Core-Stable Kidney Exchange via Altruistic Donors

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.

💡 Research Summary

The paper tackles a fundamental stability problem in multi‑organization kidney exchange markets, where hospitals or national programs may withhold easily matched donor‑patient pairs, causing the core of the exchange game to be empty. An empty core means that some coalition of organizations can improve its outcome by running a separate exchange, threatening participation and overall efficiency.

To remedy this, the authors introduce the notion of a supplemented core: the central exchange platform is allowed to add a limited number of altruistic (non‑directed) donors to the pool. In the expanded compatibility graph that includes these altruists, a core allocation can be found, guaranteeing that no coalition can profit by deviating.

The paper studies two widely used models of the compatibility graph:

1. Random graph model – donor‑patient pairs are assigned to a finite set of types (blood group, PRA level, etc.) and edges are drawn independently with type‑specific probabilities.
2. Type‑representation model – the graph is described deterministically by a small number t of compatibility types.

For pairwise exchanges only (cycles of length two), the authors prove that the number of altruists needed is exactly the maximum number of independent odd cycles in the mutual‑compatibility graph. An independent odd cycle is a vertex‑disjoint odd‑length cycle that has no mutual‑compatibility edges to any other such cycle. In the random‑graph setting this quantity grows only as O(log |V|), while in the type‑based setting it is bounded by ⌊t/3⌋. Consequently, realistic markets (with a handful of types) require at most a few altruists, often just one or two.

When short cycles of bounded length Δ (e.g., 3‑ or 4‑cycles) are allowed, the required altruists per organization drop to (t + 1)(Δ − 1). Thus, if the number of types is fixed, each organization needs only a constant number of altruists, and the total number of altruists for the whole market remains constant when the number of participating organizations is also bounded (as is typical for European cross‑border exchanges).

The authors also establish worst‑case lower bounds: without any structural assumptions, Ω(|V|) altruists may be necessary, showing that the positive results rely on realistic graph properties.

From a computational perspective, the existence proof relies on Scarf’s lemma, which is not directly constructive. The paper therefore proposes a practical heuristic based on integer programming: pre‑enumerate feasible cycles, limit the size of blocking coalitions, iteratively add coalition‑blocking constraints, and inject altruists whenever infeasibility is detected. This approach solves instances with up to 500 pairs in seconds on modest hardware, even when restricting attention to coalitions of up to four organizations.

Extensive simulations using data‑driven generators that mimic real blood‑type distributions and PRA incompatibilities confirm the theory. In over 97 % of randomly generated instances the weak core is already non‑empty; in the rare cases where it is empty, a single altruistic donor restores core stability. Even for the stricter strong core and transferable‑utility core, fewer than 1 % of the patient pool in altruists suffices. Moreover, imposing the usual lexicographic objectives (maximize total transplants, then minimize cycle length, then prioritize same‑type matches) before adding altruists incurs virtually no loss in objective value; typically fewer than three altruists are needed to achieve a core‑stable outcome.

The paper’s findings have clear policy implications. Although current U.S. and European programs involve only a few hundred altruistic donors compared with several thousand patient‑donor pairs, the analysis shows that strategically placed altruists can dramatically improve market stability without sacrificing efficiency. This suggests that exchange platforms should actively recruit and allocate altruistic donors not merely to extend chains but also to enforce coalition‑proofness.

In summary, the work provides (i) a rigorous characterization of how many altruistic donors are sufficient for core stability under realistic graph models, (ii) a scalable algorithmic framework to compute such stable allocations, and (iii) empirical evidence that the required number of altruists is negligible in practice. These contributions bridge a gap between theoretical cooperative‑game stability and the operational design of modern kidney exchange programs.