The Equivalence of Strong Positive Association and Strategy-proofness EITAN MULLER AND MARK

Consider a group that must select one alternative from a set of three or more alternatives. Each member casts a ballot that the voting procedure counts. For a given alternative X, let two ballot profiles C and D have the property that if a member ranks alternative x above alternative y within C, then he also ranks x above that y within D. Strong positive association requires that if the voting procedure selects x when the profile is C, then it must also select x when the profile is D. We prove that strong positive association is equivalent to strategy-proofness. It therefore follows that no voting procedure exists that satisfies strong positive association, nondictatorship, and citizens’ sovereignty. Define a group to be a set N whose ) N 1 elements are the group’s members. They select one element from the set of alternatives, S, by each casting a ballot and then using a voting procedure to count the ballots. A ballot Bi is a strict ordering of the elements within S, e.g., Bi = (xyz) where S = (x, y, z> and x is ranked highest, y second highest, and z lowest. Indifference is not allowed. A voting procedure is a single-valued function v(B, ,..., BJ that evaluates the profile of ballots and selects one element of S as the group’s chosen alternative. Each member i E N has preferences Pi over the set of alternatives S, Preferences, like a ballot, are a complete, asymmetric, and transitive ordering of S. A member’s preferences Pi describe what he truly desires. For example, Pi = (xyz) denotes that individual i most prefers that the group’s choice be x, next prefers that it be y, and least prefers that it be z. An alternative notation for the preference ordering Pi = (xyz) is xPiy, xPiz, and yPiz, where xPi y means individual i prefers x toy. Similarly an alternative notation for the ballot Bi = (xyz) is xBiy, etc. Beyond completeness, asymmetry, and transitivity we place no restrictions, such as single-peakedness, on either