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If the social choice rule g selects from one up to k alternatives (but not more), then there exists a coalition H of k individuals such that for each profile r, the choice set g(r) is the collection of the top-most alternatives in the orderings of the individuals in H. Consequently, g is independent of the preferences of individuals not in H, forcing a disagreeable trade-off: Either some choice sets are very large, or most individuals never have any say in the social choice.

Gibbard (1973)and Satterthwaite (1975)have shown that, subject to a range condition, strategy-proofness implies dictatorship for resolute social choice procedures, i.e., for procedures where the choice set always contains just a single alternative. This paper characterizes social choice procedures that allow more than one alternative to be selected. There are two quite different reasons for relaxing resoluteness. First, although resoluteness might be quite desirable, non-dictatorship is even more desirable; and then we ask a trade-off question: If we relax resoluteness slightly, allowing small non-singleton choice sets, is there then a way to construct strategy-proof social choice rules that are far from dictatorial? Or is it the case that getting far from dictatorship forces some choice sets to be quite large? Beyond this trade-off approach, which assumes that resoluteness is desirable, sometimes we actually want to chose more than one alternative. First, we might be interested in a social choice procedure that represents a preliminary stage in a process, say a piano competition, that ultimately chooses a singleton outcome, but it is actually desirable to have several alternatives selected at the preliminary stage. Secondly, even where we are settling final outcomes, we may desire that a non-singleton set of alternatives be selected in some situations2: The International Mathematical Society will select up to four Fields Medalists to be announced at their next Congress. Imagine a mathematician asked to take part in the process of determining the recipients of the four 1998 Fields Medals. One may feel comfortable writing down a ranking of the say 25 candidates proposed but feel baffled by the request to rank the 12,650 quadruples of candidates. We will assume that individuals have a complete ordering of X itself. An individual is assumed to be able to compare any two subsets of X, and the comparison will have to be consistent with the ordering of X in the sense specified in the Section 2. Section 2provides the basic notation and definitions. In Section 3we prove that the only strategy-proof rules that select from 1 to k alternatives identify a set of k individuals and then select the set of their top-most alternatives. Ching and Zhou (1997)derive dictatorship from strategy-proofness, and without any condition on the size of the choice sets. Because the rules characterized by our theorem are not dictatorial (except when kmax(g)=1), the Ching–Zhou notion of strategy-proofness is very demanding when g is not resolute. Using a less demanding definition of strategy-proofness, Duggan and Schwartz (1997)prove that there is an individual whose most-preferred alternative always belongs to the set of alternatives selected by the social choice function. Our theorem provides more information, but our strategy-proofness requirement is more exacting than that of Duggan and Schwartz. Baigent (1998)examines strategy-proofness for non-resolute social choice rules, and in his case the domain is restricted to the set of dichotomous preferences — an alternative is either acceptable or not: Y is preferred to Z if every member of Y is acceptable and some members of Z are not acceptable, of if some members of Y are acceptable and every member of Z is unacceptable.