¹ Arrow, Kenneth, Social Choice and Individual Values (New York: John Wiley and Sons, Inc., 1964), p. 1 Google Scholar.

² It is with some discomfort that I leave aside the problem of distribution in this paper. The issue is attacked directly in a later effort, “Risk Spreading and Distribution,” Kennedy School of Government, Discussion Paper No. 10 (August, 1972). There I argue that a profound belief in the efficacy of the outcome of a perfect market gives one insights into distributional questions. One can consider the problem of drawing the social contract equivalent to that of a group of future citizens starting in some initial position of equal potential (they face the same lottery on future possibilities). They draw up a contingent claims agreement, the magnitudes of payment to depend on their future fortune in securing endowments of goods and capabilities (that is the states of the world). In retrospect, we may decide that it is our duty to redistribute in the manner that would be prescribed by the hypothetical agreement that antedated our present position. For a philosophically compatible, but perhaps policy inconsistent view, see Rawls, John, The Theory of Justice (Cambridge, Mass., Harvard University Press, 1971)Google Scholar.

³ Thomas Schelling has commented at great length, interpreting and questioning the significance of Pareto optimality as a criterion for social choices. He states, “the significance of failure to achieve Pareto-optimality merely means that the situation is improvable … it is the degree of improvability that determines how important the nonoptimality is. Pareto optimality is not like virginity or justice …. Small departures are of small interest, large departures are of large interest.” Personal communication, July 12, 1972.

Economists have long recognized that Pareto optimality by itself is a hopelessly incomplete guide to normative and descriptive considerations of social choice. An indication of this author's response is given by the conclusion to this paper.

⁴ For the conditions, see Arrow, K., Social Choice, pp. 22–33 Google Scholar.

⁵ Sen, Amartya K., Collective Choice and Social Welfare (Holden-Day, Inc., San Francisco), pp. 41–42 Google Scholar.

⁶ Following Pareto optimality, efficiency requires that a switch away from the selected candidate would lower the welfare of at least one voting individual. If the selected candidate had the highest total score, then there could be no other candidate who received an equally high individual score from each voter. A switch from the man with the highest total would therefore involve a loss for at least one voter.

Special difficulties arise if lotteries on alternatives are permitted; and rank-order schemes need not lead to Pareto-optimal outcomes. See Zeckhauser, Richard, “Majority Rule with Lotteries on Alternatives,” The Quarterly Journal of Economics, 83 (November, 1969), 696–703 CrossRefGoogle Scholar.

⁷ For a good introduction to this subject, see Raiffa, Howard, Decision Analysis (Reading, Mass.: Addison-Wesley, 1968)Google ScholarPubMed.

⁸ The ex-ante expected utility approach can be employed to make evaluations of social welfare. See Zeckhauser, Richard, “Determining the Qualities of a Public Good—A Paradigm on Town Park Location,” Western Economic Journal, 11 (March, 1973), 39–60 Google Scholar. For an application of this principle in the social decision context, see Richard Zeckhauser, “Majority Rule With Lotteries on Alternatives.” Fishburn, Peter formalizes this argument in “Lotteries and Social Choices,” Journal of Economic Theory, 5 (October, 1972), 189–207 CrossRefGoogle Scholar. Peter C. Ordeshook employs the principle to define candidates' mixed strategies as lotteries on social states should they get elected. See “Pareto Optimality in Electoral Competition,” The American Political Science Review, 65 (December, 1971), 1141–1145 CrossRefGoogle Scholar.

⁹ Appropriate historical attribution would give much of the credit for this utility concept to Frank P. Ramsey.

¹⁰ If we extend this concept to the market setting, we discover that market procedures are unique with respect to the information required for efficiency: individuals' marginal rates of substitution and producers' marginal rates of transformation among valued goods.

¹¹ Say v_A is only slightly greater than v_B , but much greater than v_c . If C had less probability of being elected if the individual voted for B rather than A, it might be reasonable for him to vote contrary to his true preferences.

¹² For further discussion of this matter, see R. Zeckhauser, “Majority Rule with Lotteries on Alternatives.”

¹³ This paper is addressed to situations where a single outcome is selected, and all individuals vote to help determine that outcome. The model does not apply, for example, to the selection of a legislature. (With a stretch of the imagination, we might construe voters to be choosing among all possible legislatures, but usually an individual voter ballots to help determine only a small part of that body.) Strict proportional representation is the closest (but still imperfect) legislative equivalent to the random dictator system. The number of seats secured is proportional to the number of votes received.

Proportional representation will guarantee honest balloting if two conditions are satisfied. (1) Each voter's utility for the legislature must be an additive separable function of his utilities for each elected representative. (Otherwise, his preferred vote might depend on the candidates others were voting for.) (2) All ballots cast in the first round must count in the determination of the final legislature. This requires that representation be broken down into arbitrarily small units, and that there be no elimination of candidates. (Condition (2) can be violated if there are strict party interests, and if each voter is indifferent among the candidates of his party.) See Thompson, Mark and Zeckhauser, Richard, “Proportional Representation,” mimeo (1971)Google Scholar. A referee suggests that Kramer also has unpublished work on this subject.

¹⁴ See Sen, A., Collective Choice, pp. 166–168 Google Scholar for a discussion of single-peakedness where individuals' preferences are defined over a single dimension. There is no equivalent to the single-peakedness criterion if two or more dimensions enter individuals' preferences; cyclical majorities will be the rule.

¹⁵ The concluding theorem of the paper employs the dictatorship concept negatively. It states that no system can simultaneously be nondictatorial and at the same time guarantee that other desirable standards are met. The more restricted the definition of dictatorship, then, the stronger my final result. This result would still be achievable if the term “Pareto-optimal” in the above definition were replaced with “unanimously most preferred.” The argument for the definition in the text is that it captures more of what we may mean by dictatorship in a general context.

It should be noted that Pareto-terrible is not a perfect counterpart to Pareto-optimal. There are situations where all outcomes are Pareto-optimal and none Pareto-terrible. The converse is not true.

¹⁶ On a less grandiose level, there are a number of ways that single social-choice decisions can be formulated to deal with a large number of issues. Issues can be compounded. The issues at hand might be whether Smith or Jones should be mayor and whether or not the new high school should be built. This can be viewed as a single decision with four possible outcomes. Alternatively, we might amalgamate issues by employing a lottery. A coin will be flipped to see whether a voting mechanism will select a mayor or decide on a high school. Here too there are four possible outcomes, each composed of two contingent decisions.

¹⁷ Another numerical example may be helpful. Assume that 1 marks his ballot the same way whether his cardinal utility values are 1,.5,0 or 1,0,.2. Then if individual 2 has preferences 0,1,.9, for example, a nondictatorial Pareto-optimal outcome cannot be assured. The third candidate can never be included for individual 1's first set of preferences and must always be included for his second set of preferences.

¹⁸ I strongly believe that it is possible to substantiate the arguments in the remainder of this paper without these assumptions. But the whole area is most elusive, and I have not been able to discover a convincing proof for this assertion.

¹⁹ It may be possible to construct voting schemes that are not unique for each individual taken alone, but are unique for all individuals taken together. This would require that individuals have some capability to monitor each others' preferences and ballot markings. For example, individual 1 might mark his ballot the same way when he has preferences V ₁′ and individual 2 marks X ₂′ as he does when he has preferences V ₁,″ and individual 2 marks X ₂″. The processing system can take 2's markings into account in deciphering individual 1's preferences.

If each individual's cardinal preferences are to be revealed, it is sufficient that given other individuals' markings there be a one-to-one correspondence between an individual's markings and his preferences.

²⁰ The uniqueness requirement is equally well satisfied so long as there is a one-to-one mapping between z and y. Any scheme that induces such a mapping has the key structural aspects that are discussed below in relation to this scheme.

²¹ I am indebted to Kenneth Arrow for this point among others.

²² To deal with discontinuous cases, merely replace (f[x + Δx] − f[x])/Δx wherever f′(x) appears, and similarly for g′(x). Second derivatives can be handled in parallel fashion.

²³ This analysis could be carried out equally well for a world where other individuals' ballot markings were uncertain parameters. A subjective probability distribution f(γ) would be placed over that parameter. Thus, efficiency condition [C] would become

Other conditions would be modified equivalently; the basic results would be the same. Where there is perfect information on other individuals' preferences and ballot markings, the case considered in the text,

²⁴ One scheme that elicits true preferences for this two-individual case allows either individual to allocate 50 per cent of the outcome probability. Each individual takes half his probability as given and deals with the rest as he would in Figure 1, with the vertical scale relabeled with .5 placed where there is now a 1.

²⁵ If y = z = 1, then p ₃ can be 0. As z can no longer increase, p ₃ need not further decrease in value.

²⁶ New variants of Theorem IV could be developed by altering the strength of different definitions, and interchanging existential and universal modifiers. I suspect that Pareto improvements may be possible: stronger results with the same definitions, or stronger definitions leading to the same results.

²⁷ I am indebted to Thomas Schelling, who suggested that I emphasize this theme in my conclusion and provided me with helpful insights. This theme reiterates the central message of my volume, Studies in Interdependence.

²⁸ Many observers of the political scene would argue that our present social choice procedures are very far from aggregative processes that rely on individuals' preferences. See Banfield, Edward, “‘Economic’ Analysis of ‘Political’ Phenomena; A Political Scientist's Critique” (mimeo.), 1967 Google Scholar, and a forthcoming joint piece by Banfield and Zeckhauser on the same subject.