Review of 'Arrow's Theorem'

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Review: [Untitled] Reviewed Work(s): Arrow's Theorem: The Paradox of Social Choice. by Alfred F. Mackay W. D. Hart Mind, New Series, Vol. 92, No. 367. (Jul., 1983), pp. 471-472. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28198307%292%3A92%3A367%3C471%3AATTPOS%3E2.0.CO%3B2-2 Mind is currently published by Oxford University Press.

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BOOK REVIEWS

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Arrow's Theorem: T h e Paradox of Social Choice. By ALFRED F. MACKAY. New Haven: Yale University Press, 1980. Pp. ix f 143. k9.15. Professor MacKay has given us a very good little book about Arrow's Theorem and some related questions about social choice. H e begins by stating Arrow's second list of four conditions on any social c h ~ i c efunction. These are unrestricted scope (the value of the function must be a unique, irreflexive, transitive and connected ordering of the alternatives for a n y profile of individual preference rankings of the alternatives; these rankings must also be irreflexive, transitive and connected), the Pareto principle (when each individual prefers one alternative to another, the first must lie hlgher in the composite social ordering than the second), nondictatorship (no social welfare function may automatically copy a single individual's preferences whatever the preferences of others), and independence of irrelevant alternatives (how two alternatives rank socially depends on how individuals rank those alternatives, not on how they rank other alternatives). As is well-known, some thirty years ago Kenneth J. Arrow proved that there can be no social welfare function satisfying these conditions. MacKay examines the background to Arrow's result, provides a proof of it suitable for presentation to undergraduates studying philosophy, and attempts to assess the significance of it for social philosophy. It seems to me that the most interesting sections of MacKay's book deal with failures of transitivity and interpersonal comparisons of utility. It has long been known that an individual's indifference between alternatives need not be transitive. I might be indifferent between a pint of beer and one and one thousandth pint of beer, because I cannot tell the difference. But pile up a thousand such miniscule differences, where I cannot distinguish adjacent glasses of beer, and I certainly need not be indifferent between the first (one pint) and the last (two pints). MacKay urges, quite sensibly in my view, that this intransitivity of indifference is really just an overlay on an intransitivity in our capacities for perceptual discrimination; thus its significance for the intrinsic structure of desire is at best unclear. But MacKay, like most philosophers, seems unaware of quite another sort of counter-example to the transitivity of (even rational) preference. (I learnt this example from S. D . Guttenplan. It is reported in the December 1970 Mathematical Games column of T h e Scientzjic American by Martin Gardner, who credits it to Bradley Efron.) Imagine four cubes, A, B, C, and D , the six faces of each being marked with certain numbers. T h e faces of A are marked with two zeros and four fours, those of B with six threes, those of C with four twos and two sixes, and those of D with three ones and three fives. First you pick one of the four cubes and then I select one of the remaining three. We will then play a long series of games in which we both roll our cubes, and whoever's cube shows the lower number on its top face pays the other one pound. Some simple calculations show that the probability that A heats, B is two thirds, and that similarly B beats C, C beats D and D beats A, each with probability two thirds. Here it might seem rational for you to have intransitive preferences between A and B, B and C, C and D , and D and A. Moreover, while MacKay might say that if we could discriminate between the glasses of beer described above, we

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BOOK REVIEWS

would not be indifferent between them (since we prefer more to less of goods), no analogous response seems apposite here. I t has long been objected to the independence of irrelevant alternatives that it forbids some modes of taking comparisons of intensities of preference into account in making social choices. Suppose you rank x above y and I rank y above x, but we both rank both below getting a lot and little money, and we both rank both above losing a little and a lot of money. If we were the only members of this society, then it might seem sensible to sum our opposed but apparently weak preferences between x and y as social indifference between them. Independence of irrelevant alternatives says that if my preferences stay the same, but you come to prefer your best of x and y to getting a lot of money, and to prefer losing a lot of money to your worst of x and y, then socially x and y should still be indifferent. T h i s might seem wrong since it ignores the apparent intensity of your preference as compared with mine. Arrow is part of that ordinalist tradition which seeks to avoid even a whiff of interpersonal comparisons of intensity of preference. Here MacKay develops a fascinating analogy between making interpersonal comparisons of utilities and comparing the performances of athletes in different events in the decathlon. I not only learnt something from MacKay's book; I also enjoyed it very much. I have no hesitation in recommending it to philosophy students interested in social choice. UNIVERSITY COLLEGE. LONDON

W. D. HART

Education and the Individual. By BRENDA COI-[EN. London: Allen and Unwin, 1981. P p . i x + 104. E8.50, P.B. Eg.50. Surrey's department of philosophy is rapidly gaining a reputation as the source of a number of books and articles on education which challenge contemporary egalitarian thinking in that field. This, the latest of these philosophical Black Papers, seeks 'to reassert the importance of the individual and the family in the area of education'. I n succeeding chapters it supports a selective system of education against an egalitarian one; argues in favour of independent schools alongside state schools rather than a state monopoly; defends parents' rights t c educate their children according to their own values; discusses the nature of religious education on the way to supporting the toleration of religious schools; extends arguments about indoctrination in religion to the areas of morality and politics; argues for a Millian rather than a Marxian ideological background for educational thinking; outlines the nature of education in a liberal society; and among the various educational rights enshrined in current international agreements puts more weight on the 'negative' right of parents to choose their children's education than on the 'positive' right of children to be educated. Philosophy of education has now more or less given u p its bid of a few years ago to become a quasi-autonomous branch of philosophy on all respectable fours with, say, philosophy of science or aesthetics. Much, but by no means all, of its current work is in the area of applied political