  A common assumption in modern microeconomic theory is that choice should be
rationalizable via a binary preference relation, which \citeauthor{Sen71a}
showed to be equivalent to two consistency conditions, namely $\alpha$
(contraction) and $\gamma$ (expansion). Within the context of \emph{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,
$\hat\alpha$ and $\hat\gamma$, which are defined in analogy to $\alpha$ and
$\gamma$, and find that a choice function is set-rationalizable if and only if
it satisfies $\hat\alpha$. Moreover, a choice function satisfies $\hat\alpha$
and $\hat\gamma$ 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.
