Partitioned Matching Games for International Kidney Exchange

TL;DR

The paper introduces partitioned matching games as a new cooperative-game framework tailored to international kidney exchange, where each round’s transplant capacity must be fairly allocated among participating countries. It establishes deep links with the existing $b$-matching paradigm through polynomial reductions, enabling a unified view of core-related questions across these models. The authors prove a dichotomy: for width $c\le 2$ the core-related problems P1--P3 are solvable in polynomial time, while for $c\le 3$ core-emptiness and related tasks become co-NP-hard, with extensive hardness results for uniform and sparse cases. They also develop algorithmic tools, notably Lex-Min, to compute strongly close maximum matchings in the uniform case and provide a comprehensive exploration of the computational landscape, including exact equivalences to classic problems like Partition and 3-Partition in hard instances. The work thus offers both theoretical foundations and practical mechanisms for fair, stable distributions of transplants in IKEP simulations and related multi-agent settings.

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 $G=(V,E)$ 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.

Figures (8)

• Figure 1: An example BBJPX of a matching game $(N,v)$ on a graph $G=(V,E)$, so $N=V$. Note that $v(N)=7$ and that the core of $(N,v)$ is nonempty, e.g. the allocation $x=(\frac{1}{2}, \frac{3}{2}, \frac{3}{2}, 1, 2, \frac{1}{2})$ belongs to the core.
• Figure 2: Left: a directed compatibility graph $\overline{G}=(V,A)$ with a positive edge weighting $w$ for a certain round of an international KEP; note that $\overline{G}$ has a directed $4$-vertex cycle, so in the corresponding KEP even a $4$-way exchange could take place. Right: the corresponding undirected compatibility graph $G=(V,E)$ (which is used when only $2$-way exchanges are allowed).
• Figure 3: A vertex $v\in V$ with the three pendant triangles ($a_v,c_v,d_v$). Thick edges are edges in $M$. In this case, we also say that $v$ is matched "down" to all of $a_{v,1},a_{v,2},a_{v,3}$.
• Figure 4: Left: a $b$-matching game $(N,v)$ with six players, where $b\equiv 1$ apart from $b(2)=2$ and $b(5)=3$, so $b^*=3$. Note that $v(N)=10$ (take $M=\{12,35,45,56\}$). Right: the reduction to the partitioned matching game $(\overline{N},\overline{v})$. Note that $|\overline{N}|=14$ and $c=b^*$ (example taken from BBJPX.)
• Figure 5: Left: a partitioned matching game $(N,v)$ with three players and width $c=3$. Note that $v(N)=7$. Right: the reduction to the $b$-matching game $(\overline{N},\overline{v})$. Note that $|\overline{N}|=9$ and that for every $i\in \overline{N}$, $b(i)\leq c$ (example taken from BBJPX).

Theorems & Definitions (21)

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