Incentive properties for ordinal mechanisms

We study three incentive properties for ordinal mechanisms: (i) strategy-proofness, which requires that no agent gain by misrepresenting his preferences; (ii) adjacent strategy-proofness, which requires that no agent gain by switching the rankings of two adjacent alternatives; and (iii) mistake monotonicity, which requires that the welfare of each agent weakly decrease as he reports increasingly bigger mistakes. Each of these properties has three versions, depending on whether preferences over sure alternatives are extended to preferences over lotteries by the stochastic dominance, downward lexicographic, or upward lexicographic extension. We identify conditions on the preference domain that guarantee the equivalence of these properties. The universal domain and the domains of single-dipped and single-peaked preferences satisfy our conditions.