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.

中文翻译：

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分数匹配中激励兼容性和稳定性问题的研究

在经济学和计算机科学文献中都对稳定匹配进行了广泛的研究。然而，大部分工作只考虑整数匹配。稳定分数匹配的研究是最近才出现的，而且很少。本文报告了对匹配机制的激励兼容性这一重要但尚未探索的主题的首次调查，以找到稳定的分数匹配。我们将注意力集中在严格偏好下的匹配实例上。首先，我们进行了重要的观察，即存在没有产生稳定分数匹配的机制是激励兼容的匹配实例。然后，我们描述了允许唯一稳定分数匹配的匹配实例的受限设置。具体来说，我们表明，当且仅当给定的匹配实例满足我们所谓的条件互优先属性（CMFP）时，匹配实例才会存在唯一的稳定分数匹配。对于这类实例，我们证明了每个产生独特稳定分数匹配的机制都是（a）激励兼容和（b）抵抗联盟操纵。我们还提供了一个多项式时间算法来计算稳定的分数匹配。该算法使用了迄今为止在稳定匹配研究中未使用的嫉妒图。我们证明了每个产生独特稳定分数匹配的机制都是（a）激励兼容和（b）抵抗联盟操纵。我们还提供了一个多项式时间算法来计算稳定的分数匹配。该算法使用了迄今为止在稳定匹配研究中未使用的嫉妒图。我们证明了每个产生独特稳定分数匹配的机制都是（a）激励兼容和（b）抵抗联盟操纵。我们还提供了一个多项式时间算法来计算稳定的分数匹配。该算法使用了迄今为止在稳定匹配研究中未使用的嫉妒图。