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 = V1 ∪ • • • ∪ Vn. The player set is N = {1, . . . , n}, and player p ∈ N owns the vertices in Vp. The value v(S) of a coalition 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 |Vp| = 1 for all p ∈ N , then we obtain the classical matching game. Let c = max{|Vp| | 1 ≤ p ≤ n} be the width of (N, v). We prove that checking core non-emptiness is polynomial-time solvable if c ≤ 2 but co-NP-hard if 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 nonemptiness 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.
The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties-it always exists and lies in the core (if the core is nonempty), and it is unique. The nucleolus is considered as the most 'stable' solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n ≤ 15). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the 'balancedness' of at most (n − 1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.
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