Motivated by the real-world problem of international kidney exchange (IKEP), recent literature introduced a generalized transferable utility matching game featuring a partition of the vertex set of a graph into players, and analyzed its complexity. We explore the non-transferable utility (NTU) variant of the game, where the utility of players is given by the number of their matched vertices. Our motivation for studying this problem is twofold. First, the NTU version is arguably a more natural model of the international kidney exchange program, as the utility of a participating country mostly depends on how many of its patients receive a kidney, which is non-transferable by nature. Second, the special case where each player has two vertices, which we call the NTU matching game with couples, is interesting in its own right and has intriguing structural properties. We study the core of the NTU game, which suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players. We prove computational complexity results about the weak and strong cores under various assumptions on the players. In particular, we show that if every player has two vertices, then the weak core is always nonempty, and the existence of a strong core solution can be decided in polynomial time. Moreover, one can efficiently optimize on the strong core. In contrast, it is NP-hard to decide whether the strong core is empty when each player has three vertices. We also show that if the number of players is constant, then the non-emptiness of the weak and strong cores is polynomial-time decidable, and we can find a minimum-cost core solution in polynomial time.

翻译：受国际肾脏交换（IKEP）这一现实问题的启发，近期文献引入了一种广义的可转移效用匹配博弈，其特点是将图的顶点集划分为若干参与者，并分析了其计算复杂性。我们探讨该博弈的非可转移效用（NTU）变体，其中参与者的效用由其匹配的顶点数量决定。我们研究此问题的动机有两点。首先，NTU版本可以说是国际肾脏交换计划更自然的模型，因为参与国的效用主要取决于其有多少患者获得肾脏，这在本质上是不可转移的。其次，每个参与者具有两个顶点的特殊情况（我们称之为带配偶的NTU匹配博弈）本身具有研究价值，并展现出有趣的结构性质。我们研究NTU博弈的核心，它恰当地捕捉了IKEP的稳定性概念，因为它排除了任何可能的参与者联盟偏离所提出解决方案的动机。我们在不同参与者假设下证明了关于弱核心与强核心的计算复杂性结果。特别地，我们证明若每个参与者有两个顶点，则弱核心始终非空，且强核心解的存在性可在多项式时间内判定。此外，可以在强核心上进行高效优化。相比之下，当每个参与者有三个顶点时，判定强核心是否为空是NP难的。我们还证明，若参与者数量为常数，则弱核心与强核心的非空性可在多项式时间内判定，并且我们能在多项式时间内找到最小成本的核心解。