How an Economist Solved the Kidney Exchange Market Design Problem

There's no shortage of kidneys because everyone has two, but the price is zero by law. Making markets illegal doesn't make them go away—black markets exist in many places. Repugnant transactions include same-sex marriage, slavery, indentured servitude, and interest on loans—things that some people want to engage in while others don't think they should be allowed. Many transactions become repugnant when money is added. Most people have some sense that there are times when money shouldn't be involved, even if they disagree about which specific transactions. Market design involves understanding these boundaries and making incremental improvements to existing systems.

Repugnant transactions are activities some people want to engage in while others believe should be prohibited. Examples include prostitution, surrogacy, and paid kidney donation. Laws banning such transactions often fail because they create black markets operating outside legal oversight. In surrogacy cases, Swedish couples may travel to California (where it's legal) to have children, then face difficulties returning with offspring. Similarly, drug prohibition creates black markets where quality control is absent, leading to higher consumer risks. The key insight is that prohibition may achieve aesthetic goals at significant economic and human cost. Paid plasma donation presents another interesting case: countries prohibiting payment rely on voluntary donation, which studies show is actually more costly than paid programs due to marketing and outreach expenses. The United States pays plasma donors and exports billions annually in plasma-derived pharmaceutical products. Countries prohibiting payment must import these products at higher cost. This suggests that prohibition may achieve aesthetic goals at significant economic cost, though ethical considerations about commodifying body parts remain contentious.

The addition of money to otherwise morally acceptable transactions transforms them into 'taboo trades' or 'repugnant markets,' as demonstrated by the stark contrast between free kidney donation (heroic) and commercial kidney sales (criminal). This transformation occurs because money introduces perceived corruption, exploitation, and societal degradation concerns, even when empirical evidence of actual harm is lacking. The term 'repugnant' is problematic because it suggests irrational emotional reactions, whereas 'taboo' better captures the socially constructed nature of these moral boundaries.

Repugnant transactions are those some want to engage in but others believe shouldn't—examples include kidney sales, surrogacy, and prostitution. What's repugnant varies culturally: surrogacy is illegal in Germany but legal in California; prostitution is legal in Nevada but illegal in most U.S. states. Market designers must consider not just how to organize existing markets but which markets society should allow. Implementation faces political challenges—hospitals have financial incentives to keep transplants local, and referring patients for transplantation takes time away from dialysis revenue. Success requires convincing stakeholders they'll do better as everyone else does better. The kidney exchange story shows that victories in increasing transplant numbers are part of a larger war being lost against the growing waiting list.

This section explores the tension between economically efficient solutions and societal moral objections. Kidney sales represent a classic repugnant transaction—potentially saving lives through increased supply but violating moral intuitions. Iran's legal kidney market provides a natural experiment, though evidence is limited. The discussion extends to healthcare economics, noting that Medicare mandates dialysis regardless of cost, creating substantial waste. A proposed reform redirects savings from reduced dialysis toward incentivizing donations. The broader lesson is that changing public attitudes toward repugnant transactions takes time, and policy must balance efficiency against moral concerns while acknowledging that different people draw ethical lines at different points.

The maximum matching problem can be formulated as an integer linear program: (1) Variables: Xi = 1 if edge i is in the matching, 0 otherwise; (2) Objective: Maximize sum of Xi (size of matching); (3) Constraints: For each pair of adjacent edges (e1,e2), X1 + X2 ≤ 1 (cannot select both); (4) Integrality: Xi ∈ {0,1}. This demonstrates that many combinatorial optimization problems can be expressed as LPs.

The dynamic matching problem requires maintaining a large node-disjoint set of edges (matching) as edges are added or deleted. The approximation ratio measures how close the maintained matching is to the maximum matching size μ(G). Fractional matchings relax the integrality constraint by assigning non-negative values to edges such that each vertex's incident values sum to at most one. The quality is measured by the L1 norm divided by μ(G). Known results include slightly worse-than-half approximation in poly-log time, better-than-half in sublinear time, and one-minus-epsilon approximations in decremental settings. The key challenge is converting these fractional solutions to integral matchings efficiently.

Solomon's local sparsification marks edges probabilistically to achieve (1+ε)-approximation with O(√m) update time. Kernels impose degree constraints on excluded edges, achieving exactly 2-approximation. A (b,ε)-kernel has max degree b and excludes edges where neither endpoint has near-b degree. Any kernel contains a (2+ε)-approximate matching through fractional matching analysis and integrality gap arguments. Two adversary models exist: oblivious adversaries determine sequences in advance, while adaptive adversaries choose updates based on execution history. Using dynamic fractional matching algorithms combined with rounding techniques, we achieve (2+ε)-approximate matchings with constant amortized randomized time against adaptive adversaries, and polylogarithmic worst-case deterministic time. This demonstrates how fractional matching maintenance enables practical dynamic matching algorithms.

The course addresses four main problems: maximum matching in bipartite graphs, maximum matching in general graphs, minimum cost perfect matching in bipartite graphs, and minimum cost perfect matching in general graphs. Bipartite graphs are easier than general graphs. The maximum-minimum principle underlies many efficiently solvable combinatorial optimization problems. Linear programming connects to these problems. The course covers classical results from the 1970s-80s.

In dynamic matching problems where vertices arrive online and must be matched within a fixed time window, the batching algorithm achieves a competitive ratio of approximately 0.27 (1/3.58) against the optimal offline solution, and this bound is tight since no algorithm can achieve better than 0.5 competitive ratio; under adversarial arrival models, randomized algorithms using patient/impatient vertex classification achieve 0.25 competitive ratio, with the optimal competitive ratio remaining an open question between 0.25 and 0.5.

School choice involves centralized mechanisms matching students to K-12 schools. Students rank schools in preference order; schools have priority orders (walk zone, sibling priority) and capacity constraints. Key design goals include: (1) Individual rationality (no unacceptable assignments), (2) Stability (no justified envy), (3) Strategy proofness (truthful preferences are dominant), (4) Pareto efficiency, and (5) Respecting improvements of school quality. The deferred acceptance algorithm produces stable matches by having students propose to first choices, schools tentatively hold best applicants, and rejected students move to next choices.

School choice systems initially suffered from strategic manipulation where schools could see students' rank order lists, incentivizing families to misrepresent preferences. The deferred acceptance algorithm solves this by having students apply to first-choice schools, with schools tentatively accepting without immediate rejection. Rejected students then apply to subsequent choices while schools re-sort all applicants by priority (sibling status, proximity) without considering application timing. Before computerized clearinghouses, New York City left 30,000 students unmatched annually who had to be administratively assigned to schools. After implementation, only 3,000 remained unmatched. The system eliminated incentives for principals to withhold capacity, effectively increasing available school seats.

This section introduces mechanism design, a field addressing how to allocate scarce resources fairly when markets cannot function. The speaker explains that school choice mechanisms allocate students to schools based on preferences and qualifications. The Boston mechanism is presented as a simple approach that assigns students to their first-choice school if they qualify, then proceeds to subsequent choices. However, this mechanism can produce inefficient outcomes where students who strategically misrepresent preferences end up with worse assignments. The concept of justified envy is introduced, occurring when a student who was not assigned to a preferred school could have been assigned if they had reported different preferences. The deferred acceptance mechanism (Gale-Shapley algorithm) is presented as a solution achieving both Pareto efficiency and strategy-proofness. The section explains that no mechanism can simultaneously achieve strategy-proofness, Pareto efficiency, and stability, highlighting fundamental trade-offs in mechanism design.

School choice involves centrally-run matching systems where students submit ranked preferences for schools outside their local area. Priority rules resolve conflicts: students with siblings AND in walk zones have highest priority, followed by siblings-only, walk-zone-only, and then others. Random numbers break ties within priority classes. A matching assigns each student to a school without exceeding quotas. A student's priority is violated if they prefer a school with an empty seat or if a preferred school has a student with lower priority. Stability requires no priority violations. The student optimal stable matching exists and is the best stable outcome for students, but it is typically not Pareto efficient. This creates a fundamental tension between stability and efficiency that motivates research into alternative matching mechanisms.

Mechanism design applies mathematical principles to create fair allocation systems in complex markets. The American medical school admissions market was redesigned using a centralized computerized clearing house after unraveling caused early offers and strategic behavior. Students now submit ranked preferences alongside employers, with a computer algorithm matching everyone simultaneously on Match Day. Similarly, New York City redesigned its school choice system to prevent strategic manipulation: previously, families couldn't safely list their true first-choice schools because rejection meant losing access to second choices. The new mechanism ensures equal likelihood of getting any choice regardless of its ranking, enabling honest preference expression and improving outcomes for all participants.