¹ The analogous economic case would work as stated; but to complete its resemblance to our imaginary three-Power world, we should add explicitly the condition that Red competes with Yellow for a market from which Blue is excluded.

² See Ash, M. A., “An Analysis of Power, with Special Reference to International Politics,” World Politics, 111, No. 2 (January 1951), pp. 218–37.CrossRefGoogle Scholar My analysis is not in tended to depend upon any of that article's conclusions. But in 1948 I was shown another of Mr. Ash's MSS., which still remains unpublished; and from that and in conversations with him I learned the approach and much of the terminology used in the first part of this study.

³ “Rational” here has nothing to do with “moral.” The owner of a firm would have rational warrant, in this sense, for attempting monopoly if he could be sure he would be risking no permanent reduction in the value of his capital. As elsewhere in this article, the element of prediction in Principle 4 below is conditional upon the assumption of rationality—the assumption, broadly, that each Power's program of aims and estimates will involve neither lacunae nor contradictions. That this is false in most cases need not wholly destroy the theory's usefulness as a canon.

⁴ “Trading” suggests a far more passive role for Yugoslavia than in fact she adopted. I had in mind the possibility of a lesser Power's shaking loose from one bloc and being then incorporated by another, some quid pro quo ensuring the tacit consent of both.

⁵ This requirement of equal strength is needed to make sense of the previous requirement concerning odd number. Since together they yield a comparatively simple model of the international system, I shall continue to use them throughout the earlier part of the analysis. But the practical point indicated by this theoretical gambit consists in that temptation to polarize itself which afflicts any world whose constituent Powers could fall readily into a pair of equal alliances. That is a thing that might happen to worlds of odd-numbered as well as of even-numbered Powers, provided that not all the Powers were of a size. In fact, the most asymmetrical of distributions may approach polarization through division into a larger but looser and a smaller but tighter alliance—see Principles 1 and 3. Generally, asymmetry affords greater possibilities of self-equilibration than does symmetry, only through the opportunities—not offered in symmetrical worlds—for exercise of pressure to break up the polarization.

⁶ Wight, Martin, Power Politics, Looking Forward pamphlet, No. 8, London, Royal Institute of International Affairs, 1946, pp. 24, 54.Google Scholar

⁷ “Securing Peace Through Military Technology,” Bulletin of the Atomic Scientists, XII, No. 5 (May 1956), pp. 159–64; and “Design for Deterrence,” Ibid., pp. 164–65.

⁸ Sherwin, , loc. cit., p. 161.Google Scholar

⁹ Ibid., p. 160.

¹⁰ Publications during 1957's first few months may appear to have given the lie to the above assertions of last year. We now know that artificial satellites may be able to transmit information about quite small areas located anywhere, and that unjammable homing devices have been perfected for planes and guided missiles (cf. the next paragraph of text). Further, it seems possible that even ballistic missiles might be equipped to home in during the later stages of their flight. At the same time, however, there are foreshadowed anti-missile systems capable of dealing with some significant proportion of ballistic missiles. Assuming that these factors of opposite tendency are likely to phase in together, we may be sure that the probability of any single ballistic missile's eliminating even one other remains fairly small, while that of its destroying vastly many times its own worth of economy is still rather larger. Such a contrast, of course, maintains the deterrent situation. But the prospect of these innovations, many of them employing expensive electronic equipment, impinges upon the findings of this study in another way: by multiplying the money-and-skill costs of optimum deterrent forces, it raises the entrance fee for the deterrent Powers' club well beyond most nations' resources. Such a possibility has been allowed for in Principles 9 to 12 (see below). Principles 13 to 15, on the other hand, must now tend to represent a nerely formal limit to the deterrent situation, since they can begin to apply only when there are scores of deterrent Powers.

¹¹ Dr. W. A. O'N. Waugh, of Canberra University College, has shown me that the above argument and the formula immediately following leave out of account the extent to which random effects may cause deviations from my “expected numbers” of hits in reprisal. My formula suggests, e.g., that if P(h) equals “one in a hundred,” just a hundred projectiles could be relied on to knock out a launching site, and that V times a hundred would knock out V sites—clearly, an overconfident assumption! I have thus understated consistently both the case for Amster's type of strategy, and the deterrent situation's prospects of enduring. This means that the tendencies (discussed on p. 514, and formalized in Principles 13 to 15, p. 525) toward reintroduction of alliances into the deterrent situation, toward its polarization, and finally toward its reversion to a balance of power are much weaker and longer-run than I had supposed. I shall try to adjust the picture by footnotes.

Dr. Waugh writes in correction: “It is possible to derive the probabilities for the number of bombs to be anticipated in reprisal after a volley has been fired at an opponent's sites, and these give rather more information than an argument based purely on ‘expected numbers,’ which would yield an approximate expression only in certain circumstances—for example, if there were as many sites as missiles, and each missile were aimed at a different site. We can correct your formula as follows. A particular missile can only be aimed at one of the sites. Suppose that m missiles are aimed at each site, so that mV missiles are let off at once. Interpret P(h) as the probability that a particular missile hits its target. Suppose that after this onslaught each surviving site fires off a missile in reprisal, each aimed at a different city (to avoid the waste of duplication) and each having probability P(c) to hit its target. Then the number of reprisal hits on cities will be a ‘binomial variable’ with mean

Other schemes such as firing successive shots until a site was eliminated would yield different results but might not fit the description of a ‘volley.’”

¹² Both “distances,” as here estimated, understate the aggressor's difficulties (see the previous footnote). Dr. Waugh comments: “In this numerical example, t is the expected number of hits in reprisal. If the aggressor fires 50,000 missiles, aiming 100 at each of the victim's sites, our corrected formula makes this expected number approximately 116. Use of the ‘normal approximation’ indicates that the number of reprisal hits will exceed 88 with probability greater than 99.7%. To reduce the expected number of reprisal hits to 3, it would be necessary to aim about 455 missiles at each site, i.e. to fire about 230,000 in the initial volley.”

¹³ Let n connote the number of Powers, and let us neglect the consideration, very favorable to Amster, that one chance in ten of a hydrogen bomb on one's cities would not deter much less than the certainty of the same thing. Then for each Power in a coalition of aggressors

ADDENDUM. This formula also leaves out of account deviations from random effects (see footnote 11). Yet its general point—the larger the number of deterrent powers, the shorter the “distance” required by aggressors—remains valid. So does the comparison between the figures in the paragraph this note refers to and those in the paragraph before that, since the extent of underestimation is the same in both.

¹⁴ See Morse, P. M. and Kimbal, G. M., Methods of Operations Research, New York, 1951, pp. 64–77Google Scholar; Lanchester, F. W., Aircraft in Warfare: The Dawn of the Fourth Arm, London, 1916Google Scholar, passim; Driggs, Ivan, “A Monte Carlo Model of Lanchester's Square Law,” Operations Research, IV, No. 2 (April 1956), pp. 148–52.CrossRefGoogle Scholar

¹⁵ I must make it clear that this statement is true only when both belligerents have world-ranging missiles, and when deterrent defense is impossible. The size of a front or engagement in the past has been conditioned partly by the limits of the range of fire, which, as Lanchester has shown, is the precondition of gaining military advantage by concentration.

¹⁶ Quoted in New York Times, July 15, 1956, p. 22.

¹⁷ Loft-bombing enables the dropping of atomic bombs by planes without destruction of the planes themselves.

¹⁸ In terms of the imaginary diagram described on p. 497f.: global deterrent relationships in a world for which geographic location has lost military significance are in theory to be plotted along the line between the points of “total alliance” and “total opposition,” since there is no longer a point of “total indifference.” Indeed, only those two remaining points are needed for the global relationships of a peaceful deterrent state; while for the unlikely deterrent war, the level of deterrent forces allocated against each distinct foe will intersect the line to form a part proportional to the number of foes the Power depicted is fighting against.

¹⁹ Trends of 1957 suggest that this is a feature of prime significance, whose implications should perhaps have been built directly into our theory of the deterrent state. Suppose that military planners were to be offered many and divers novel weapon-systems, and told that fundamental research foreshadowed—without specifying—the possibility of many more. Now, though they could guess at the several probabilities that their potential enemy would adopt one or another of the systems already envisaged by them, they would have to allow also for the possibility of his commitment to systems they had not conceived of. Should their estimates of that possibility grow very large, military planning would become impossible—what range of contingencies could they be instructed to prepare for? (The deterrent situation itself can be defined as one in which planning for traditional victory is impossible.) If this happened, they would presumably recommend either immediate attack or a complete understanding with the enemy—in either case, the sudden end of deterrence. Recent breakthroughs in electronics and space-study provoke such speculations.

²⁰ Japan should have been included in the list of candidates for deterrent status. I probably underestimated the willingness of, e.g., the United States to help other Powers over the deterrent threshold, which will be higher than I had supposed; see footnote 10.

²¹ See also footnote 10.

²² “The objective of deterrence, as well as the allied objective of achieving victory should war be forced upon us, will be served as much as it will ever be served when there are on hand sufficient atomic weapons to defeat an enemy just once, and this regardless of the number he may have. When this point is reached, the competition, if it still exists, will no longer be one of simply making weapons and weapons material; it will become, instead, mainly one of designing, developing, manufacturing, and training men to use new and effective means for delivering weapons or preventing their delivery against us and our armed forces.” Titterton, E. W., Facing the Atomic Future, Melbourne and London, 1956, p. 333.Google Scholar

²³ Another six months later the great blocs, though no longer “monolithic,” appear to be patching their cracks. I still think the Berlin blockade brought us much nearer to world war dian did Suez and Hungary.