[Paper Review] Partitioned Matching Games for International Kidney Exchange

Márton Benedek, Péter Bíró+3 more|arXiv (Cornell University)|Jan 30, 2023

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper introduces partitioned matching games as a novel cooperative game-theoretic model for international kidney exchange programs, where countries collaborate to maximize transplant outcomes under fairness constraints. It proves that checking core non-emptiness is polynomial-time solvable when each country has at most two donors (width ≤ 2), but becomes co-NP-hard when widths exceed 2, establishing a sharp complexity threshold for stability in international transplant coordination.

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.

Motivation & Objective

• To model international kidney exchange programs as cooperative games where countries pool donor-patient pairs to maximize transplants.
• To ensure stability of international collaborations by analyzing core non-emptiness and fair allocation mechanisms.
• To establish the computational complexity of core-related problems in partitioned matching games, especially with respect to the width of the partition.
• To link partitioned matching games to b-matching games and resolve a gap in the complexity classification of core computation for b-matching games.
• To investigate the feasibility of finding matchings close to a prescribed fair distribution of transplants across countries.

Proposed method

• Define a partitioned matching game (N,v) on a weighted graph G=(V,E) with vertex partition V=V₁∪⋯∪Vₙ, where each player p owns vertices in Vₚ.
• Define the value v(S) of a coalition S⊆N as the maximum weight of a matching in the subgraph induced by vertices owned by players in S.
• Establish a connection between partitioned matching games and b-matching games by transforming the graph to model b-matching constraints.
• Use a reduction from the 3-Partition problem to prove co-NP-hardness of core non-emptiness when the width c≥3.
• Prove polynomial-time solvability of core non-emptiness when c≤2 via a reduction to b-matching games with b≤2.
• Apply the ellipsoid method to show that if core non-emptiness is decidable in polynomial time, then core allocation and core membership testing are also polynomial-time solvable.

Experimental results

Research questions

• RQ1 Under what conditions is the core of a partitioned matching game non-empty, and how does this depend on the maximum size of any player’s vertex set (width c)?
• RQ2 Is there a polynomial-time algorithm to determine whether a given allocation is in the core for partitioned matching games?
• RQ3 Can a maximum weight matching be found that is weakly close to a prescribed fair distribution of transplants across countries?
• RQ4 What is the computational complexity of core-related problems (core non-emptiness, membership, and allocation) for partitioned matching games of bounded width?
• RQ5 How are partitioned matching games related to b-matching games, and can results on one class be transferred to the other?

Key findings

• Checking core non-emptiness for partitioned matching games is polynomial-time solvable when the width c≤2.
• Checking core non-emptiness becomes co-NP-hard when the width c≥3, establishing a sharp threshold at width 3.
• The paper resolves a complexity gap in b-matching games by showing that core non-emptiness is co-NP-hard for b≥3, while polynomial for b≤2.
• The problem of finding a maximum weight matching that is weakly close to a prescribed fair allocation is polynomial-time solvable when edge weights are uniform (i.e., all weights equal).
• For non-uniform weights, the problem of finding a weakly close maximum weight matching becomes NP-hard, even for sparse graphs.
• The results imply that core-related problems (P1–P3) are co-NP-hard for partitioned matching games of width c≥3, and likely co-NP-complete, though this remains open.