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-4 DIFFICULTY IN THE CONCEPT OF SOCIAL WELFARE KENNETH J. ARROW^ Stanford University I. INTRODUCTION cia1 choice in any given situation, e.g., N A capitalist democracy there are a religious code. essentially two methods by which The last two methods of social choice, I social choices can be made: voting, dictatorship and convention, have in typically used to make "political" de- their formal structure a certain definite- cisions, and the market mechanism, ness absent from voting or the market typically used to make "economic" de- mechanism. In an ideal dictatorship, cisions. In the emerging democracies there is but one will involved in choice; with mixed economic systems, Great in an ideal society ruled by convention, Britain, France, and Scandinavia, the there is but the divine will or perhaps, same two modes of making social by assumption, a common will of all choices prevail, though more scope is individuals concerning social decisions, given to the method of voting and to so that in either case no conflict of indi- decisions based directly or indirectly vidual wills is involved. The methods on it and less to the rule of the price of voting and of the market, on the mechanism. Elsewhere in the world, other hand, are methods of amalga- mating the tastes of many individuals and even in smaller social units within in the making of social choices. The the democracies, the social decisions methods of dictatorship and convention are sometimes made by single individ- are, or can be, rational in the sense uals or small groups and sometimes that any individual can be rational in (more and more rarely in this modern his choice. Can such consistency be world) by a widely encompassing set attributed to collective modes f choice, of traditional rules for making the so- where the wills of many peopk are in- 'This paper is based on research carried on at volved? the RAND- Corporation, a project of the United It should be emphasized here that States Air Force, and at the Cowles Commission the present study is concerned for Research in Economics and is part of a longer studv. "Social Choice and Individual Values." to with the formal aspects of the foregoing be published by John Wiley & Sons as a Cowles question. That is, we ask if it is for- Commission monograph. A version was read at the mally to construct a procedure December, 1948, meeting of the Econometric So- ciety. I am indebted to A. Kaplan, University of passing a set indi- California at Los An~eles,and 1. W. T. Younss.-, - . vidual tastes to- a Latt tern~ of~ social~ de- University of Indiana, for guidance in formulating cision-making, the procedure in ques- the problem, and to A. Bergson and A. G. Hart, Columbia University, and T. C. Koopmans, Cowles tion being required to certain Commission and the Universitv of Chicago.u, who natural conditions. An illustration of have read the manuscript and made valuable com- the problem is the following well-known ments on both the presentation and the meaning. Needless to say, any error or opacity remaining is "paradox of voting'" Suppose there is a the responsibility of the author. community consisting of three voters THE COKCEPT OF SOCIAL ITELFARE 329 and this con~munitymust choose among iinply rational behavior on the pa~tof three alternative modes of social action the com~llunityand which will be satis- (e.g., disarmament, cold war, or hot factory in other ways?" war). It is expected that choices of this If we adopt the traditional identifica- type have to be made repeatedly, but tion of rationality with maximization sometimes not all of the three alterna- of some sort, then the problem of tives wiil be available. In analogy with achieving a social maximum derived the usual utility analysis of the indi- from individual desires is precisely tile vidual consumer under conditions of problem which has been central to the constant wants and variable price-in- field of welfare economics.~owever, come situations, rational behavior on the search for a clear definition of opti- the part of tile connmunity would mean mum social welfare has been plagued that the con~lllunityorders the three by the difficulties of interpersoial com- alternatives according to its collective parisons. The emphasis, as is well preferences once for all and then known, has shifted to a weaker defini- chooses in any given case that alter- tion of optimum, namely, the determi- native anlong those actually available nation of all social states such that no which stands highest on this list. A individual can be made better olf with- natural way of arriving at the collec- out making someone worse OH. As Pro- tive preference scale would be to say fessors Bergson, Lange, and Sai~~uelson that one alternative is preferred to have argued, though, the weaker defini- another if a majority of the corninurlity tion cannot be used as a guide to social prefer the first alternative to the sec- policy; the second type of welfare eco- ond, i.e., would ciloose the first over the nomics is only important as a preii~ni- second if those were the only two alter- natives. Let "1, B, and C be the three 'It may be added that the method of decision sketched above is essentially tint used in deiiber- I, 2, alternati~res,and and 3 the three ative bodies, where a whole ranse of ~:!ternati\-es individuals. Suppose individual I pre- usually comes up for decision in the form of suc- ftrs -4 to rF; and B to C (and therefore cessive pairnise comparisons. The phenomenon de- scribed in the test can be seen in a pure for~ilin A to C), individual 2 prefers 13 to C the disposition of the proposals before recent Con- and C to '1 (ai:d therefore B to A), and gresses for federal aid to state education. the three individual 3 prefers C to A a~ldA to B alternatives being no federai aid, federal aid to pub- (and therefore C to 23). Then a major- lic schools only, and federal aid lo both public and parochial schools. ity prefers il to R, and a majority pre- The problem of collective rationality has been fcrs B to C. lye may therefore say that discussed by Professor Frank H. Knight, but chieily the comm~~nityprefers A to B and B to in terms of the socio-psychological prerecjuisiies: C. If the community is to be regarded see "The Planful Act: The Possibilities and Limita- tioils of Collective Rationality," in Frcrdoilz and as behaving rationally, we are forced to R~forl7z(New York: Harper & Bros., 194;), pp. say that il is preferred to C. But, in 335-69, esp. PP. 346-65). fact, a majority of the community pre- 'See P. -4. Samuelzon, For~nrlatiotzsoj Eco~zoi~zZ'c fers C to A.2 So the method just out- Analysis (Cambridge, Mass.: Harvard University Press, rgd~),chap. viii; .4. Bergson (Burk), ".4 lined for passing from individual to col- Reiormulation of Certain .\specis of LVeliare Eco- lective tastes fails to satisfy the condi- nomics," Quarterly Joz~r;zal of Econo~rlics, LI1 tion of rationality as we ordinarily un- (~gjS),310-34; 0. Lange, "The Foundations of derstand it. Can we find other methods Welfare Economics," Econo:?z~trica,X (1942)) 21 j-28; M. W. Reder. Sttldies iw the Theory of of aggregating individual tastes which Welfare Economics (New York, 1947)) chaps. i-v. nary to the determination of a genuine conzpensation is not actually paid. social maximum in the full sense. E.g., Apart from the ethical difficulties in the under the usual assumptions, if there is acceptance of this prin~iple,~there is a an excise tax imposed on one commod- formal difficulty which was pointed out ity in the initial situation, it can be by Professor Scito~szky:~it is possible argued that the removal of the tax ac- that simultaneously x should be pre- corllpanied by a suitable redistribution ferred to y and y be preferred to x. Just of income and direct tax burdens will as in the case of majority voting, this improve the position of all individuals method of aggregating individual pref- in the society. But there are, in general, erences may lead to a pattern of social many redistributions which will accom- choice which is not a linear ordering of plish this end, and society must have the social alternatives. Note that in some criterion for choosing among them both cases the paradox need not occur; before it can make any change at all. all that is said is that there are prefer- Further, there is no reason for confining ence patterns which, if held by the indi- the range of possible social actions to vidual members of the society, will give those which will injure no one as com- rise to an inconsistent pattern of social pared with the initial situation, unless choice. Unless the trouble-breeding in- the status quo is to be sanctified on dividual preference patterns can be ethical grounds. All we can really say is ruled out by a priori assumption, both that society ought to abolish the excise majority voting and the compensation tax and make some redistribution of in- principle must be regarded as unsatis- come and tax burdens; but this is no factory techniques for the determina- prescription for action unless there is tion of social preferences. some principle by which society can The aim of the present paper is to make its choice among attainable in- show that these difficulties are general.

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