Strategy-proofness and Single-crossing I Am Grateful to Two Anonymous Referees and the Coeditor, Martin Osborne, for Helpful Comments and Corrections on an Earlier Draft. I Also Thank

This paper analyzes strategy-proof collective choice rules when individuals have single-crossing preferences on a finite and ordered set of social alternatives. It shows that a social choice rule is anonymous, unanimous, and strategy-proof on a maximal single-crossing domain if and only if it is an extended median rule with n − 1 fixed ballots distributed over the individuals' most preferred alternatives. As a by-product, the paper also proves that strategy-proofness implies the tops-only property. It also offers a strategic foundation for the so-called " single-crossing version " of the Median Voter Theorem, by showing that the median ideal point can be implemented in dominant strategies by a direct mechanism in which every individual reveals his true preferences. 1. I In social choice theory, a collective decision making process is usually represented by a social choice rule. A social choice rule associates a unique alternative from the set of feasible alternatives to every possible list of preferences of the individuals in the society. A social choice rule is said to be strategy-proof if no individual can ever benefit from misrepresenting his true preferences. A fundamental result in social choice theory , known as the Gibbard (1973)–Satterthwaite (1975) Theorem, shows that, if the set of alternatives contains at least three possible outcomes and individual preferences are not restricted in any particular way, then every strategy-proof social choice rule is dictatorial. That is, there is an individual whose preferences always dictate the final choice regardless of the other individuals' preferences. The Gibbard–Satterthwaite Theorem holds under the so called universal domain assumption , which means that every profile of complete and transitive preference relations is an admissible element of the domain of the social choice rule. In many economic and political applications, however, preferences satisfy additional properties. A case in point is the single-peaked property.