  Stable matchings have been studied extensively in social choice literature.
The focus has been mostly on integral matchings, in which the nodes on the two
sides are wholly matched. A fractional matching, which is a convex combination
of integral matchings, is a natural extension of integral matchings. The topic
of stability of fractional matchings has started receiving attention only very
recently. Further, incentive compatibility in the context of fractional
matchings has received very little attention. With this as the backdrop, our
paper studies the important topic of incentive compatibility of mechanisms to
find stable fractional matchings. We work with preferences expressed in the
form of cardinal utilities. Our first result is an impossibility result that
there are matching instances for which no mechanism that produces a stable
fractional matching can be incentive compatible or even approximately incentive
compatible. This provides the motivation to seek special classes of matching
instances for which there exist incentive compatible mechanisms that produce
stable fractional matchings. Our study leads to a class of matching instances
that admit unique stable fractional matchings. We first show that a unique
stable fractional matching for a matching instance exists if and only if the
given matching instance satisfies the conditional mutual first preference
(CMFP) property. To this end, we provide a polynomial-time algorithm that makes
ingenious use of envy-graphs to find a non-integral stable matching whenever
the preferences are strict and the given instance is not a CMFP matching
instance. For this class of CMFP matching instances, we prove that every
mechanism that produces the unique stable fractional matching is (a) incentive
compatible and further (b) resistant to coalitional manipulations.
