Condorcet winners and social acceptability

We say that an alternative is socially acceptable if the number of individuals who rank it among their most preferred half of the alternatives is at least as large as the number of individuals who rank it among the least preferred half. A Condorcet winner may not necessarily be socially acceptable. However, if preferences are single-peaked, single-dipped, or satisfy the single-crossing property, any Condorcet winner is socially acceptable. We identify maximal families of preferences that guarantee that Condorcet winners are socially acceptable.