PRINCETON UNIVERSITY

CENTER OF INTERNATIONAL STUDIES

STAFF

Frederick S. Dunn, Director

Klaus Knorr, Associate Director

Faculty Associates

Gabriel A. Almond
Cyril E. Black
Gordon A. Craig
William Ebenstein
Edgar S. Furniss, Jr.
William W. Lockwood
Wilbert E. Moore

Research Associates

Arthur Lee Burns
Robert J. C. Butow
Miriam Camps
Bernard C. Cohen
Herman Kahn (2nd term)
Morton A. Kaplan
Dankwart A. Rustow
Glenn H. Snyder

Research Assistant

Sidney Verba

Editorial Assistant

Jean MacLachlan

Recent Policy Memoranda

No. 14. *Is the American Defense Effort Enough?* by Klaus Knorr

No. 15. *The European Common Market and Free Trade Area: A Progress Report*,
by Miriam Camps

No. 17. *The First Year of the European Economic Community*, by Miriam Camps

No. 18. *The Free Trade Area Negotiations*, by Miriam Camps

Recent Research Monographs

No. 1. *Deterrence by Denial and Punishment*, by Glenn H. Snyder

No. 2. *Some Problems in the Strategic Analysis of International Politics*, by Morton A. Kaplan

Copies of the above memoranda and monographs can be obtained from the
Center of International Studies, Woodrow Wilson Hall, Princeton, N. J.,
at 50 cents per copy.

[LC]

## Introduction

A general study of the nth-country problem brought to light the question dealt with in this monograph: if some third nation were to try and deceive two nuclear Powers into attacking each other, how strongly might either one's prognosis of the other's response to the attempted deception rationally motivate it to launch pre-emptive attack? "Rational motivation" is all that will here concern us; when "fears," "worry," and similar expressions occur in the following pages, they shall do so merely in order to show the color of various reflections and hypotheses, and shall bear no affective or psychological connotation.

Under our heading of "catalytic war," we consider only the mischief-making of some third party. The phrase is sometimes used instead to describe the possibility -- one more likely to be realized -- of a third Power's inadvertently causing conflict between others. The analysis of such a possibility has little in common with ours, which rather resembles those concerned with surprise attack.

Among the latter, Professor T. C. Schelling's *The Reciprocal Fear of Surprise Attack*¹ analyzes a problem which resembles that of catalytic war nearly enough to invite comparisons and contrasts between his approach and ours. But his ingenious and path-breaking study turns upon features which we shall ignore. It has the character of a systematic, game-theoretical treatment, applicable for the most part to situations generated by the possibilities of mechanical error -- e.g., of accidents with warning- or firing-systems. For our part, we concentrate entirely upon predicaments created by the human power to worry about the dangers consequent upon other people's worrying. However, this monograph owes so much to Schelling's that I ought to indicate why I have not followed his procedures even more closely.

I shall be concerned only with that kind of situation in which the prospect of alternative outcomes would leave both the intended victims of a third Power's trickery with every motive for holding their fire -- with one exception. This is the case in which either one, because of highly suggestive evidence provided by his warning-system, had come so strongly to believe the other to be launching an attack that his former reasons for holding fire had given way to pressing reasons for sending off as effective a mission as he might, chiefly in order to blunt his attacker's capacity to mount further

1. P-1342, The RAND Corporation, Santa Monica, Cal., April 16, 1956.

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strikes, but also as punishment for aggression. Thus, for both sides, "joint non-attack" would be decisively the best outcome, while the receiving of retaliation for having launched a successful counter-strike would be only less disastrous than the suffering of a first strike without reply, since the enemy could then go on striking with impunity. Striking first and "successfully" cannot in that case be a rational option for either of the parties, since each is supposed to be assuming that the other even so would be able to hit back intolerably.

Mr. Daniel Ellsberg has pointed out that I have thus assumed a very stable balance of terror. That is true; therefore I should perhaps suggest a kind of technological situation that might afford the stabilizing motives which I assume to operate.

Let two Powers, Blue and Red, both possess two types of vehicle for deterring thermonuclear attack. One type reacts and travels quickly, is reasonably accurate, and is rather the more vulnerable. The other type reacts and travels more slowly, is less vulnerable, and has two subclasses -- one more, and the other less, accurate than the quick reactors. If, per absurdum, Red decided to attempt a surprise attack, he would do best by using all his quick reactors to take out most of Blue's quick reactors, sending the more accurate of his slower vehicles against the sources or locations or routes of all of Blue's slower vehicles. If Blue got wind of such an attack too late to pre-empt but in time to lose his quick reactors, he in turn should send those against the locations and routes of Red's slower vehicles. The latter -- more likely to be on their way -- would be poorer targets for Blue than Blue's own slower vehicles had previously been for Red. As soon as both Powers fully understood that both were fighting, each would need to consider the advantages and disadvantages of switching more of its slow reactors to missions against cities. Now, the later stages of such a war -- say, from the second hour on -- are much less calculable than the earlier stages, so that even Red would have to envisage a distinct possibility (depending upon Blue's last-minute decisions) of losing all his cities. On the other hand, Blue, if he should delay and lose his quick reactors, would be very likely to lose most of his slow ones as well, and then suffer more counter-city retaliation than would Red if some of Blue's retaliation should have managed to get home against Red cities. Moreover, the intact forces of the third Power (which by definition takes part in any catalytic situation) would reinforce Blue's motives -- in the first instance, against striking at all unless he were sure that Red was doing so; and in the second instance, against losing his slower vehicles (his principal means of deterring both Red and the third Power) to an unopposed follow-up attack by Red.

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The stage of technology presupposed in this example, and the resultant stable balance of terror, would not be altogether implausible for the era of possible catalytic war which we are here imagining. Under certain other technologies, any third-party promotion of catalytic trouble would be too risky. For instance, if each nuclear Power had first-strike capability against the rest of the world together, trouble-making would be suicidal. But it is hard to imagine a weapon-system which would make this possible. If each could disarm any single one of the others by a surprise first strike (e.g., if all quick-reacting weapons were highly vulnerable), each would of course find highly plausible the hypothesis that someone was attacking him and this would aid the catalytic trouble-maker. Yet each Power would know that to use up its forces against one opponent would leave both alike helpless before any other, and there would be a strong inducement to seek allies, thus approximating the world situation to a classical balance of power. Many stages of weapon-technology in this way preclude this problem of catalytic war.

It remains true, however, that the problem will be the more severe, the more quick-reacting weapons are vulnerable to each other, for they make first strikes plausible, and a quick retort the best strategy if one thinks that the other is striking first. Professor Schelling's work, especially where he refers to requirements for warning-systems (pp. 17-20), is most apposite at this point. But our present study is confined more to the question whether the trouble-maker could hope to capitalize upon his victims' having an indefinite and cumulative regress of reciprocal fears of being struck first.

I have therefore designed the scheme of this study so that no accidental war, occasioned by error in the warning-system, is possible -- indeed, I have to suppose that the third Power must go to some trouble in his faking the appearance of attack by one of the others. Therefore I need not consider anything that might properly be called a "probability" (in the strict technical sense of the term) that one side or the other will attack irrationally. But Schelling's models are contrived so as to fit both the case in which there is some finite probability that each side's warning-system may announce an attack though none is being made, and also the case in which, along with such a probability, there is open to each side the technical possibility of responding "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" (p. 17). The latter apparently represents an actual feature of warning-systems -- viz.,

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the more sensitive, the more likely they are to err by announcing a non-existent attack. Schelling has therefore to introduce the conception of a real probability -- the probability that the other fellow will attack when we should not. He devotes the first half of his study to disabusing us of "the intuitive idea that initial probabilities of surprise attack become larger -- through a 'multiplier' effect -- as a result of this compound-ing of each person's fear of what the other fears" (p.2). I agree that the intuitive idea is mistaken (though for quite different reasons), but I also think it a mistake to try to expose the error, as Schelling does, by making a model which postulates some real maximal and minimal probability that one side "will in fact attack when he selects (should select) a strategy of no-attack" (p.5); for as Schelling himself points out, "The outcome of this game—which he assumes to start 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" (p.7).

My analysis of catalytic war, therefore, makes no use of proba-bilities, except in an example of how not to deal with the subject (below, pp. 15-16). A probability may be assigned legitimately to an hypothesis only if it belongs to an array that one believes to have (or is prepared to treat as having) exhausted the logical possibilities. But the essence of any situation -- surprise attack or catalytic war -- which can evoke a regress of reciprocal fear surely consists in either side's being able to go on from any given level of imputation to the next. If Blue dismisses the possibility that Red believes him to be attacking, he can always en-tertain the hypothesis that Red believes him to believe that Red believes him to be attacking -- and so on indefinitely. The logical possibilities are thus not exhaustible,² and therefore no probability in the strict sense can be assigned.

2. Other competitive dilemmas in which the possibilities are not ex-haustible can be invented -- e.g., variations on the Prisoners' Dilemma. It is now perhaps well-understood that the probabilistic resolution of dilemmas differs greatly from their rationalistic resolution. A certain case discussed by Schelling (ibid., pp.17 et seq.) interestingly combines the two (in brief, by supposing that each side, learning of the genuine probability that the other's warning-system will sound the alarm when it should not, responds "rationally" by increasing the sensitivity -- and ipso facto the probability of false alarm -- in its own warning-system); but such a combination depends upon the distinction between machine behavior, which sometimes has probabilistic aspects, and human behavior, which consists of emotional and cogitative responses and is not probabilistic in the same sense.

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Of course, it is possible for the Supreme Commander of each Power to regard the other's behavior as probabilistic. But if he does so, he will himself have to decide upon a ne plus ultra for that risk of attack by the other which he is prepared to tolerate; for he cannot otherwise derive a rule for deciding whether or not himself to counter-attack. In gaming terms, the numbers in the pay-off matrix from which he reads "rational instructions" whether or not to attack are in part dependent upon his own preferences for "gambling," and those in turn depend upon the extent of his prospects as he imagines them, and also, within the limits of ration-ality, upon his decision. Reiterating uncertainties about the matrix are at the heart of the surprise-attack problem, and cannot be occluded into probabilities of irrational behavior. In later dealing with the problem, under the assumption of imperfect information, Schelling recognizes this (see p. 23, the central paragraph, and pp. 24-25, especially the passage in parentheses) and points out further (as I follow him in doing, see p. 16 below) that if each player regards the other's but not his own behavior as probabilistic, a multiplier effect operates. He also shows that where each player of an uncooperative surprise game "projects" the other's behavior there can be no guarantee of a unique solution, though stability at the point of "mutual attack" is a possible candidate (pp. 23-26). It is difficult to see how this approach can be squared with his assumptions of the first part, which he justifies on the grounds that "we have to put at least some data into the problem for expectations and conjectures to work on" (p. 6). My point is that these data should not be probabilities of action or behavior, but physical capabilities plus human recognition of the com-petitive-cooperative relationship between humans.

I have eschewed games analysis of our problem for another reason as well. Destruction of a whole nation may be among the outcomes of a catalytic-war crisis. This, I suggest, introduces an incommensurability which makes game theory inapplicable to our problem (and to several others concerned with modern war). National annihilation, from the national leader's viewpoint, would not be merely the worst outcome we could conceive -- it would be, for him in his official role, the absolute evil, which would include all others (many private persons would similarly regard destruction of the human race as the absolute evil). Now, between a non-negligible possibility -- even a quite improbable one -- of the absolutely evil outcome, and a definite possibility -- even a certainty -- of some finite evil, no trade-off can be determined. Hence some of the criteria for the application of game theory seem to be absent.

There is another, less recondite argument to the same effect. We begin our account of the catalytic situation by supposing at least one of two Powers to be receiving a signal suggestive of the other's attacking. The first Power

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has a choice of only two "strategies" -- to attack likewise, or to hold his fire. If he knew the other were attacking, the pay-offs for such a situation would constrain him to attack. If he knew the other were not attacking, a different set of pay-offs would give him every reason to hold his fire. He may thus be thought of as having (i) a choice of strategies which is common to two different games, plus (ii) evidence which supports the hypothesis that he is playing a particular one of these games. Unfortunately, other things beside evidence bear on that hypothesis -- namely, his guesses about the other's guesses...and so on indefinitely. This means that only if the evidence, considered by itself, is conclusively in favor of the hypothesis will he be able definitely to determine the "value" of one game against that of the other. Therefore he cannot apply game theory.

From another viewpoint, he might be regarded as playing just one quite definite game, in which his opponent has three alternatives: (i) never to strike, (ii) to be about to strike, and (iii) to have struck already. Apart from the oddity of one of the opponent's strategies being in the past tense, some features of such a competitive situation exclude it from orthodox game-theoretical treatment. For, while no measurable probabilities can be assigned to the occurrence of the several outcomes, there is some evidence favorably relevant to the opponent's having chosen his third alternative, and also an indefinitely extensible series of conceptions which, if imputed to him, would support his choosing the second alternative.

Some day, the game theorists may develop their formal analyses to cover such situations. Meanwhile, we must spell them out laboriously in plain English -- the method of this study.

That method has just one advantage -- for a while yet, the decision-makers will have to think out their responses to apparent attack in plain English. Warning-systems still yield some signals which are ambiguous. Ordinary language, though cumbersome, offers less risk than do formal systems of our prejudging and distorting the problem because of restrictions imposed by the axioms of a system. And when we have understood the issue thoroughly by discursive methods, the designing of an appropriate formal system should not be so difficult.

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I

Analysts concerned about the nth-country problem are beginning to talk of possible "catalytic war." They assume that there may be -- among several nuclear Powers -- at least two, each of whom has specially prepared itself to hit back quickly if the other gives decisive evidence of launching a surprise attack, and in any case prefers the certainty of a war in which it manages to attack first, or even simultaneously, to the certainty of one in which it receives the first attack. A third Power

Single:
Level 1
(C for Catalyst) somehow produces the appearance of an imminent surprise attack by Power A on Power B. The latter responds by a real attack against A. This is the simplest method of catalytic war.

A more effective and complicated method would be for C to arrange, for the benefit of both his dupes, apparent surprise attacks by each against the other. Each might then be expected to respond by finishing off the other; whereas B might have survived the simpler treatment with no worse damage than it would have suffered by way of retaliation had it deliberately launched an effective surprise attack against A.

Double:

Level 1

But even the simpler strategy looks diabolically subtle. For though it should fail to deceive at the first level, it might succeed at some remoter altitude of ratiocination. Suppose that B is not itself deceived: by some means it knows that the signals from its early-warning system are not evidences of an attack by A, though without being able to tell who is responsible for them. Nevertheless B may reason, "Power A picks up the signals from my warning-system at least as soon as I do myself; but he has no means of telling how I interpret them. He is quite likely to assume that I will interpret them as signs of an attack by him, justifying retaliation; and therefore try to attack first himself" -- and therefore try to attack first himself.

Single:

Level 2

Now is this the most a deceiver can hope to play on; or may he be cashing in on an infinite regress, even with his single pass at B? Suppose again that B is not deceived, but this time has information that A both picks up the signals from B's warning-system and also has access to B's rules for interpreting them. Might not B nevertheless reason, "Power A should discover that I am interpreting these signals as no evidence of an attack by him. Yet he may, for all I know, believe that I believe that he hasn't learnt how I interpret the warning signals I receive (which of course he knows I know he picks up); and he may therefore reason that I will reason, 'Power A picks up the signals from my warning-system at least as soon as I do myself; but he doesn't know how I interpret them. He may well assume

Single:

Level 3

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that I will interpret them as signs of an attack by him, justifying retaliation; and therefore try to attack first himself' and therefore try to attack first myself -- and therefore try to attack first himself" -- and therefore try to attack first himself?

Whether of practical importance or not, an infinite regress could indeed be generated by reflections of this character. For, so long as every halt on a new level of confidence requires a new piece of information, and so long as from each new level of doubt -- however vestigial -- concerning the other's certainties (and therefore concerning his motives) one acquires a new motive -- however weak -- for attack, there can be no bar imposed by formal logic against forever raising "the next question." There is nothing in formal logic against expressing complete confidence about the other's hundred-and-third-stage confidence, and some doubts about his hundred-and-fourth. The only thing is that before the other can have either confidence or doubt at Level 103, he must have both found the fears of Level 102 plausible and yet also at length dismissed them. But if, like the rest of us, he has already cut out at Level 2 or 3, then he can be no source of danger at the fourth or any higher level.

Each level of doubt, then, involves a factual assumption about the extent of the other's foresight -- which includes and goes beyond that involved at the next lower level. But the cost of sending off an irrevocable attack remains constant, while one's speculation about its forestalling advantages grows ever more fantastic; so that at some level (unless surprise attack were absolutely costless, in which case it would have been launched as soon as practicable anyhow) one's risk that the other will succeed in attacking first is not worth the premium -- i.e., the consequences of one's own attacking first -- that would have to be paid to insure against it. Thus, if one has set one's mind at rest, concerning the grounds for fear, one level or more below that at which the risk would not justify the premium, one has not the least part of a sound reason for responding with immediate attack to higher-level thoughts of the possibility of being attacked first.

## II

Let us reconsider, at this point in the argument, the case "Single: Level 3" as outlined above; and let us postulate further that B has definitely ruled out the possibility that A has additional motives for attack, arising from fourth- or higher-level doubts. This seems to be where the study

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of rationale should begin, since the third is the lowest level at which automated
response by either side to the sheer physical signals from the warning-systems
would be quite impossible: here, surely, whatever happens has to be set in
train by imagination and insight. If, unaugmented from other levels, motiva-
tion toward attack sometimes emanates from the considerations which arise
at this level, and provided that, in principle, considerations from lower levels
may be added to those from a higher (i.e., provided such motivations are
cumulative), then there may be other occasions on which considerations
appropriate to this third level (though necessarily even less considerable a
source of risk than when impinging alone) could augment motivation from
Level 2, or from Levels 1 and 2 together, and provide just that additional
weight of argument required to swing the balance of judgment from "no
attack" to "attack." Or, if these motivations in a rational vis-à-vis turn
out to be not cumulative but, as we shall maintain, alternative and exclusive
of each other, then a clear conception of the relative scales of motivations
that, special circumstances apart, might develop on this level should enable
us to see whether the theoretical possibility of infinite regress of doubt should
also be a matter of lively practical concern.

I suppose that almost anyone who attends to the matter must feel that a
third-level worry can have, at best, no stronger a base than has a worry of
the comparable second level -- i.e., "He believes I believe he believes I
believe he is attacking" cannot be more plausible than "He believes I believe
he is attacking." Indeed, since the one incapsulates the other and therefore
expresses a more complex hypothesis, we would usually suppose it to be
initially less plausible -- in the absence of any specific evidence, it should
worry one less than the second-level hypothesis. Is this merely because
both employ the concept of "belief"; or do we also feel that "He believes
that I believe that he is attacking" is a more plausible, and therefore a more
serious, ground for worry than "He is certain that I am certain that he believes
I believe that he is attacking"? I think we do also feel that the latter is less
plausible (even though, in some other contexts, reiterating that I am certain
that he is certain..., and so on, adds no new information). For "my" worry,
in the case we have been imagining, is generated by a predictive rule to the
effect that if someone becomes certain about information which would pro-
vide him with a conclusive reason for taking some line of action, he will act
accordingly. Now this predictive rule is no doubt confirmed more often than
not -- it has been deduced as part of a pattern supposed to evince rational
behavior. But for that very reason we do not assume it to be unfailing, even
when applied to persons otherwise rational. "My" vis-à-vis, then, may be
rationally expected to suppose it to be just possible that I may be certain he
is attacking and yet not attack myself, and also that I may be certain that he
is certain... and yet not draw the obvious conclusion or act accordingly.

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[LC]

One may have the best of reasons for an act, and yet not perform it. Thus, even where the third level comprises certainties about states of belief at the second level, the third-level worry has a hypothetical base initially less plausible than the second-level worry. And when that source of diminution is compounded with the one arising from the regress of beliefs, it will follow that third-level hypotheses about beliefs are necessarily less plausible than the second-level kind.

Let us now press a little further, to decide whether we have completely accounted for the common-sense impression that third-level worries are far-fetched. We shall suppose all doubts to have been quieted by conclusive evidence that in this case the vis-à-vis is "certain that...." Further, we shall suppose (in practice, an impossible condition) that there is no question but that he will act as rationally expected. You are certain, that is, (i) that he is certain that you believe that he believes that you believe that he is attacking -- and therefore (ii) that he is attacking. Given (ii), of course, you have every warrant for launching a counter-attack to blunt his capacities; the question is, how strong is that therefore?

Consider the burglar in the basement, whom Mr. Schelling makes much play with:³ are you to justify your shooting of him on the sole grounds (i) that you were certain he was certain you believed that he believed that you intended to shoot him anyway (though you knew he knew you intended no such thing), and (ii) that you knew him to be quite ruthless in pursuit of his own rational interest? In some countries, where the law does not license the killing of burglars even though they be armed, you would be very lucky if you got off on a plea of justifiable homicide: even the case a fortiori arguable -- that you had shot because you were certain he was certain you intended to shoot anyway (though, as you could prove, you had intended no such thing) -- would make a somewhat recondite defense. After all, the burglar would be a peculiar one -- not at all the ordinary rational extrovert who just happens to be an armed burglar -- if he would shoot you because he was certain you believed that he believed that you intended to shoot, knowing as he did that you had no such intention. One can perhaps imagine a paranoiac saying, "Poor fellow, he wouldn't even shoot a burglar if he could help it, but he's just like the rest of them, always convinced that we Burglars are all paranoiacs and are persuaded that they are a ruthless lot, anxious to shoot us as soon as we show our heads -- so I've just got to shoot him." But is it possible to imagine a convincing parallel reasoning

3. *Ibid.*, p. 1 and pp. 14-16.

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by someone without a kink? At this third level of reflection, he has to be
envisaging you as near enough the paranoid end of the spectrum to suspect
suspicions qua suspicions, and yet far from shooting burglars qua burglars.

Playing about like this in Mr. Schelling's basement, we may have
stumbled on a further source of common-sense skepticism about third-
and higher-level worries: they are not merely far-fetched, but also
paradoxical. By assuming rational suspicions elaborately quieted in a
primary way, but as a consequence all the more rampant in those abstruse
realms, one would be building hypothesis upon hypothesis not merely inde-
pendent of, but decidedly unfavorable to, each other. (A true logical or
semantic paradox is one which, if it be supposed true, must be false; and
if false, true. This type might be considered an epistemological paradox,
since to assume it is to cast doubt upon the presuppositions required to
generate it. If in the long run invalidated by its paradoxical character,
it will be so only by degrees -- a mild dose of the complaint merely makes
the proposition more interesting.) Clearly, the more levels in the hypo-
thesis, the more paradoxical -- and therefore the less worry it generates.

We may therefore conclude:

(1) If a third-level worry is a matter of a chain of beliefs -- "B believes A believes B believes..., etc." -- then for that reason alone we might doubt whether it would justify the _certain_ evils of preventive war.

(2) Even if it involves a chain of certainties, it cannot be as good a reason for attack as would be the immediate certainty that the other was attacking, if only because people need not do what they seem rationally compelled to do.

(3) Finally, one would have to assume a vis-à-vis primarily innocent
but perversely ingenious enough to carry on his worries to such a point, and --
though thus suspicious -- quite capable of having his suspicions quieted by
factual evidence or by rational considerations at lower levels: and this is
to assume a complicated type of opponent. It should be noticed that objec-
tion (2) is cumulative with (3), and (1) cumulative with both.

III

Third-level worries, then, are highly unlikely to have rational bases
justifying preventive war or pre-emptive attack when derived only from
imputation of motives; though it may be barely possible that quite recherche

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information would provide some sort of justification. If it turns out that our worries themselves are not cumulative but alternative, we should hardly expect the theoretically possible infinite regress of worries to have much practical effect.

Well, then; should we think of these worries as summing to a total of worry about being attacked; or, are they alternatives, so that the gravest of them is the most you have to concern yourself with? To spell it out: are there rational grounds for supposing that (i) if Level 3 -- or some higher Level n -- really does in some case provide distinct motive power on its own account, and (ii) if your assumption is that the other fellow thinks as far ahead as that level, and (iii) if in the particular case you have not dismissed as utterly implausible the arguments for attack that might have presented themselves to him at lower levels -- say, Level 1 or 2 -- then such weight as the higher-level argument may have will be cumulative with that of the lower-level arguments in favor of attack? Or does each level instead provide a strict alternative, so that if two of them are equally serious but each more serious than any of the others, it is as if you had only one of that pair to worry about?

I am supposing, in the terms of the "single deception" examples, that Power B is neither wholly convinced that the warning signals do not come from a real attack by A, nor wholly convinced, even if they do not, that A will not assume that B might interpret them as such, nor at all convinced that, even if A does not assume so, he will go on to assume that B assumes he has assumed it. My supposition, if it be coherent, presents an array of some of B's alternate hypotheses about A's current behavior: A is attacking, or instead is assuming something, or instead is assuming the negative of that along with something else positive, and so forth. (It will be seen that these hypotheses of B's are exclusive of each other -- if one of them is true of A's behavior, the others must be false -- though of course, on a fourth hypothesis, A could be confused about the whole affair!) On the other hand, each succeeding alternative hypothesis incapsulates its predecessor as a cogitation you suppose imputed to you in the course of a cogitation that you are imputing to the other fellow, and thus depends for its plausibility upon the degree of plausibility of the hypothesis involved in that predecessor (and of that in its predecessor, and so back to Level 1), though not more than one of these hypotheses can be true.

The situation then is that each of two or more factual hypotheses, disjunctive but severally plausible, lends at least some plausibility (though, as we shall see, a successively diminished degree of it) to

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the alarming hypothesis that the other fellow has started to attack or is about to do so. But a peculiar logical relationship makes it difficult to think about the subject: for if you knew that the other fellow was certain that you interpreted the signal as indicating an attack from him (while you also knew -- what he did not know you knew -- that for some reason he was physically prevented from attacking you), you could infer that any higher-level worries would be otiose -- he would already be expecting you to attack. Whenever, on the other hand, you have less than knowledge, or he, as you consider, has less than certainty, then your doubt as to his belief that you have been deceived (or, more strongly, your belief as to his belief that you were not deceived) leaves open the possibility for you of a higher-order worry -- that nevertheless he may believe that you believe that he will assume you deceived; and that, if it should turn out no more than a strong possibility, would leave open the further possibility that he may believe that you believe that he believes that you believe that he will assume you deceived; and so on.

Our present sub-problem -- cumulative worries or alternative worries? -- thus seems to boil down to whether Power B would be rational in imputing to Power A, as a supposedly rational vis-à-vis, a cumulative or an alternative soliloquy. Cumulative: "Given information about the information that B is receiving from his warning-system, the risks involved in an immediate counter-attack by me seem greater than the danger that he has made a mistake and consequently may be 'counter-attacking,' as he would regard it. But I must also consider the possibility that, supposing he doesn't make that mistake, he may think that I may think that he has, and therefore attack. Of course, that is a remoter possibility, but it is another distinct ground for attack on his part; logic therefore demands that I suppose him to add it to the previous possible ground. But that seems to increase the risk that he will attack, making the total risk an amount I cannot tolerate, so I must counter-attack." Alternative: the same as for cumulative, down to "...and therefore attack"; but after that, A is supposed by B to continue as follows: "The latter is a more remote possibility (and any higher level would be remoter still), so I need not consider it. For it is a possibility which only arises if I suppose he has not made the initial mistake of thinking that I have attacked him; and if he has not made that initial mistake, then I can completely discount the risk of his attacking because of it, and can concentrate on this admittedly lesser second-level risk -- that he thinks I think he thinks I am attacking; therefore it seems that I need not counter-attack."

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Now as far as I can judge, the alternative soliloquy expresses the sounder reasoning, and so our sub-problem is answered. ⁴ But the flaw in the "cumulative" argument is not so obvious that it might not escape notice. Indeed, most colleagues who have read this paper think the "cumulative" argument right and the "alternative" wrong. So B must allow for the possibility that A, not noticing it, may be led, by what seems to him rational argument, toward a counter-attack. Thus, at the very moment when we seem to have taken care of everything, another infinite regress may break out -- i.e., B, who thinks worries are not rationally cumulative, may fear that A's worries will accumulate irrationally; or he may fear that A, though believing worries not to be cumulative, will fear that his (B's) worries may accumulate irrationally; or... and so on.

4. Mr. Ellsberg has suggested to me that I have neglected the strongest argument on the cumulative side. This is *not* that concerning the reasons which B imputes to A, since in the "single deception" case, A has no Level 1 worries to bother him. It is rather that concerning the relation between B's Level 1 worries and his Level 2 worries (or, on a different line of reasoning, between B's Level 1 worries and his Level 2-and-above worries compounded). Are *these* levels of worry cumulative?

At first glance, it would seem that one must answer, "Yes": *either*
(a) A is already attacking, *or* (b) he is going to attack because he thinks
B is going to attack, *or* (c) he will not be attacking at all. This seems to
give one an ordinary calculation in probabilities -- P(a|b|c) -- which is
additive.

I think *not*, because B's Level 1 worries concern what has or has not happened, and his Level 2-and-above worries (if any) concern only what may or may not be about to be done. In respect of Level 1, B has just to make up his mind whether this evidence provided by his warning-system is to be taken as evincing that A has attacked. That cannot be affected by any hypothesis he may form as to what A will be thinking if he has not attacked. Worries about this latter cannot arise unless and until B decides that his warning-system does not show A to have launched an attack, and therefore -- given the prior assessment of the consequences of war -- that A has not launched an attack. Until Level 1 worries are silent, Level 2

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The catalytic warmonger saves his reputation for subtlety just as he seems about to lose it. For when the possibilities seem multifarious that the other fellow will err in judgment or will miscalculate, one is tempted to simplify the issue and say, "There must be some all-up probability that he will attack even though he shouldn't, which equals the sum of all the probabilities that he will go wrong for some particular reason or cause. I shall make a guess at that all-up probability and act accordingly. And since the consequences of my getting in the first strike, grim as they are, are preferable to those of being hit first, I shall err if I must on the side of caution," -- determining at the same time that when one guesses, with allowances on the side of caution included, that the all-up probability of his attacking exceeds such-and-such, then one will try to strike first.

Now, if one side -- say, B -- in the catalytic-war situation thinks along these probabilistic lines, he will seem to himself to have cumulative reasons for striking. If, amongst reasons considered severally, there is one which B would consider almost to warrant his counter-attacking, then, given that he considers them jointly, they will seem to him to add up to a warrant for counter-attacking. But if B thinks probabilistically and cumulatively, and "makes allowances on the side of caution," why should not A also? If both think that way, attack by at least one will be more "probable" than if just one does.

Footnote 4 continued

worries cannot be heard. Thus, the type of worry which is based on the thought of how one's opponent might react on finding out that one had received a warning signal is rationally not cumulative with the type caused by the question whether the warning signal shows that he has attacked already. You should not suffer the one until you have dismissed the other.

Similarly with B's worries, from Level 2 upward, about A's worries. Since it is logically impossible for A to think, "B may believe that I believe that he believes that I have struck already," without (i) having thought, "He may believe that I have struck already," or (ii) having already decided, "He fully believes -- wrongly -- that I have struck already, and is therefore himself striking now," then it is not rational for A to have more than one worry at a time. Therefore it is not rational at any one time for B to have more than one worry (from Level 2 up) about the at-most-single worry that A may have. Therefore B's higher-level worries should not be cumulative.

[LC]

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Suppose the last two sentences of the previous paragraph enter B's head, though they are in the long run inconsistent with his probabilistic approach. The picture of the total situation which then he well might form is one in which each side's guess is adjusted upward to "allow for" the others. This is a picture of the familiar multiplier effect -- i.e., it illustrates an unstable case of reciprocal response and cannot stabilize below an explosive level.⁵

The moral seems to be that in a catalytic-war situation it is safer to have an opponent (or partner) whom one can count on to regard oneself as an autonomous strategist and not as a subject for estimates of "probability." ⁶ I have tried to show, throughout this analysis, that though a careful dissection reveals layer upon layer of reasons for suspecting the other fellow, yet none of the more subtle reasons is in the normal case (i.e., where neither side has any positive reason to believe the other irrational or perverse) as weighty as any of the more obvious ones, and that in any case they are alternative and not cumulative possibilities.

It should, moreover, have become evident that such plausibility as the several regresses of worries may have must diminish very quickly with improvements by either side of the technical features of weapon systems -- in particular, of warning-systems (directed toward increasing, not their range and sensitivity, but their power to distinguish genuine from bogus attacks) and, more importantly, of retaliatory forces (directed toward reducing their vulnerability and to increasing their penetrative capacities). As far as concerns the nth-country problem as a whole, one clear conclusion is therefore derivable:

5. Cf. with the situation described by Schelling as "dynamic adjustment" and "parametric behavior" (op.cit., pp. 21-23).

6. In qualification, I am prepared to concede that there may be a **minimum** figure for the probability that any human being suffers from one of the physiogenic mental diseases which do appear to have a fairly consistent distribution in any population. But as this would not be also a **maximum** figure for the "probability" of irrational action, it hardly helps the probability theorists. Professor Schelling also suggests that the decision of a **collective** executive having divergent values and a voting system might be at least quasi-probabilistic but yet not non-rational (ibid., pp. 16-17). This suggestion raises one of social science's most crucial and controversial problems; but even if we were to grant it, we might still suppose that collectives **sometimes** act as though they were individuals.

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Only a deterrent force both highly protected and easily capable of penetrating the active defenses of the population of a first-striking enemy will unequivocally increase the security of a would-be deterrent Power, in a world of more than a few multi-lateral deterrent Powers.

A nuclear-deterrent strategy depending upon low-confidence measures is no doubt comparatively safe at present for the United Kingdom, since no one has motive and opportunity for catalyzing a nuclear war between Britain and any present nuclear Power. But, if nuclear capabilities were to spread very wide, a country such as Israel would be unwise to depend upon weapons that permitted her no more than a low-confidence technique.

## IV

Instead of drawing further speculative conclusions, I had better tie up an end left loose at the beginning of this analysis -- i.e., the question of double deception, at various levels. We have so far considered single de-ceptions, in which the catalytic Power simulates a surprise attack by one of his victims on the other. Now, when he simulates attacks by each against the other, does this reopen possibilities of practically important infinite regresses that we have decided are closed in the case of single deception?

We may at once notice in this situation a new ambiguity which does not compound with others but which might lend itself to unscrupulous exploitation if the nations come to regard catalytic war itself as a real possibility. Suppose that A and B suspect C as a would-be catalyst; and suppose that each of the two has access to the early-warning signals which the other receives. B might reproduce signals to itself, somewhat suggestive of an attack by A, and at the same time direct against A a real attack -- with the expectation that A, knowing itself not to have launched any attack against B, and therefore construing the coincidence of apparent signals to B and signals from its own system as evidence of an attempt by C at catalytic war, and anxious moreover to disabuse B of any impression of danger, would neither counter-attack nor alert its defenses, thus leaving itself as vulnerable as possible to B's first strike.

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Such a strategy's efficacy for B would depend upon two necessary conditions: (1) A's strong expectation of a catalytic attempt by C (and lack of expectation of B's exploiting that expectation); and (ii) B's ability to eliminate by complete surprise the possibility of retaliatory hits from A, supposing there to be a considerable difference between the result of complete surprise and the result if A should alert his defenses and get off some retaliatory mission. Again we notice the grave dangers attendant upon possessing a deterrent which would not survive the most complete surprise or penetrate alerted defenses of population. A deterrent which, on the other hand, would do both of those things would also deter Power B from exploiting even the strongest expectations by A of a catalytic attempt by C. Otherwise, it will follow that A's recognition of this possibility of exploitation by B will increase for him the plausibility of the hypothesis that any signal on his warning-system (whether in fact genuine or produced by C) indicates a real attack by A. Mutatis mutandis, the same applies to A; and the effectiveness of a catalytic feint by C is thus likely to be much enhanced. Thus the would-be catalyst and the would-be exploiter of anti-catalytic measures can each trade upon the victim's fear of the other.

Since the state of military technology presupposed throughout this study would make it very dangerous to miscarry or be found out in such a Machiavellian trick, I shall now discount the possibility that one side or another will exploit fears of catalytic war along the lines suggested in the paragraph before last; and I shall return to the question of differences between single and double deception. We can treat of this only by running through the several technical situations in which a catalytic attempt might be made:

Double:
Level 1

(a) Neither side in fact has access to what appears on the other's warning-system. This case, from the point of view of either side alone, is of course equivalent to Single Level 1 -- the prime question for each is "Do these signals really mean an attack on me?" Worries equivalent to those of Single Levels 2 and 3 could arise as alternatives for either side if it entertained the possibility that its own warning-system was under observation by the other. From the point of view of C, if he takes a "probability" approach to the behavior of his intended dupes, it will follow that he has a better chance than he would with single deception of provoking an attack by one or the other, and some chance (contrasted with none at all on a single-deception attempt in the assumed technical situation) of bringing off attacks by both. To be precise: if probabilities did apply here, the probability that either A or B would attack should equal the probability that at least A would attack, plus the probability that at least B would attack, minus the probability that both would attack.

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[LC]

**Double: Level 2**
(b) One side only has access to the other's warning-system. Here, for the side which has no access, the situation is again equivalent to Single Level 1 (unless it believes that the other has access to both systems). The Power which has access -- since we are discounting the possibility of the other's exploiting a fear of catalytic war -- will *ipso facto* possess near-to-conclusive evidence that the signals are tricks by C or some fourth Power (for the Power with access knows that what it picks up from the other's warning-system is no evidence of attack by itself). It is therefore in the position of considering only the following situation, which does not correspond exactly with any single-deception level: it knows that the other Power has not already launched an attack, but it must consider the possibility that the other might attack in the belief that it has been attacked already. (The Power with access need go to no higher levels unless it is deceived into believing that the other side has access to its system.) From the point of view of C, the situation is the same as when neither has access, except that, on the basis of earlier discussion in this paper, the side having access is now less likely to attack. If, on the other hand, one Power that has access mistakenly thinks that the other has access reciprocally, it will fall into a position very much like B's in Single Level 2 above.

**Double: Level 2, plus 2**
(c) Both sides have access to the other's warning-system. The possible outcomes of this situation are bizarre in their difference, which depends upon whether one or both realize that both have access. If both realize this, the catalytic strategem has failed, for each should be able to infer, and will know that the other should be able to infer, from the fact that the signals have appeared on both screens within a short space of time, that they must be the creation of some third Power.

If, however, only one realizes that both have access, he will be in a position equivalent to that of someone on Single Level 2 alone: he will know that the other has not already attacked and knows that he has not already attacked, but he will also have occasion to think that perhaps the other believes that he believes that the other does not know that. This may give him a motive for alerting his defenses and trying to get off a massive counter-strike, for we have had no reason to think that Level 2 worries are as far-fetched as those of Level 3.

Summing up the issues of double deception, we see that at Double Level 1, where there is no access, the situation for each of the dupes is precisely that of Single Level 1, but that the catalytic Power has a better chance of achieving his aim. At Double Level 2, the non-access Power is virtually at Single Level 1, while the Power with access is in a novel position not as explosive as Single Level 1. But where only one has access yet mistakenly believes the other has also, or where both have it but only one

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knows this, the situation for the knowledgeable one is equivalent to Single
Level 2 alone; and where both have access and know that both have it
(Double Level 2 plus 2), the strategem fails.

We can now state explicitly the most interesting conclusion of our
analysis. In considering single deceptions, we saw that warning-systems
might conduce to catalytic trickery; but that its scope would be reduced by
systems better able to distinguish real from simulated attacks, and by
numerous and protected deterrents. We now see that since reciprocal
knowledge would defeat any attempt at double deception:

> public access to each other's warning systems, or common dependence upon a public warning system, would notably reduce the possibility of a catalytic war's being brought about by double deception.

This may further suggest that negotiations (such as those recently adjourned)
concerning reduction of the possibility of surprise attack might have pacifi-
catory consequences that should be felt after the present virtual duopoly of
nuclear force has been infringed on.

World Politics

*A Quarterly Journal of International Relations*

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CONTENTS—Vol. XI, No. 3 (April 1959)

Southeast Asia as a Problem Area in the Next Decade, by Guy J. Pauker

International Relations: The Long Road to Theory, by Stanley H. Hoffmann

The Soviet Bloc and the Underdeveloped Countries, by Stanley J. Zyniewski

Israel's Capacity to Govern, by Marver H. Bernstein

Review Articles

Strategic Surrender: The Senate Debate and the Book, by James E. King, Jr.

Brainwashing and Totalitarianization in Modern Society, by Edgar H. Schein

Geography at Mid-Twentieth Century, by William Warntz

Partisanship and Policy, by Joseph C. McKenna, S.J.

The West and the Middle East, by Firuz Kazemzadeh

Cohesion and Commitment in Employers' Organizations, by Heinz Hartmann

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