  We study a generalization of the classical stable matching problem that
allows for cardinal preferences (as opposed to ordinal) and fractional
matchings (as opposed to integral). After observing that, in this cardinal
setting, stable fractional matchings can have much higher social welfare than
stable integral ones, our goal is to understand the computational complexity of
finding an optimal (i.e., welfare-maximizing) or nearly-optimal stable
fractional matching. We present simple approximation algorithms for this
problem with weak welfare guarantees and, rather unexpectedly, we furthermore
show that achieving better approximations is hard. This computational hardness
persists even for approximate stability. To the best of our knowledge, these
are the first computational complexity results for stable fractional matchings.
En route to these results, we provide a number of structural observations.
