Abstract 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 $$G=(V,E)$$ G = ( V , E ) with an edge weighting w and a partition $$V=V_1 \cup \dots \cup V_n$$ V = V 1 ∪ ⋯ ∪ V n . The player set is $$N = \{ 1, \dots , n\}$$ N = { 1 , ⋯ , n } , and player $$p \in N$$ p ∈ N owns the vertices in $$V_p$$ V p . The value v ( S ) of a coalition $$S \subseteq N$$ S ⊆ N is the maximum weight of a matching in the subgraph of G induced by the vertices owned by the players in S . If $$|V_p|=1$$ | V p | = 1 for all $$p\in N$$ p ∈ N , then we obtain the classical matching game. Let $$c=\max \{|V_p| \; |\; 1\le p\le n\}$$ c = max { | V p | | 1 ≤ p ≤ n } be the width of ( N , v ). We prove that checking core non-emptiness is polynomial-time solvable if $$c\le 2$$ c ≤ 2 but co--hard if $$c\le 3$$ c ≤ 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.