Core-Stable Kidney Exchange via Altruistic Donors

TL;DR

This work addresses core stability in multi‑organization kidney exchange by showing that the core can fail, but can be stabilized through a supplemented core that adds altruistic donors via a centralized platform. It develops two graph models (random and type‑based) and analyzes both pairwise and cyclic exchanges, deriving upper bounds that depend on network structure: the maximum number of independent odd cycles $ u(\\\mathcal{G})$ for pairwise graphs (with $O(\\\log |V|)$ in random graphs and $t/3$ in type representations) and a $( riangle-1)n(t+1)$ bound for cyclic exchanges with cycle limit $ riangle$. The authors provide a constructive approach using Scarf’s lemma and a rounding technique, supplemented by efficient integer‑programming heuristics to compute core‑stable allocations in practice, often needing far fewer altruists than worst‑case theory predicts. Empirically, simulations on realistic kidney‑exchange instances show that the weak/core is almost always nonempty, with minimal altruist requirements (often $<1\%$ of the patient pool) and only modest needs even for stronger core notions, while preserving lexicographic objectives used in real programs. Overall, altruistic donors can play a pivotal stabilizing role in large‑scale, multi‑institution kidney exchange without sacrificing efficiency or clinical goals.

Abstract

Kidney exchange programs among hospitals in the United States and across European countries improve efficiency by pooling donors and patients on a centralized platform. Sustaining such cooperation requires stability. When the core is empty, hospitals or countries may withhold easily matched pairs for internal use, creating incentive problems that undermine participation and reduce the scope and efficiency of exchange. We propose a method to restore core stability by augmenting the platform with altruistic donors. Although the worst-case number of required altruists can be large, we show that in realistic settings only a small number is needed. We analyze two models of the compatibility graph, one based on random graphs and the other on compatibility types. When only pairwise exchanges are allowed, the number of required altruists is bounded by the maximum number of independent odd cycles, defined as disjoint odd cycles with no edges between them. This bound grows logarithmically with market size in the random graph model and is at most one third of the number of compatibility types in the type-based model. When small exchange cycles are allowed, it suffices for each participating organization to receive a number of altruists proportional to the number of compatibility types. Finally, simulations show that far fewer altruists are needed in practice than worst-case theory suggests.

Figures (5)

• Figure 1: A partition exchange economy with pairwise exchanges that has no core but admits a $1$-supplemented core. The three organizations are represented by vertices with distinct shape (circle, triangle, square). The black star vertex in the upper-left corner represents $V^0$. The bold matching shows a $V^0$-supplemented core.
• Figure 2: The average number of altruists needed in each simulation setting.
• Figure 3: The maximum number of altruists needed in each simulation setting.
• Figure 4: The percentage of instances with no weak/strong/TU core in each setting. The TU core data points are not visible in the figures, because each data point exactly overlaps with the strong core case.
• Figure 5: An illustration for Theorem \ref{['thm:general-empty-core']} with $\Delta = 5$. A type-A graph is on the left and a type-B graph is on the right. The index $i$ is such that $i\equiv 0$ mod 2 and $i\equiv 1$ mod 3. Player 1 owns the red, player 2 the blue and player 3 the green vertices.