Set-Rationalizable Choice and Self-Stability

One of the fundamental assumptions in modern microeconomic theory is that choice should be rationalizable via a binary preference relation. \citeauthor{Sen71a} showed that rationalizability is equivalent to two consistency conditions on choice, namely $\alpha$ (contraction) and $\gamma$ (expansion). Within the context of social choice, however, rationalizability and similar notions of consistency have proved to be highly problematic, as witnessed by a range of impossibility results, among which Arrow's is the most prominent. Since choice functions select \emph{sets} of alternatives rather than single alternatives, we propose to rationalize choice functions by preference relations over sets (set-rationalizability). We also introduce two consistency conditions, $\widehat\alpha$ and $\widehat\gamma$, which are defined in analogy to $\alpha$ and $\gamma$, and find that a choice function is set-rationalizable if and only if it satisfies $\widehat\alpha$. Moreover, a choice function satisfies $\widehat\alpha$ and $\widehat\gamma$ if and only if it is \emph{self-stable}, a new concept based on earlier work by \citeauthor{Dutt88a}. The class of self-stable social choice functions contains a number of appealing Condorcet extensions such as the minimal covering set and the essential set, yet excludes other well-known Condorcet extensions as well as all scoring rules.