The Gibbard random dictatorship theorem: A generalization and a new proof

This paper proves stronger versions of the Gibbard random dictatorship theorem using induction on the number of voters. It shows that when there are at least three voters, every random social choice function defined on a domain satisfying a Free Triple at the Top property and satisfying a weak form of strategy-proofness called Limited-Comparison Strategy-proofness and Unanimity, is a random dictatorship provided that there are at least three alternatives. The weaker notion of strategy-proofness requires truth-telling to maximize a voter's expected utility only for a limited class of von NeumannMorgenstern utility representations of the voter's true preference ordering. In the case of two voters, an even weaker condition on the domain and a weaker notion of strategy-proofness are sufficient for the random dictatorship result.