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合适模型，其中每轮可用的肾脏移植总数需要以"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$G=(V,E)$和划分$V=V_1 \cup \dots \cup V_n$上。玩家集$N = \{1, \dots, n\}$，玩家$p \in N$拥有$V_p$中的顶点。联盟$S \subseteq N$的收益$v(S)$是$G$中由$S$中玩家拥有的顶点诱导子图的最大匹配权重。若对所有$p\in N$均有$|V_p|=1$，则得到经典匹配博弈。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，核非空性检测可在多项式时间内求解，但当$c\leq 3$时则为co-NP难问题。这是通过精确定位与已知的$b$-匹配博弈类的关系，并完善$b$-匹配博弈核非空性检测的复杂度分类来实现的。针对我们的应用场景，我们证明了在可能存在的多个最优解中选择一个能使肾脏移植分配尽可能接近指定公平分配方案的一系列复杂度结果。