On the Coexistence Of  Stability and Incentive Compatibility in Fractional Matchings

The topic of stability of fractional matchings has started receiving attention only very recently with incentive compatibility in this context receiving scarce attention. Our paper studies the incentive compatibility of mechanisms finding stable fractional matchings. Agent preferences are expressed as cardinal utilities. We exhibit matching instances for which no stable fractional matching mechanism is approximately incentive compatible. We then characterize the class of matching instances with unique stable fractional matchings. We first show that a unique stable fractional matching exists if and only if the given matching instance satisfies the conditional mutual first preference (CMFP) property. To this end, we provide an algorithm that ingeniously uses envy-graphs finding a non-integral stable matching whenever the preferences are strict and the given instance is not in CMFP. For this class of CMFP matching instances, we prove that every mechanism that produces the unique stable fractional matching is incentive compatible.