  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.
