Majority judgment and majority criterion

Majority criterion says that an alternative should not be chosen if there is another alternative which majority of voters prefer to it. It is well known that this criterion is too strong: there is no social choice rule that satisfies it when there are at least three alternatives. In this paper, we show that majority judgment, reformulated as a social choice rule, satisfies a weaker variant of majority criterion, referred to as shuffling majority criterion. In addition, we show that if a social choice rule satisfies this axiom and another one concerning non-manipulability, it should pick only those alternatives with the highest "median grade".