Strategy-Proofness on the Condorcet Domain

A social choice rule g selects a member of a given set of alternative X as a function of individual preferences. The Gibbard-Satterthwaite theorem establishes that if preferences are unrestricted and the range of g has at least three members, only dictatorial rules are strategy-proof. However, if the domain of g is the set of profiles at which there exists a strong Condorcet winner, Campbell and Kelly have shown that majority-rule is the only non-dictatorial strategy-proof rule for an odd number of individuals when the range of g contains at least three alternatives. Dasgupta and Maskin consider the case of a continuum of voters as a means of circumventing the issue of parity. Although their analysis provides an approximation for a sufficiently large (but finite) set of individuals, no exact analysis exists for an arbitrary even number of individuals. We are therefore interested in characterizing the family of strategy-proof social choice rules over the Condorcet domain for an even number of individuals. We provide a full characterization when individual preferences are strict linear orderings, and prove several propositions concerning strategy-proof rules when individual preference orderings are permitted to be weak linear orders