The NTU Partitioned Matching Game for International Kidney Exchange Programs

Motivated by the real-world problem of international kidney exchange (IKEP), [Bir\'o et al., Generalized Matching Games for International Kidney Exchange, 2019] 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. 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. We study the core of this 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, which can be considered as an NTU matching game with couples, then the weak core is always non-empty, and the existence of a strong core solution can be decided in polynomial time. 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.