With as few as eight individuals and five alternatives, there are 561, 304, 372, 286, 875, 579, 077, 983 strategy-proof social choice rules.

There are two feasible alternatives. We impose a weak version of anonymity on an arbitrary social choice function, which selects a single alternative at each profile. It implies majority rule in the presence of neutrality and monotonicity.

X is infinite and social preference is quasitransitive. Subset Y of X is sufficient for {x,y} if x and y can be socially ordered with individual preference information over Y alone. If there is an integer β such that every pair of alternatives has a sufficient set with at most β members then...

The Condorcet rule on the domain of profiles at which there exists a unique Condorcet winner is the unique rule satisfying anonymity, neutrality, and strategy-proofness.