On domains that admit well-behaved strategy-proof social choice functions

In this paper, we investigate domains which admit “well-behaved”, strategy-proof social choice functions. We show that if the number of voters is even, then every domain that satisfies a richness condition and admits an anonymous, tops-only, unanimous and strategy-proof social choice function, must be semi-single-peaked. Conversely every semi-single-peaked domain admits an anonymous, tops-only, unanimous and strategyproof social choice function. Semi-single-peaked domains are generalizations of singlepeaked domains on a tree introduced by Demange (1982). We provide sharper versions of the results above when tops-onlyness is replaced by tops-selectivity and the richness condition is weakened.