We introduce partitioned matching games as a suitable model for international kidney exchange programmes, where in each round the total number of available kidney transplants needs to be distributed amongst the participating countries in a "fair" way. A partitioned matching game $(N,v)$ is defined on a graph $G=(V,E)$ with an edge weighting $w$ and a partition $V=V_1 \cup \dots \cup V_n$. The player set is $N = \{1, \dots, n\}$, and player $p \in N$ owns the vertices in $V_p$. The value $v(S)$ of a coalition $S \subseteq N$ is the maximum weight of a matching in the subgraph of $G$ induced by the vertices owned by the players in $S$. If $|V_p|=1$ for all $p\in N$, then we obtain the classical matching game. Let $c=\max\{|V_p| \; |\; 1\leq p\leq n\}$ be the width of $(N,v)$. We prove that checking core non-emptiness is polynomial-time solvable if $c\leq 2$ but co-NP-hard if $c\leq 3$. We do this via pinpointing a relationship with the known class of $b$-matching games and completing the complexity classification on testing core non-emptiness for $b$-matching games. With respect to our application, we prove a number of complexity results on choosing, out of possibly many optimal solutions, one that leads to a kidney transplant distribution that is as close as possible to some prescribed fair distribution.

翻译：我们提出分区匹配博弈作为国际肾脏交换项目的适切模型，其中每轮可用的肾脏移植总数需以"公平"方式在参与国之间分配。分区匹配博弈$(N,v)$定义在带边权$w$的图$G=(V,E)$及顶点划分$V=V_1 \cup \dots \cup V_n$上。玩家集合为$N = \{1, \dots, n\}$，玩家$p \in N$拥有$V_p$中的顶点。联盟$S \subseteq N$的收益值$v(S)$为$G$中由$S$中玩家拥有的顶点所诱导子图内最大权匹配的权重。若对所有$p\in N$均有$|V_p|=1$，则退化为经典匹配博弈。令$c=\max\{|V_p| \; |\; 1\leq p\leq n\}$为$(N,v)$的宽度。我们证明当$c\leq 2$时核心非空性可在多项式时间内判定，而$c\leq 3$时则为co-NP难问题。这一结果通过揭示其与已知的$b$-匹配博弈类的关联，并完成$b$-匹配博弈核心非空性检验的复杂度分类而获得。针对实际应用，我们证明了在诸多最优解中选择尽可能接近预定义公平分配的肾脏移植分布方案时的若干复杂度结论。