Reflections on Vote Manipulation

The notion of non-manipulability (or: strategy-proofness) used in the famous Gibbard-Satterthwaite theorem is too strong to make useful distinctions between voting rules. We explore alternative definitions and suggest how these can be used to classify voting rules. If A is a finite set of alternatives, with |A| > 2, then an A-ballot is a linear ordering of A. Let {1, . . . , n} be a set of voters. An (A, n)-profile is an n-tuple of A-ballots. If P is an (A, n)-profile, then P can be written as ( 1, . . . , n). i, the i-th component of profile ( 1, . . . , n), is the ballot of voter i. i expresses “what voter i wants.” If P(A) is the set of all (A, n)-profiles, for given n ∈ N, then a function V : P(A)→ A is a resolute voting rule for A, and a function V : P(A)→ P(A) is a voting rule for A. Let P ∼i P′ express that P and P′ differ only in the ballot of voter i. Then the standard definition of non-manipulability is given by: