trend toward mutual attack, and that constitutes a dynamically stable withdrawal to a less menacing alert status, one that yields neither side a dangerous advantage in the process, and that is within the physical capabilities of the forces concerned. The second requirement is that compliance somehow be observable, verifiable, and provable. We cannot carry out our part of the bargain unless we have trustworthy means for monitoring the other side’s compliance, and the same is true for them. Conceivably we would have an interest in cheating; but it is overwhelmingly more probable that we should wish in these circumstances for a cheat-proof monitoring system that we could submit to, so that if we did comply with our part of the bargain the other side would have no doubt about it. The problem is essentially one of contract enforcement.

XREF: This is classic Thomas Schelling game-theory/contract-enforcement thinking on verification and mutual monitoring — connects to trust, prisoner's dilemma, and commitment problems in cooperation more broadly.

For the purpose of being at least somewhat prepared for crises and unforeseen situations, there is a good argument for instituting some flexible stand-by arrangements for communicating with potential enemies and inspecting each other. In particular there is a good argument for overbuilding an inspection system relative to such use as has been agreed on. Having standby capacity to enlarge or intensify the system, or to augment it with additional facilities and inspectors, may have a good deal to do with the usefulness of the system in time of crisis.

XREF: Connects to deterrence theory and arms control literature; overbuilt inspection capacity is a signaling device for crisis readiness.

There is only a hope—no presumption—that even with great ingenuity and the best of diplomacy we and the Russians could find cooperative measures to arrest a trend toward instability.

So we may get stability without cooperation, or we may not find it even with cooperation. Still, some kind of cooperation with the Russians, or mutual restraint, formal or informal, tacit or explicit, may prove to make a significant difference in the stability of the balance of terror; and the stakes of course are very high. So although we cannot be sure that a deliberate policy of collaborating to make each side’s retaliatory forces invulnerable would make any difference, we have to consider that it might and to ask ourselves whether in fact we should want a perfectly stable balance of deterrence if we had the option before us.

QUESTION: Author questions whether a perfectly stable balance of deterrence is even desirable. This is worth interrogating further — what would be lost with perfect stability?

It can be argued that except under the most extreme provocation we would shrink from any retaliatory strike that had no significant chance of eliminating or softening the Russian return strike. According to this argument, a pair of invulnerable SAC’s is a pair of neutralized SAC’s; and while that might be the best kind in a completely bi-polar world, it is a luxury that we could not afford in the existing world—a world in which there is a large “third area” in which we wish to deter Russian aggression by a threat more credible than that of mutual suicide.

XREF: This anticipates the concept of mutual assured destruction (MAD) and how a second-strike capability undermines credibility of retaliation — echoes of deterrence theory debates.

The threat of massive retaliation, if “massive” is interpreted to mean unlimited retaliation, does indeed lose credibility with the loss of our hope that a skillfully conducted all-out strike might succeed in precluding counter-retaliation. But if we were ever to consider limited or graduated reprisals as a means of putting pressure on the Russians to desist from actions intolerable to us, or to consider extending a limited local war inside Russian borders in a way that maintained the presence of local military action but was really intended to work through the sanction of civilian pain and the threat of more, this kind of retaliatory action, and the threat of it, might enjoy increased credibility with a reduction in the vulnerability of both sides’ strategic forces. It does, paradoxically, for the same reason that all kinds of limited war might become less inhibited as the possibility of all-out surprise attack became unavailable. The risk involved in a bit of less-than-massive retaliation should be less than it is now because the fear of an all-out strike in return should be a good deal less.

While the threat of all-out punishment may lose credibility with the achievement of invulnerability by both sides’ retaliatory forces, the threat of limited retaliation may well gain it. Whatever the net effect, we cannot deprecate a world of invulnerable SAC’s simply by reference to the need for third-area deterrence; it has to be demonstrated that one particular deterrent threat (the massive one) is more potent than the other (limited) one.

DEFINE: Distinguishes between massive and limited retaliation as distinct deterrence mechanisms whose credibility shifts with force invulnerability. Builds on: "Limited retaliation gains credibility as all-out strike risks diminish."

With the development of small-size, small-yield nuclear weapons suitable for local use by ground troops with modest equipment, and with the development of nuclear depth charges and nuclear rockets for air-to-air combat, the technical characteristics of nuclear weapons have ceased to provide much basis, if any, for treating nuclear weapons as peculiarly different from other weapons in the conduct of limited war. It has, of course, been argued that there are political disadvantages in our using nuclear weapons in limited war, particularly in our using them first. Even those who consider a nuclear fireball as moral as napalm for burning a man to death must recognize as a political fact a worldwide revulsion against nuclear weapons.

XREF: Connects to classic Cold War limited-war deterrence theory and debates among strategists like Kahn, Schelling, and Brodie about the 'nuclear taboo' and firebreak.

An economist can argue—with the same persuasiveness as those who argue that “pinpoint”-delivered small-yield weapons are just another form of artillery—that equipment and manpower are fungible resources in a military campaign, that air intervention is not “really” different from ground intervention, that military intellect is as important as leg muscle for troops that lack leadership and planning skill. The controversy about redefinition of service functions in the light of modern weapons, and about the usefulness of defining military-service functions in terms of the means of locomotion, suggests that an air-ground distinction or a naval-ground distinction rests on nothing but tradition. But the point of all this is that, in limiting war, tradition matters.

XREF: Connects to debates about defense procurement and joint service doctrine in contemporary military thinking and organizational theory.

In fact, what we are dealing with in the analysis of limited war is tradition. We are dealing with precedent, convention, and the force of suggestion. We are dealing with the theory of unwritten law—with conventions whose sanction in the aggregate is the need for mutual forbearance to avoid mutual destruction, and whose sanction in each individual case is the risk that to breach a rule may collapse it and that to collapse it may lead to a jointly less favorable limit or to none at all, and may further weaken the yet unbroken rules by providing evidence that their “authority” cannot be taken for granted.

What makes atomic weapons different is a powerful tradition that they are different. The reason—in answer to the usual rhetorical question—why we do not ban bows and arrows on the grounds that they too, like nuclear weapons, kill and maim people, is that there is a tradition for the use of bows and arrows, a jointly recognized expectation that they will be used if it is expedient to use them. There is no such tradition for the use of atomic weapons. There is instead a tradition for their nonuse—a jointly recognized expectation that they may not be used in spite of declarations of readiness to use them, even in spite of tactical advantages in their use.

DEFINE: Introduces the concept of a 'tradition of nonuse' as the operative distinction for why atomic weapons are treated differently from conventional arms.

The need for qualitatively distinguishable limits that enjoy some kind of uniqueness is especially enhanced by the fact that limits are generally found by a process of tacit maneuver and negotiation. They are jockeyed for, rather than negotiated explicitly. But if the two sides must strike a "bargain" without explicit communication, the particular limit has to have some quality that distinguishes it from the continuum of possible alternatives; otherwise there is little basis for the confidence of each side that the other acknowledges the same limit. Even a parallel of latitude, or an international date line, or the north pole, may have this quality when no other natural, plausible, "obvious" point or line is available for expectations to converge on.

XREF: Echoes Schelling's focal-point theory of tacit coordination — limits converge where some salient, qualitatively distinct anchor exists.

My argument is that there are particular limits—simple, discrete, qualitative, "obvious" limits—that are conducive to a concerted choice; those who specify other kinds of limits, I predict, can find few partners or none at all whose limits coincide with theirs.

If on the contrary we wish to enhance the tacit understanding we have with our enemies that nuclears are a class apart and subject to certain reservations, agreement on nuclear test suspension (or even just extensive discussion of such an agreement) will probably contribute to the purpose.

XREF: Connects to theories of nuclear deterrence and tacit norms in arms control literature.

A second conclusion is that the principal inhibition on the use of atomic weapons in limited war may disappear with their first use. It is difficult to imagine that the tacit agreement that nuclear weapons are different would be as powerfully present on the occasion of the next limited war after they had already been used in one. We can probably not, therefore, ignore the distinction and use nuclears in a particular war where their use might be of advantage to us and subsequently rely on the distinction in the hope that we and the enemy might both abstain. One potential limitation of war will be substantially discredited for all time if we shatter the tradition and create a contrary precedent.

XREF: Connects to the concept of taboo norms and precedent-setting behavior; the idea that a single breach destroys a norm echoes sociological research on norm cascades. Builds on: "Nuclear test discussions reinforce tacit norms distinguishing nuclear weapons."

If we are to avoid adding a whole new dimension to our payoff matrix, in the form of discount rates, we must suppose that the game is terminated soon enough so that nothing like the interest rate enters the picture. We do not want to have to consider the time at which agreement is reached, in addition to the agreement itself. This is more than a matter of convenience; the game ceases to be "moveless," except in very special cases, unless we make this stipulation. For, if the players' time preferences take any shape except that of a continuously uniform discount rate, the game itself changes with the passage of time and a player can, in effect, change the game itself by failing to reach agreement.

Next, what do we mean by "agreement"? For simplicity, suppose that each player keeps (or may keep) his current "official" offer recorded in some manner that will be visible to the referee when the bell rings. Perhaps he keeps it written on a blackboard that the other player can see; perhaps he keeps it in a sealed envelope that is surrendered to the referee when the bell rings; perhaps he keeps it punched into a private keyboard that records his current offer in the referee's room. When the bell rings, the blackboard is photographed, the envelope surrendered, or the keyboard locked, so that the referee needs only to inspect the two "current" offers as they exist at midnight to see whether they are compatible or not.

From this follows the significant feature. The last offer that it is mechanically and legally possible for a player to make is one that he necessarily makes without knowing what the other player's final offer is going to be; and the last offer that a player can make is one that the other player cannot possibly respond to in the course of the game. Prior to that penultimate moment, no offer has any finality; and at that last moment players either change or do not change their current offers, and whatever they do is done in complete ignorance of what each other is doing, and is final.

At this point it looks as though the cooperative feature of the game is irrelevant. The players really need not show up until 11:59; in fact they do not need to show up at all. The preplay communication and ability to reach binding agreements, which were intended to characterize the game, prove to be irrelevant; the cooperative game as a distinct game from the tacit game does not exist.

XREF: Connects to game theory concepts about when communication and commitment actually matter versus when they're superfluous to the equilibrium.

The tacit game now has a perverse incentive structure. There is no rational reason for either player to demand less than the whole of the available reward; each knows this and knows that the other knows it. There is no incentive to reduce one's claim because any residual dispute costs the player no more than he would lose if he reduced his claim to eliminate the dispute. The single equilibrium point yields zero for both players.

If the rules require that full and exhaustive agreement be reached, then in the tacit game the players are dependent on their ability not only to divide the total value of the objects in coordinated fashion but to sort out the 100 individual objects into two piles in identical fashion. If, then, one of the players has demanded specific items worth 80 percent of the total and the other player has refused, the former has an advantage in the tacit game. The only extant proposal for dividing the 100 objects is the one player's specification of 80 that would satisfy him; the chances of their concerting identically on any other division of the 100 objects, equal or unequal between them, may be so small that they are forced for the sake of agreement into accepting the only extant proposal in spite of its bias.

DEFINE: Illustrates how focal points (Schelling's coordination mechanism) force agreement onto extant proposals even when biased.

It should be emphasized that the fruitful alternative to symmetry is not the assumption of asymmetry, but just non-symmetry, admitting both symmetry and asymmetry as possibilities without being committed to either as a foregone conclusion.

DEFINE: Distinguishes non-symmetry (neutral openness to either possibility) from asymmetry (a committed assumption of imbalance), offering a third conceptual stance.

Suppose we were to analyze the game in which there is $100 at the end of the road for the player who can get there first. This game of skill is not hard to analyze: the money goes to the fastest, barring accidents and random elements. We can predict rational behavior (running) and the outcome (money to the fastest). Ties will occasionally occur; but they will occur at the end of a race and will not be taken for granted at the outset. We need an auxiliary rule to cover ties, but it need not dominate either the game or the analysis.

The perfectly move-symmetrical cooperative game seems a little like that foot race. Bargaining in the one case is as unavailing as leg-work in the other; every player knows in advance that all moves and tactics are foredoomed to neutralization by the symmetrical potentialities available to his opponent. The interesting elements that we might inject in the bargaining game are meaningless if perfect symmetry, and its acceptance as inevitable by both players, are imposed on the game by its definition.

What should we add to the game to enrich it if the assumption of symmetry is dropped? There are many "moves" that are often available, but not necessarily equally available to both players, in actual game situation. "Moves" would include commitments, threats, promises; tampering with the communication system; invocation of penalties on promises, commitments, and threats; conveyance of true information, self-identification; and the injection of contextual detail that may constrain expectations, particularly when communication is incomplete.

DEFINE: Identifies a taxonomy of non-symmetric game moves: commitments, threats, promises, communication tampering, information conveyance, and contextual constraint injection.

To illustrate, suppose in the earlier cooperative game there is a turnstile that permits a player to leave but not to return; his current offer as he goes through the turnstile remains on the books until the bell rings. Now we have a means by which a player can make a "final" offer, a "commitment"; whoever can record an offer favorable to himself and known to the other, and leave the room, has the winning tactic. Of course it may win for either of them; but this may mean that we end up with something like a foot race, and the one closest to the turnstile wins.

DEFINE: Illustrates commitment as the ability to make a final, binding offer that stays on the books, reshaping the strategic picture of bargaining.

It remains true that one wins when he gets to the turnstile first only through the other's cooperation, only by constraining the other player's choice of strategy. He does not win legally or physically by going through the turnstile; he wins strategically. He makes the other player choose in his favor. It is a tactic in a game of strategy, even though the use of it may depend on skill or locational advantage.

The cooperative game is rich and meaningful when "moves" are admitted; and much of the significance of the moves will vanish if complete symmetry in their availability to the players is stamped into the definition of the game. It is the moves that are interesting, not the game without moves; and it is the potential asymmetry of the moves that makes them most interesting.

DEFINE: Distinguishes the game's formal definition from the moves available to players, where the interesting strategic content lives. XREF: Connects to game theory (von Neumann) and mechanism design, where rules and player asymmetries drive strategic behavior.

What I am going to argue is that, though symmetry is consistent with the rationality of the players, it cannot be demonstrated that asymmetry is inconsistent with their rationality, while the inclusion of symmetry in the definition of rationality begs the question.

We must be careful not to make symmetry part of the definition of rationality; to do so would destroy the empirical relevance of the theory and simply make symmetry an independent axiom. We must have a plausible definition of rationality that does not mention symmetry and show that asymmetry in the bargaining expectations would be inconsistent with that definition.

DEFINE: Clarifies that rationality must be defined without reference to symmetry, so that symmetry can be derived as a consequence rather than assumed.

Nor can we trip them up if we ask them how they arrived at their expectations. Any grounds that are consistent would do, since any grounds that each expects the other confidently to adopt are grounds that he cannot rationally eschew. Consistent stories are all they need; and if they say that a sign on the blackboard said A-$80, B-$20, or that they saw in a bulletin that two other players, named Aʹ and Bʹ, split $80-$20, and that they confidently perceived that this was clear indication to both of them of what to expect—that this was the only "expectable" outcome—we cannot catch them in error and prove them irrational. They may be irrational; but the evidence will not show it.

XREF: Connects to coordination games in game theory and theories of common knowledge, where self-fulfilling expectations create equilibria immune to rational critique.

If two players jointly expect a priori the same outcome, and confidently recognize it as their common expectation, they must have the intellectual power to pick a particular point in common. If the whole $100 can be divided to the nearest penny, there are 9,999 relevant divisions to consider, one of which would have to be picked simultaneously but separately by both players as their expectations of the outcome. But how can two people concert their selections of one item out of 9,999, in the sense that their expectations focus or converge on it, except with odds of 9,999 to 1 against them? The answer must be that they utilize some trick, or clue, or coordinating device that presents itself to them. They must, consciously or unconsciously, use a selection procedure that leads to unique results. There must be something about the point they pick that distinguishes it—if not in their conscious reasoning, at least in our conscious analysis—from the continuum of all possible alternatives.

DEFINE: Introduces the concept of a coordinating device or 'focal point' that lets players select among many alternatives without communication.

In a game that has absolutely no details but its mathematical structure, in which no inadvertent contextual matter can make itself appreciated by a player as something that the other can appreciate too, there may be nothing to work on but a continuum of numbers. And all the numbers can be sorted according to whether they correspond to symmetrical or asymmetrical divisions. If all numbers but one represent an asymmetrical split, then sheer mathematical symmetry is a sufficient rule and a supremely helpful one in concerting on a common choice.

DEFINE: Explains Schelling's concept of using mathematical symmetry as a focal point when games have no contextual clues.

Thus a normative theory of games, a theory of strategy, depending on intellectual coordination, has a component that is inherently empirical; it depends on how people can coordinate their expectations. It depends therefore on skill and on context. The rational player must address himself to the empirical question of how, in the particular context of his own game, two rational players might achieve tacit coordination of choices, if he is to find in the game a basis for sharing an a priori expectation of the outcome with his partner.

And, if the Nash point appeals to us powerfully in the bargaining game, it must do so because we are confident that it appeals equally to our partner who in turn we believe to be aware that our views coincide. It must therefore appeal to us in the pure coordination game as a unique point that the partner will consider to be obviously obvious.

DEFINE: Explains how a Nash point becomes salient in coordination games — through recursive mutual awareness of its obviousness.

What does this prove or suggest? I am not arguing for the Nash point. I am arguing rather that the appeal of the Nash point to a game theorist (as introspective game player) may be the reverse of the sequence I have just run through. It may be the focal quality of the Nash-point in the pure coordination game—the unequivocal usefulness of a uniquely defined symmetry concept, when no nonmathematical impurities are available to help—that makes it a controlling influence in the tacit and terribly cooperative boundary-line variant of the game; that in turn makes it a reliable guide in the less demanding tacit bounded-area variant of the game; and that in turn takes the heart out of any player in the explicit bargaining game who might hope that expectations could focus anywhere else.

In other words, by postulating the need for coordination of expectations, we seem to have a theoretical basis for something like the Nash axioms. What a theory like Nash’s needs is the premise that a solution exists; it is the observable phenomenon of tacit coordination that provides empirical evidence that (sometimes) rational expectations can be tacitly focused on a unique (and perhaps efficient) outcome, and that leads one to suppose that the same may be possible in a game that provides nothing but mathematical properties to work on. The Nash theory is vindication of this supposition—complete vindication if it dominates all competing mathematical solutions in terms of mathematical esthetics. The resulting focal point is limited to the universe of mathematics, however, which should not be equated with the universe of game theory.

DEFINE: Clarifies how focal points (tacit coordination) ground the Nash equilibrium concept, connecting experimental observation to abstract game theory.

In a game as abstract as the matrix in Fig. 28, there is little to guide them but the numbers; and between the alternative rules of picking the lesser pair or the greater, the latter probably has more plausibility.

What the players need is some signal to coordinate strategies; if they cannot find it in the mathematical configuration of the payoffs, they can look for it anywhere else. And strategies may occur in such fashion, or with such labels or connotations, as to provide a potential basis for ordering them or sorting them that rational players find useful.

The “clues” under discussion in this paper would then appear to be so much free communication to be taken advantage of; and it is an empirical question what free communication a rational player should be able to find and take for granted. Just as esthetic or syntactic constraints on a language help to eliminate garbles in a badly transmitted message, esthetic or dramaturgical constraints, casuistic or geometric constraints, can help to eliminate ambiguousness in a situation where tacit concerted choice is required.

One argument might be that, in the absence of any way of knowing whether to aim for (i, I) or (ii, II), one should consider what insurance he can fall back on. The row chooser gets nothing if he wrongly chooses the upper row, he gets 5 if he wrongly chooses the lower row, “wrong” meaning that he fails to rendezvous with his partner for 10. He might then choose the lower row arguing that he does so because he will at least get 5 if he does not get 10, and his chances of getting 10 are no worse with this choice. Perhaps this is all that “rationality” requires of him; but it might be more perceptive to reason as follows. “Comparing just (i, I) and (ii, II) my partner and I have no way of concerting our choices. There must be some way, however, so let’s look for it. The only other place to look is in the cells (ii, I) and (i, II). Do they give us the hint we need to concert on 10 apiece? Yes, they do; they seem to “point toward” (ii, II). They provide either a reason or an excuse for believing or pretending that (ii, II) is better than (i, I); since we need an excuse, if not a reason, for pretending, if not believing, that one of the equilibrium pairs is better, or more distinguished, or more prominent, or more eligible, than the other, and since I find no competing rule or instruction to follow or clue to pursue, we may as well agree to use this rule to reach a meeting of minds.”

XREF: Connects to Schelling's focal point theory and coordination games, where salient features of a matrix guide players toward particular equilibria without communication.

It is probably in the realm of more-than-two-person games that coordination theory is most relevant of all, games involving the formulation of coalitions. Study of the signals and communication channels in coalition formation appears to be a fruitful meeting ground for game theory and sociology.

SEED: A potential article angle: how coalition-signaling dynamics in multiplayer games can illuminate organizational politics and alliance-building inside teams. Blends game theory with real-world workplace dynamics.

Also, as far as the power to deter is concerned, there is no such thing as "too large" a threat; if it is large enough to succeed, it is not carried out anyway. A threat is only "too large" if its very size interferes with its credibility. Atomic destruction for small misdemeanors, like expensive incarceration for overtime parking, would be superfluous but not exorbitant unless the threatened person considered it too awful to be real and ignored it.

XREF: Relates to deterrence theory in game theory and Cold War mutual assured destruction doctrine; the credibility paradox resembles Schelling's work on threats.

Mutual defense treaties among strong and weak nations might best be viewed in this light, that is, not as undertaken to reassure the small nations nor in exchange for a quid pro quo, but rather as a device for surrendering an embarrassing freedom of choice.

Some trucks carry signs that say, "Alarm and lock system not subject to the driver's control." The time lock on bank vaults serves much the same purpose, as does the mandatory secret ballot in elections. So does starting an invasion with a small advance force that, though too small and premature to win the objective, attaches too much "face" to the enterprise to permit withdrawal: the larger force can then be readied without fear of inviting a purely deterrent threat. At many universities the faculty is protected by a rule that denies instructors the power to change a course grade once it has been recorded.

DEFINE: This example cluster illustrates the concept of precommitment devices across domains — locks, secret ballots, military advance forces, and grading rules all make withdrawal or reversal costly or impossible.

During 1950, the Economic Cooperation Administration declared its intention to reward Marshall Plan countries that followed especially sound policies, and to penalize those that did not, through the device of larger or smaller aid allotments. But since the base figures had not been determined, and since their determination would ultimately involve judgment rather than formulas, there would be no way afterwards to see whether in fact the additions and subtractions were made, and the plan suffered from implausibility.

The threat may seem to be a promise if the pledge behind it is only one's reputation with his adversary; but it is not a promise from which the second party can unilaterally release the threatener, since he cannot convincingly dissociate his own future estimate of the threatener from the latter's performance.

DEFINE: Distinguishes a reputation-backed threat from a genuine promise, clarifying that the receiver cannot release the threatener because both are entangled in the same reputational evaluation.

Inability to assume an enforceable promise, like inability to perform the activity demanded, may protect one from an extortionate threat. The mandatory secret ballot is a nuisance to the voter who would like to sell his vote, but protection to the one who would fear coercion.

XREF: Connects to democratic theory and discussions of voting integrity, as in the Federalist Papers and contemporary debates about voter intimidation.

Also, the pure games cannot admit “side payments.” If one of the partners in a pure common-interest game threatens to sabotage the effect unless he is paid—assuming that the communication and enforcement structure of the game makes this possible—a conflict of interest is introduced; in effect, the point denoting the payment of a bribe would appear to the upper left or lower right of another point or points on the upward-sloping line, producing the configuration of a mixed game. And if one of the players in a pure-conflict game can threaten damage or offer compensation to induce his opponent to yield in this game, there is scope for bargaining; there is no longer a relation of pure conflict, and the points denoting the threatened damage or promised compensation would lie off the downward-sloping line. In other words, all pertinent potential outcomes must be allowed for.

DEFINE: Explains how introducing side payments (bribes or compensation) shifts a pure common-interest or pure-conflict game into a mixed game configuration, redefining the game's structural category.

It deserves to be emphasized that nonzero-sum games can as properly be classed under theory of partnership as under theory of conflict; and for providing insight into problems like that of limiting war, there is merit in using words that bring out the common interest of the adversaries and the “bargaining process” involved in the military maneuvers themselves. As will be seen in Chapter 9, even the problem of surprise attack is logically equivalent to a problem in partnership discipline.

XREF: Conceives of war and conflict through game-theoretic partnership framing, connecting to Schelling's larger bargaining and limited-war theory.

And this suggests an alternative orientation of those experiments in which subjects are instructed to make guesses, throughout a long random sequence of red or green lights, whether red or green will come up next. Subjects apparently persist in guessing on the basis of some pattern they think they perceive, an “irrational” mode of behavior given their knowledge that the sequence is generated by a random device. But, as Herbert Simon points out, “Man is not only a learning animal; he is a pattern-finding and concept-forming animal” (“Theories of Decision-Making in Economics and Behavioral Science,” American Economic Review, 44:272). Why not, then, add to the experiment a cooperating pattern-maker, who generates the signals subject to various constraints and random interferences, and let the persistent pattern-seeking subject use his skill in finding the pattern planted by a cooperative partner rather than spend it futilely on random series?

XREF: Connects to the classic gambler's fallacy and research on apophenia and pattern-seeking cognition (e.g., Kahneman's work on the narrative fallacy). FICTION: A cooperative pattern-maker who hides a pattern inside randomness could be a premise for a story about hidden design vs. apparent chaos.

A partial randomized strategy may also be used to reduce an area of conflict. Suppose two people, seated at North and East sides of a card table, are to move to another card table adjacent that is identically oriented, must choose without communication what seats they will take at the other table, and will win prizes of $1 apiece if they pick adjacent seats. This is an easy coordination problem; but let us subvert the incentives, by giving an additional $2 premium to the player who is on the other's right in the event they succeed in sitting next to each other. This game has no equilibrium point; interests do not converge; there is no seating arrangement that would not give one an incentive to move. (Each may wish that he could promise to sit on the other's left, but cannot.) A random strategy yields each player a minimax value of $1. But, if each decides where he would sit in the pure common-interest game, then flips a coin to see whether he does sit there or sits opposite, the players guarantee that they neither choose the same seat nor sit opposite each other and share equal chances of winning the premium. This is an equilibrium pair of (mixed) strategies, worth an expected value of $2 apiece.

XREF: Connects to game theory concepts like Schelling points, mixed strategies, and coordination games — echoes common-interest vs. mixed-interest coordination problems.

In ordinary language, "threat" is often used also for the case in which one merely points out to an adversary, or reminds him, that one would take action painful to the adversary if the latter fails to comply, it being clear that one would have incentive to do so. To "threaten" to call the police on a trespasser is of this sort, the threat to shoot him is not. But it seems better to use a different word for these cases—I suggest "warning" rather than "threat"—because the "threat" either is superfluous, and does not constitute a move, or it conveys true information and relates to situations with an information structure and communication structure worth keeping distinct. In this latter case it is a mutually beneficial move, precluding a jointly undesired outcome by improving the second party's understanding. The main point of analytical similarity, between this "warning" case and that of the "threat," is in the possible difficulty of conveying true information credibly, of conveying evidence for the assertion that one would have, ex post, incentive for doing as one warns he will. As a matter of fact, if a threat is of such nature (as it often is) that the act of commitment is not contained in the act of communicating it—if the commitment precedes the conveyance of the threat, with evidence for believing it, to the threatened party—the first act in the process of threatening changes the "true" incentive structure, and the second is, in effect, a "warning."

DEFINE: Distinguishes 'threat' from 'warning' as analytically distinct communication acts based on whether a commitment precedes the conveyance.

It has been argued that an important function of the racketeer is sometimes to help enforce agreements that are beyond the law. Price-cutting in the Chicago garment trade was punishable by explosion—the fee for the explosion being paid by the price-fixing organization—according to R. L. Duffus, “The Function of the Racketeer,” New Republic (March 27, 1929), pp. 166–68.

What might be called the “legal status” of communication is nicely developed by Goffman: “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.”

DEFINE: Goffman defines hinted communication as a distinct unofficial mode of exchange governed by a tacit rule of deniability.

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.

DEFINE: Introduces the 'reaction function' concept as a generalization of commitment, threat, and promise tactics in game-theoretic strategy.

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, so that we can count on it and make some prior choice that is to his disadvantage.

XREF: Connects to game theory and commitment signalling literature, and to the paradox of credible commitments in negotiation.

That one’s position can be painfully weakened by new legal powers is poignantly suggested by one of the arguments raised against legalizing euthanasia, granting hopeless incurables the right to authorize their own removal: “What . . . would be the effect on old people with incurable infirmities who are already suspicious that those around them want to get rid of them?”

QUESTION: The euthanasia argument reveals a deeper concern: legal rights intended as empowerment can invert into pressure on the vulnerable. Worth exploring how liberty-granting laws create new coercive dynamics.

A good laboratory example of the communication-perception part of game strategy is the experiment reported by M. M. Flood, who presented his players with a 2 × 2 nonzero-sum matrix for 100 consecutive tacit plays. The special property of the matrix is that the players can win only by cooperating on a particular cell on each play, but to distribute the winnings for the 100-play sequence they must cooperate 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.

XREF: This is an early formulation of coordination in repeated games, prefiguring focal points and tacit bargaining that Schelling elaborates elsewhere.

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.

QUESTION: What kinds of hints are most successful in practice? This points to possible experimental studies on hint-taking and social perception.

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 Deutsch is pertinent. He let pairs of players play nonzero-sum games (in matrix form) tacitly for a sequence of two plays, the game providing both a “cooperative” and an “uncooperative” choice. Those who played uncooperatively against a cooperative partner had an opportunity, on the second play, to respond to the implicit offer of cooperation. But, “when their expectation of the other person’s choice was not confirmed, they tended to interpret his choice as being a function of indifference 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 regarding the intentions of the other person.”

One can, instead, interpret mixed strategies in zero-sum games as a means of introducing continuity of strategies into a discrete-strategy game that has no pure-strategy saddle point, thereby converting it into a game that does have a saddle point. In this interpretation the role of mixed strategies in zero-sum games is not so different from their role in the nonzero-sum games. One can flip a coin to keep an opponent from guessing with confidence whether it will come up heads or tails; or one may flip a coin to “average” heads and tails, to create (in an expected-value sense) a strategy halfway between heads and tails. Both interpretations are useful. If the second is somewhat more sophisticated, the first may better catch the spirit of the problem as it presents itself to a game player. And the first reminds us that the problem, even with randomization, is still to prevent the opponent’s anticipation of our actual strategy choice, and that the machinery of choice, the procedures for recording and communicating a choice, and any advance preparations required by the outcome of the random process, must remain inaccessible to his intelligence system.

DEFINE: Clarifies two interpretations of mixed strategies: averaging creates a halfway strategy, and randomization prevents opponent anticipation.

If in order to prove that one is committed to a threat, or that one is in fact capable of fulfilling the threat, one must display evidence of the commitment or the capability to the other party, the evidence may be of a kind that necessarily yields information helpful to the second party in combatting the threat. To prove to an enemy that one has a potent weapon that can overcome his defenses we might have to demonstrate the weapon or some aspect of it, or provide technical knowledge to prove the weapon feasible; to do so may aid him greatly in preparing a defense against it.

XREF: This is the classic deterrence credibility paradox — the costly signaling problem, related to Schelling's work on commitment but with the twist that signaling can backfire by leaking capability information.

The tactic may be the less risky, the more automatic the mechanism is; the more automatic it is, the less incentive the enemy has to test my intentions in a war of nerves, prolonging the period of risk.

In the author’s opinion the dispatch of United States troops to Lebanon in 1958 was not only both risky and successful but successful precisely because of the risk—a risk that the Communists could lessen or aggravate according to their response.

“Prisoner’s dilemma” refers, in game theory, to a configuration of payoffs that gives both players dominant incentives—in the absence of an enforceable agreement to the contrary—to choose strategies that together yield both players a less desirable outcome than if both had made opposite choices. The name derives from the problem of two prisoners, separately interrogated, who may confess to a moderate crime in common or accuse each other of a heavy crime, an accuser going free unless himself accused, the accused one or ones receiving heavy sentences.

If R can, at his own expense, improve his “enemy’s” alert system, or if he can blunt his own surprise capacity in a visible way, to hold Qr below the limit, he should do so. The principle is the same as that of two partners, somewhat distrustful, who keep two separate private padlocks on the partnership vault. If one could not afford a padlock, the other should provide it to him at his own expense; only then can they do business together.

XREF: Echoes game theory and assurance-game logic; the padlock metaphor parallels Schelling's cooperative security ideas.

There is an important asymmetry in the problem as formulated here. We have allowed for the possibility that one may shoot when he shouldn’t and the other knows it—the “nervousness” case—but not for the possibility that one may not shoot when he ought to, and the other knows it. (There may be some chance that the burglar has wet ammunition or forgot to load his gun, and I may know that there is such a chance, he may know that I know it, and so forth.) This possibility would apparently be stabilizing, tending to reduce the likelihood of a decision to attack as well as the exogenous likelihood of inadvertent or irrational attack.

SEED: This asymmetry between nervousness (shooting when one shouldn't) and dud ammunition (no shot when one should) is a rich game-theoretic angle for explaining when deterrence works vs. fails.

Economists may find the situation reminiscent of two producers who both allocate their limited productive resources between two commodities. One commodity, “security against false alarm,” involves external economies; the other, “security against surprise,” involves external diseconomies.

XREF: Plainly connects cognitive attention allocation to microeconomic production theory and externalities.

Arthur Lee Burns, of the Australian National University, has discussed some interesting problems of a three-or-more person world. The deliberate provocation of war between two parties, by a mischievous third party, is a possibility when an overt act of ambiguous authorship can be introduced into the reciprocal-suspicion model; and the analysis takes on additional richness when one considers warning systems that, for technical reasons or by reason of joint custody, permit one or both of the central players to witness what is coming in on the other’s radar screen.

SEED: This is a ripe premise for a thriller or espionage story — a mischievous actor engineering a false-flag provocation to spark conflict between two powers.

A military historian, commenting on the alleged “historical truth” that there has never yet been a weapon against which man has been unable to devise a counterweapon or a defense, reminds us that “after five centuries of the use of hand arms with fire-propelled missiles . . . no adequate answer has yet been found for the bullet” (Bernard Brodie, The Absolute Weapon [New York, 1946], pp. 30–31).

QUESTION: Does this still hold given modern body armor and active protection systems? The claim dates to 1946 and may be dated.

This assumes that he fires his missiles all together or that, if he fires successive salvos, he has no means of reconnaissance that lets him know, on successive salvos, which particular missiles have already destroyed their targets.

DEFINE: Clarifies an implicit assumption in nuclear targeting/war-gaming models: that salvos are either simultaneous or informed by damage assessment.

This paper being about principles, not about submarines, I can perhaps be excused for pretending here that undetectability on short notice in the open sea is equivalent to invulnerability.

It should be emphasized that bargaining-game solutions that (like the Nash and Harsanyi solutions) depend on a clearly recognized zero point—that is, on an unambiguous outcome that reigns in the absence of overt agreement—cannot necessarily be applied to a cooperative game that is based on a matrix of choices. A matrix (unless perhaps all payoffs are zero except in the diagonal) does not have a zero point defined by the rules. There is consequently no “normal form” consisting of a convex region and associated zero point unless there is available a fully adequate theory that “solves” the tacit game (and does so in a manner that the players can take for granted). One may, following Luce and Raiffa (for example, page 137) take the players’ “security levels” (maximin values) as the zero point; but this is either arbitrary or based on the hypothesis that, left to themselves, the players could succeed in doing no better than this in the tacit game. The latter hypothesis, especially where there are pure-strategy efficient points (as in Braithwaite’s game, and as in the Luce-Raiffa matrix discussed in note 18 below), is a weak hypothesis that can be empirically refuted; it assumes that rational players are incapable of correlating strategies without communicating, while in fact this is something they often can do even in the face of conflicting preferences.

DEFINE: Clarifies the distinction between bargaining games with a recognized zero point and cooperative games on choice matrices that lack one.

Viewed in this way, the intellectual process of arriving at “rational expectations” in the full-communication bargaining game is virtually identical with the intellectual process of arriving at a coordinated choice in the tacit game. The actual solutions might be different because the game contexts might be different, with different suggestive details; but the 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 like 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.

DEFINE: Clarifies that the process of arriving at rational expectations is structurally identical whether coordination is tacit or through full communication, differing only in context and suggestive details.

With full information about each other’s value systems and a homogeneous set of gains to be divided, there may be an infinity of equivalent solutions, all yielding the same values to the two players, but no difficulty in agreeing on an arbitrary choice among this indifferent set. But tacit bargaining often requires a further degree of coordination, namely, a coordinated choice even among equivalent divisions of the gains. Negotiation over a boundary line in homogeneous territory is thus different from the simultaneous dispatch of troops to take up positions representing claims (as in Question 6 on page 62); such claims may overlap and cause trouble even though the terrain values claimed are consistent.

DEFINE: Distinguishes negotiation over boundaries from simultaneous claim-staking, which can cause overlap conflicts even with consistent valuations.

In this view, the theory of Nash (leading to the maximum-utility-product solution) is a response to the fact that even in the realm of mathematics there are offhand too many types of uniqueness or symmetry to provide an unambiguous rule for selection, hence a need to adduce plausible criteria (axioms) sufficient to yield an unambiguous selection. Braithwaite’s theory can be characterized the same way. The fact that the two solutions conflict implies that mathematicians may not have a sufficiently common mathematical aesthetic to satisfy the first part of the Harsanyi postulate, that is, to coordinate their expectations on the same outcome.

XREF: Connects to bargaining theory and social choice literature, and the broader debate about whether mathematics is discovered (uniquely determined) or constructed (plural).

If, for example, he bids $9.50 for a chance to play the game in Fig. 33 (implying, perhaps, a 90 percent expectation that Column will choose II), $8.65 for the game with 5 replaced by 1 (implying an 85 percent expectation of II), and $9.95 for the game with 5 replaced by 9 (implying a 95 percent expectation of II), and, finally, $5 for the game as in Fig. 32 (implying a random expectation as between I and II), we could conclude that the function, or value to the player, of the upper-right and lower-left payoffs is largely that of coordinating clue. If instead he bids amounts that imply probabilities between I and II that are invariant, or nearly so, with respect to the upper-right and lower-left payoffs, and particularly if he bids the arithmetic mean, the insurance interpretation would be indicated.

DEFINE: Distinguishes the 'coordinating clue' interpretation from the 'insurance' interpretation of payoffs via bid patterns. QUESTION: The bidding method for eliciting which interpretation applies is interesting — could be worth chasing how this could be tested empirically.

We offer a player alternative games like Fig. 33 that differ only in substituting values ranging from 0 to 9 for the 5’s in that matrix, leaving the 10’s and zeros as they are. We then ask him to “value” the games for us—to indicate how much he would pay for the opportunity to play the game with a live partner and real money payoffs. (Alternatively we ask him how much he’d pay for the privilege of playing the different variants in place of the one with 5.) If his response is fairly insensitive to variations in that particular payoff as long as it is positive, and if nevertheless he attaches a high value to the game with some positive payoff and attaches something like a random-strategy expected value for the game with zeros as in Fig. 32, we can conclude that the lower-left and upper-right payoffs are mainly of interest to him as signals.

XREF: This is a variant of revealed preference methodology; the insensitivity-to-payoff tactic connects to later behavioral economics work on separating signaling value from intrinsic value (e.g., gift-giving, status goods). QUESTION: Does this experimental design actually control for loss aversion or probability weight distortions that could also produce payoff insensitivity?

The game in Fig. 35 does have another equilibrium point, consisting of an 80:20 mixed strategy for Column and a 40:60 mixture for Row. It yields them payoffs of 3.6 apiece, and is therefore jointly dominated by the upper-left and lower-right cells.

QUESTION: How often do jointly dominated equilibria arise in real strategic settings, and do rational players reliably avoid them?

The power of similar mutually perceived signals seems to lie behind the concept of “psychological dominance” used by Luce and Raiffa to discuss the appeal in certain games of a jointly inadmissible equilibrium point.

XREF: Connects to game theory concepts (Luce and Raiffa) and coordination problems, possibly resonating with Nash equilibrium or Schelling points discussions.

A conflict-of-interest problem of this type—two cars approaching an asymmetrical narrow place in the road from opposite directions—was included in the questionnaire described in Chapter 3. The results bore out the general principle, but were omitted for brevity from Chapter 3.

XREF: Connects back to results presented in Chapter 3 of this same work, where the general principle for conflict-of-interest scenarios was established.