Set-rationalizable choice and self-stability

Rationalizability and similar notions of consistency have proved to be highly problematic in the context of social choice, as witnessed by a range of impossibility results, among which Arrow's is the most prominent. We propose to rationalize choice functions by preference relations over sets of alternatives (set-rationalizability) and introduce two consistency conditions, α and γ, which are defined in analogy to Sen's α and γ. We find that a choice function satisfies α if and only if it is set-rationalizable and that it satisfies α and γ if and only if it is self-stable, a new concept based on earlier work by Dutta. The class of self-stable social choice functions contains a number of appealing Condorcet extensions.