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Econ-178 · economics · Economics

Scaling kidney exchange

LLM status Open unverified

Statement

Original subject group: Mechanism design and market design

Audit classification: Published research agenda. Roth 2010 p.4 identifies center participation, and Ashlagi–Roth §6 item 2 explicitly asks about international collaboration. The weighted withholding mechanism is editorial.

Definitions and precise research target

Model and research target. Let countries or transplant centers $h=1,\ldots,H$ own sets $V_h$ of incompatible patient–donor pairs. A directed compatibility graph $G=(V,E)$, with $V=\bigcup_hV_h$, has edge $i\to j$ when donor $i$ can medically donate to patient $j$. Feasible exchanges are disjoint cycles of length at most $K$, together with permitted chains from nondirected donors, satisfying explicit cross-border shipping and surgery-capacity constraints. Each center may withhold pairs and conduct internal exchanges. Let $U_h(M)$ be its expected number of transplanted patients under mechanism $M$, and $U_h^{\mathrm{outside}}$ its attainable internal alternative. Fix nonnegative patient weights $w_i$ representing an openly stated equity objective. Determine mechanisms maximizing \[ E\left[\sum_{i\in V}w_i\mathbf1\{i\text{ receives a transplant}\}\right] \] while ensuring $U_h(M)\geq U_h^{\mathrm{outside}}$ and discouraging profitable pair withholding, under a specified stochastic compatibility and arrival model. Quantify separately gains from larger scale and gains from differences in biological compatibility distributions. Include exchange failures and the absence of unrestricted monetary transfers. Characterize the efficiency or equity loss needed for sustainable institutional participation and test mechanisms against realistic pool composition.

Definitions and maintained assumptions (editorial unless a source definition is named). Pair vertices are partitioned by owning centers. Add nondirected-donor vertices with no paired recipient and terminal-recipient vertices where chains may end. A feasible plan is a vertex-disjoint collection of directed cycles of length at most $K$ and directed chains with declared length, timing and surgery/shipping limits; failure realizations alter which transplants actually occur. Specify stochastic arrivals, compatibility and failure distributions and the information each center observes before submitting a subset $S_h\subseteq V_h$. A mechanism maps submissions to randomized feasible plans. Full-participation dominance means $\mathbb E[U_h(V_h,S_{-h})]\ge\mathbb E[U_h(S_h,S_{-h})+U_h^{\rm withheld}(V_h\setminus S_h)]$ for every center, subset and other submissions, under the same permitted internal-exchange timing. Interim incentive compatibility averages instead over other centers’ information. An internal outside option must be computed under identical medical and resource constraints. Patient weights are announced equity primitives; centers’ transplant-count objectives need not share them. Size and biological-heterogeneity effects require separate counterfactual pool constructions.

Variants and assumption changes

  1. One-shot center incentives (editorial). Fix graph distributions and cycle cap, prohibit money and require dominant full submission. Determine the attainable weighted-efficiency loss relative to a centralized complete-information benchmark.
  2. Dynamic international pool (source direction, editorial model). Allow arrivals, exchange failures and center exit with continuation outside options. Find participation-sustaining collaboration rules and compare matched-size pools to isolate biological heterogeneity gains.

References and source audit

  1. Roth §1 p.4; Ashlagi–Roth §6 item2 p.5467. Supports the stated research area; exact displayed specification is editorial. https://web.stanford.edu/~alroth/papers/NSF%20Grand%20Challenge.SBE%202020.%20Market%20Design.pdf
  2. Roth §1 p.4; Ashlagi–Roth §6 item2 p.5467. Supports the stated research area; exact displayed specification is editorial. https://pubsonline.informs.org/doi/pdf/10.1287/mnsc.2020.3954
  1. Alvin E. Roth. 2010. Market Design: Understanding markets well enough to fix them when they’re broken. §1 Fundamental questions, pp.3–4. https://web.stanford.edu/~alroth/papers/NSF%20Grand%20Challenge.SBE%202020.%20Market%20Design.pdf.
  2. Itai Ashlagi; Alvin E. Roth. 2021. Kidney Exchange: An Operations Perspective. §6 Open Questions and Research Directions, item2 International Exchanges, p.5467. https://pubsonline.informs.org/doi/pdf/10.1287/mnsc.2020.3954. DOI: https://doi.org/10.1287/mnsc.2020.3954.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-178-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-178

Scaling kidney exchange Participation guarantees, equity loss, and composition accounting

Problem ID: D47-174. Status: Unresolved; restricted results and reductions.

Research target and the declared one-batch model

The full target combines institutional withholding incentives, equity, exchange failures, international constraints, and dynamic participation. Roth's white paper [roth, p. 4] identifies full center submission as an institutional problem; Ashlagi and Roth [ar, Section 6, item 2, p. 5467] explicitly separate scale from biological heterogeneity. Neither source is a theorem asserting the mechanism studied below.

We analyze a restricted, completely specified model. All pairs arrive in one batch, the realized compatibility graph is known to the planner, only cycles of length at most $K\geq2$ are permitted, and there are no chains. Resources allow any collection of vertex-disjoint cycles; there is no shared capacity constraint beyond vertex disjointness. Each selected cycle succeeds in its entirety with the same probability $s\in(0,1]$, and otherwise makes no transplant. Cycle failures may be correlated, but their marginal probability is $s$ under both internal and pooled operation. A plan is selected before failures and no repair is attempted. These assumptions make expected transplant counts equal to $s$ times planned counts. There are no monetary transfers.

Center $h$ owns $V_h$. Let $O_h$ be a maximum-cardinality internal cycle packing and $b_h=|V(O_h)|$. Its outside option is $s b_h$. The union $O=\bigcup_hO_h$ is feasible, and write $b=\sum_hb_h$. Computing these packings may itself be hard for $K\geq3$; the guarantees below are structural and do not assert a polynomial clearing algorithm.

A guaranteed-participation benchmark

Reserve the cycles $O$, delete their vertices, and choose a maximum-cardinality packing $Q$ in the residual graph. Output $O\cup Q$.
Theorem (A cycle-cap bound on the cost of reservations): For every realized graph in the stated model, the reservation plan gives every center at least its internal expected outside option. If $T^*$ is the maximum number of vertices covered by any pooled packing, then \[ |V(O\cup Q)|\geq\max\{b,T^*-(K-1)b\}\geq T^*/K. \] The same $1/K$ comparison holds for expected total transplants. If patient weights satisfy $0<w_{\min}\leq w_i\leq w_{\max}$, the plan's expected weighted value is at least $w_{\min}/(K w_{\max})$ times the maximum expected weighted value.
Proof: Every reserved internal cycle is still performed with the same marginal success probability, so center $h$ receives at least $s b_h$ expected transplants. Take an optimum pooled packing $P^*$ with $T^*$ covered vertices. At most $b$ of its cycles intersect the reserved vertices, because the cycles are disjoint and each intersecting cycle contains at least one distinct reserved vertex. Deleting these cycles discards at most $K b$ vertices. The remaining cycles are a feasible residual packing, so $|V(Q)|\geq T^*-K b$ and $|V(O\cup Q)|\geq T^*-(K-1)b$. The trivial bound $|V(O\cup Q)|\geq b$ gives the maximum. If $b\geq T^*/K$ the first term suffices; otherwise the second is greater than $T^*/K$.

Linearity of expectation multiplies all counts by $s$; independence of failures is unnecessary. The algorithm's weighted value is at least $s w_{\min}T^*/K$, while any packing's expected weighted value is at most $s w_{\max}T^*$, proving the final assertion. ∎

This benchmark enforces participation conditional on full submission. It is not a dominant-strategy withholding mechanism: changing the submitted set can change the reserved internal optimum and the residual exchange opportunities. The theorem deliberately makes no such incentive claim.

Proposition (A truthful baseline and its efficiency limit): The mechanism that independently performs a maximum-cardinality internal packing for each submitting center is dominant for full submission against subset withholding and subsequent internal exchange on withheld vertices, with the same cycle cap, failure law, and resource rules. Nevertheless its ratio to pooled expected total transplants can be zero.
Proof: Fix a center's full set $V_h$ and a submitted subset $S_h$. A maximum internal packing on $S_h$ together with any packing on $V_h\setminus S_h$ is a feasible internal packing on $V_h$. Their total planned count cannot exceed $b_h$. Multiplication by $s$ proves the required expected-utility dominance, for every submission of the other centers; those submissions do not enter this mechanism. For the efficiency assertion give each of two centers one vertex and put a directed two-cycle between the vertices, with no internal cycles. The mechanism makes zero transplants, whereas pooling gives $2s>0$. ∎

Thus a participation-safe allocation and a dominant-submission rule are mathematically distinct. Combining their benefits, rather than confusing the two inequalities, is part of the remaining target.

An exact equity obstruction, including lotteries and failures

Proposition (No uniform weighted efficiency under unrestricted weights): Even with $K=2$ and the common success probability $s$, individually rational randomized mechanisms can have an arbitrarily small ratio to unconstrained weighted efficiency, irrespective of their computational power or withholding incentives.
Proof: Center $A$ owns $a_1,a_2$ and center $B$ owns $b_1$. The only cycles are $(a_1,a_2)$ and $(a_1,b_1)$. Set $w_{a_1}=w_{a_2}=1$ and $w_{b_1}=W>1$. Center $A$ has outside option $2s$. A lottery putting probability $x$ on its internal cycle and $y$ on the cross-center cycle, with $x+y\leq1$, gives $A$ expected utility $s(2x+y)$. The requirement $2x+y\geq2$ forces $x=1,y=0$: indeed $2x+y\leq2-y$. Thus every individually rational lottery has weighted value $2s$. The unconstrained optimum uses the cross-center cycle and has value $s(W+1)$. Their ratio $2/(W+1)$ tends to zero. All plans obey the same failure and medical constraints, and no payment can compensate $A$. ∎

The loss is caused by the difference between center transplant-count objectives and patient equity weights. Randomizing plans does not remove the binding participation constraint. This is a restricted impossibility result, not a prediction about actual patient weights.

Separating composition and scale in a two-type pool

To make the counterfactuals explicit, assume pairs have type $X$ or $Y$, any opposite-type pair can form a two-cycle, and no same-type cycle is feasible. Let a pool contain $x$ and $y$ vertices of the two types. Failure probability remains $1-s$ per cycle.
Theorem (Exact composition accounting and a finite-pool bound): The maximum planned transplant count is $f(x,y)=2\min(x,y)$. For two pools with imbalances $d_j=x_j-y_j$, the exact gain from pooling their fixed compositions is \[ f(x_1+x_2,y_1+y_2)-f(x_1,y_1)-f(x_2,y_2) =|d_1|+|d_2|-|d_1+d_2|. \] For an iid pool of size $N$ with type-$X$ probability $p$, let $F(N,p)$ be its expected optimum planned count. Then \[ F(N,p)=N-{\mathbb{E}}|2X-N|,\qquad X\sim\operatorname{Bin}(N,p), \] \[ \max\{0,N-N|2p-1|-2\sqrt{Np(1-p)}\} \leq F(N,p)\leq N-N|2p-1|. \] Multiply all formulas by $s$ for expected completed transplants.
Proof: Each cycle uses one vertex of each type, so at most $\min(x,y)$ cycles are possible; completeness across types attains this bound. The identity $2\min(x,y)=x+y-|x-y|$ proves the fixed-composition formula by subtraction. For the random pool substitute $x=X,y=N-X$. Convexity of absolute value gives ${\mathbb{E}}|2X-N|\geq|2Np-N|$. The triangle inequality and Cauchy--Schwarz give \[ {\mathbb{E}}|2X-N|\leq |2Np-N|+2{\mathbb{E}}|X-Np| \leq N|2p-1|+2\sqrt{Np(1-p)}. \] Subtract from $N$ and use nonnegativity. Linearity of expectation again supplies the common factor $s$. ∎

For a declared common composition $p_0$, a scale-only counterfactual is $F(N_1+N_2,p_0)-F(N_1,p_0)-F(N_2,p_0)$. At the same pooled size, a composition comparison is $F(N_1+N_2,p)-F(N_1+N_2,p_0)$. A mixture of two countries with fixed counts and different probabilities need not have binomial type counts; its corresponding expression uses the sum of their two independent binomials. Choosing $p_0$ is a substantive counterfactual, not an empirically identified decomposition.

What remains

These proofs handle batch arrivals, a transparent failure law, outside options, and one explicit biological simplification. They neither calibrate realistic blood/HLA composition nor solve dynamic exit, heterogeneous failures, shared capacity, chains, or strategic pooling. The unrestricted-weight obstruction persists even before those harder features are added. Sustainable mechanisms with quantified withholding incentives remain the principal missing step.

References

Completion Estimate

30%

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