A Study of Incentive Compatibility and Stability Issues in Fractional Matchings
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Stable matchings have been studied extensively in both economics and computer science literature. However, most of the work considers only integral matchings. The study of stable fractional matchings is fairly recent and moreover, is scarce. This paper reports the first investigation into the important but unexplored topic of incentive compatibility of matching mechanisms to find stable fractional matchings. We focus our attention on matching instances under strict preferences. First, we make the significant observation that there are matching instances for which no mechanism that produces a stable fractional matching is incentive compatible. We then characterize restricted settings of matching instances admitting unique stable fractional matchings. Specifically, we show that there will exist a unique stable fractional matching for a matching instance if and only if the given matching instance satisfies what we call the conditional mutual first preference property (CMFP). For this class of instances, we prove that every mechanism that produces the unique stable fractional matching is (a) incentive compatible and (b) resistant to coalitional manipulations. We provide a polynomial-time algorithm to compute the stable fractional matching as well. The algorithm uses envy-graphs, hitherto unused in the study of stable matchings.
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