Choices and Outcomes in Assignment Mechanisms: Agarwal, Hodgson & Somaini (2025)

What this is. The paper’s core results, the structural model it estimates (joint decisions and survival outcomes), and the two-instrument identification strategy with the defining equations: enough to know what it found and how, without reading all 44 pages. To replicate or extend, read the full source at the original.

TL;DR

The paper evaluates the deceased donor kidney allocation mechanism by estimating patient Life-Years from Transplantation (LYFT), defined as the gain in median survival from receiving a transplant. The current observational standard (Wolfe et al. (2008)) relies on hazard-ratio comparisons that cannot account for selection on unobservables. This paper uses two quasi-experimental instruments: (1) random variation in the sequence of organ offers made to each patient, and (2) a scarcity measure (number of donors or offers in the patient’s Donor Service Area) that shifts acceptance decisions but is excluded from survival outcomes. These instruments identify a joint structural model of accept/reject decisions and post-transplant and untransplanted survival.

The canonical mechanism design literature, following Roth and Sotomayor (1992), and its school choice applications, such as Abdulkadiroglu and Sonmez (2003), evaluate mechanisms based on revealed agent preferences. This paper takes a different approach: it evaluates the kidney allocation mechanism based on a downstream outcome (patient survival), because policymakers and transplantation communities focus on survival rather than preference satisfaction.

Applied to 175,640 patients on the US kidney waitlist (2000-2010), the preferred estimates find that the realized allocation achieves an average LYFT of 9.29 years among patients registered in 2005, 1.75 years above a random assignment benchmark. Most of this gain is driven by patient choice rather than priority rules: removing patient choice while keeping priority rules yields a LYFT of only 8.05 years. The maximum achievable LYFT is 14.08 years, but reaching it requires systematically transplanting healthier patients with longer expected untransplanted survival, creating a sharp conflict with prioritarianism for the sickest.

Core results

Magnitudes are as reported; locators point into the source PDF. The 2005 cohort results (R1-R4) are from Figure 4 (p. 425); the preferred-spec LYFT (R5) is from Table VII (p. 421).

#	Result	Locator	Magnitude
R1	Realized assignment achieves average LYFT of 9.29 years, 1.75 years above random	Figure 4, p. 425	Realized = 9.29; random = 7.54; difference = +1.75 years
R2	Patient choice drives 70.4% of the LYFT gain over random assignment; priority rules alone (no-choice) achieve only 8.05	Figure 4, p. 425	No-choice LYFT = 8.05; realized = 9.29; random = 7.54; choice share = 70.4% (PDF text, p. 426; no-choice achieves 29.6% of the gain over random)
R3	Maximum possible LYFT = 14.08 years, 4.8 years above realized; optimal rematching of transplanted yields only 9.93 (13.4% of the gap)	Figure 4, p. 425; text p. 426	Optimal = 14.08; realized = 9.29; optimal rematching = 9.93; patient-selection dominates
R4	Observables-only optimal assignment reaches 11.04 years, 1.8 years above realized and 3.0 below full-information optimal	Figure 4, p. 425	Observables optimum = 11.04; full optimal = 14.08; realized = 9.29
R5	Preferred specification (instruments + unobservables) LYFT = 8.93 years; observational model (no instruments) = 8.25, a 0.68-year underestimate	Table VII, col 1-2, p. 421	Preferred = 8.93 (s.e. 0.12); no-instruments = 8.25 (s.e. 0.07); positive selection on unobservables biases the observational estimate downward
R6	Positive selection on unobservables: 1-SD rise in patient selectivity reduces acceptance by 4.0 pp and raises untransplanted survival by 0.316 years	Table VI, p. 420; Figure 2b, p. 422	Acceptance effect: -0.040 (s.e. 0.001); untransplanted survival: +0.316 years (s.e. 0.062); transplanted patients’ predicted LYFT distribution shifted ~1.1 years right vs full distribution
R7	Patient heterogeneity dominates LYFT variance: patient-specific SD = 3.26 years; donor-specific = 0.99 years; match-specific = 0.41 years	p. 423; text p. 426	Variance decomposition of LYFT: patient component = 3.26 yr SD vs donor = 0.99 yr vs match = 0.41 yr; rematching alone captures only 13.4% of maximum gain
R8	Waitlist patients face high mortality and limited transplant receipt	Table I, p. 401; text p. 401	27.4% died by year five; 47.2% received a transplant by year five; 54.3% ultimately received a deceased donor kidney
R9	Kidney offers are highly skewed and some viable kidneys are discarded after repeated rejection	Table II, p. 402; text p. 401	18.9% of donors had at least one viable kidney discarded; those donors received 1,892.5 offers on average versus 547.8 overall; median = 51 offers per donor
R10	Match-specific unobservables strongly predict acceptance, while their estimated post-transplant survival correlation is not significant	Table VI, p. 420	A 1-SD increase in match value changes acceptance by +0.066 (s.e. 0.001) and +0.065 (0.001); post-transplant survival effects = -0.003 (s.e. 0.249) and +0.092 (0.245), neither statistically significant
R11	Patient choices select kidneys with higher predicted LYFT	Text p. 423, footnote 28	A 1-SD increase in match-specific LYFT raises acceptance probability by 0.24%, controlling for patient and donor fixed effects
R12	Optimal allocation based on observational estimates understates feasible LYFT	Text p. 426	Observational-model optimal rematching yields 11.75 years when valued under the preferred model, versus 14.08 years under preferred-model optimal allocation; it captures 51% of the maximum gain over realized assignment
R13	Optimal allocation changes the health and demographic composition of transplanted patients	Table VIII, p. 427	Age 60+ share falls from 26.9% realized to 16.8% optimal; diabetes falls from 33.3% to 27.7%; mean untransplanted survival rises from 7.11 to 7.61 years
R14	Prioritizing the sickest patients first lowers average LYFT	Text p. 429	Assignment that transplants patients with the lowest untransplanted survival first achieves 3.41 years of LYFT
R15	Potential top-10 offers predict transplant receipt and the donor-quality group received	Table III, p. 409	For KDPI <= 50% offers, coefficients for any, KDPI <= 50%, and KDPI > 50% or missing transplants are 0.0479 (s.e. 0.00460), 0.0602 (0.00358), and -0.0122 (0.00251); for KDPI > 50% or missing offers, the corresponding coefficients are 0.0234 (0.00386), -0.0167 (0.00292), and 0.0414 (0.00245); F-statistics = 142.6-162.7
R16	Scarcity instruments predict lower acceptance across progressively controlled first-stage specifications	Table IV, p. 410	Log(1 + donors) coefficients = -0.0434, -0.0429, -0.0393, -0.0355 (s.e. 0.00209, 0.00208, 0.00204, 0.00206); log(1 + offers) = -0.0390, -0.0387, -0.0336, -0.0316 (s.e. 0.00106, 0.00106, 0.00106, 0.00109); F-statistics = 296.8-1,361.8

Overall (paper’s conclusion). The mechanism outperforms random assignment through both patient selection and advantageous patient-kidney matching, with patient choice a key contributor relative to priority rules alone. Further gains over the realized assignment are larger from changing which patients are transplanted than from rematching kidneys among the same patients, creating a dilemma for policymakers who also wish to prioritize the sickest patients. Observational methods that do not account for selection on unobservables underestimate LYFT and the potential gains from improved allocation.

Theory / model

The model generalizes the Roy selection framework to a sequential assignment setting in which agents (patients, indexed $i$) receive offers for heterogeneous objects (organs, indexed $j$) arriving sequentially and make accept-or-reject decisions.

Unassigned outcome (eq. 3.1, p. 404): survival if patient $i$ never receives a transplant,

$$Y_{i,0} = g_0(x_i, \nu_{i,0}), \tag{3.1}$$

where $x_i \in \mathbb{R}^{d_x}$ are patient observables and $\nu_{i,0} \in \mathbb{R}$ is a patient-specific unobservable (frailty).

Assignment outcome (eq. 3.2, p. 404): survival if patient $i$ is transplanted organ $j$,

$$Y_{i,j} = g_1(q_j, x_i, \nu_{i,1}, \varepsilon_{i,j,1}), \tag{3.2}$$

where $q_j \in \mathbb{R}^{d_q}$ are organ-type observables, $\nu_{i,1}$ is a patient-specific post-transplant unobservable, and $\varepsilon_{i,j,1}$ is a match-specific shock.

Decision equation (eq. 3.3, p. 405): patient accepts organ $j$ if

$$D_{i,j} = g_D(q_j, x_i, z_i, \nu_{i,D}, \varepsilon_{i,j,D}) \in \{0,1\}, \tag{3.3}$$

where $z_i \in \mathbb{R}^{d_z}$ is the scarcity instrument (excluded from outcome equations), $\nu_{i,D}$ is unobserved selectivity, and $\varepsilon_{i,j,D}$ is a match-specific preference shock.

Patient $i$ is assigned the first accepted organ in her offer set before leaving the list (eq. (1), p. 404):

$$T_{i,j} = 1\{A_i \geq t_{i,j}\} D_{i,j} \prod_{j' < j,\, j' \in J_i} (1-D_{i,j'}). \tag{1}$$

The observed outcome is then

$$Y_i = \sum_{j \in J_i} T_{i,j} Y_{i,j} + \Bigl(1 - \sum_{j \in J_i} T_{i,j}\Bigr) Y_{i,0}.$$

Key assumptions (pp. 406-407):

Assumption 1: $\varepsilon_i$, $\nu_i$, and $z_i$ are mutually independent conditional on $x_i$ (the scarcity instrument is excludable).

Assumption 2: The potential offer sequence $J_i$ is conditionally independent of $(\nu_i, \varepsilon_i)$ given $(x_i, z_i)$ (organ arrivals are random conditional on patient priority type and geography).

LYFT (eq. 7.1, p. 421) for patient-organ pair $(i, j)$ conditional on covariates $I_{i,j} = \{x_i, q_j, D_{i,j}, \eta_j, \nu_{i,D}, \nu_{i,f}\}$:

$$\text{LYFT}(I_{i,j}) = M(Y_{i,j} \mid I_{i,j},\, Y_{i,0} \ge t_{i,j}) - M(Y_{i,0} \mid I_{i,j},\, Y_{i,0} \ge t_{i,j}), \tag{7.1}$$

where $M(Y \mid X)$ denotes the median of $Y$ given $X$ and $t_{i,j}$ is the time elapsed between patient registration and organ arrival. This conditions on the patient being alive at the time of the offer and accounts for selection on both patient observables and unobservables.

Method

Identification proceeds in three steps building on Heckman and Navarro (2007) and Imbens and Angrist (1994).

Lemma 1 (p. 411) uses variation in the offer sequence $J_i$. Let $N_i = \min\{n : D_{i,j(i,n)} = 1\}$ be the index of the first accepted offer, one greater than the number of rejected offers. For a patient with priority type $x_i$ and scarcity $z_i$, comparing patients who received offer-type sequences $(q_{j(i,1)}, \ldots, q_{j(i,n)})$ vs $(q_{j(i,1)}, \ldots, q_{j(i,n-1)})$ identifies the marginal distributions of $Y_{i,j(i,n)}$ and $Y_{i,0}$ conditional on $N_i = n$, the offer sequence, scarcity, and being alive when the nth offer arrives. This is a standard LATE argument (Imbens and Angrist (1994)) extended to the sequential setting.

Lemma 2 (p. 413) shows that offer-sequence variation identifies the choice function $g_D(\cdot)$ via its Fourier-Legendre approximation. The key quantity is the moment generating structure: for a sequence $q_j^n$ of $n$ identical organ-type offers and $k \le n$,

$$P(N_i > k \mid q_j^n, z) = \int_0^1 \varepsilon_D^k \,\mathrm{d}v(\varepsilon_D;\, q_j, z), \tag{5.1}$$

where $v(\varepsilon_D; q_j, z)$ is the CDF of rejection probabilities across patients given organ type $q_j$ and scarcity $z$. These moments identify the $(n-1)$-th order Fourier-Legendre approximation of $v(\cdot; q_j, z)$, which converges in Cesaro mean to the true CDF.

Theorem 1 (p. 414) combines the offer instrument and the scarcity instrument $z_i$ to identify the expected outcomes conditional on the selection unobservables $\nu_{i,D}$ and $\varepsilon_{i,j,D}$. The scarcity instrument “traces out” the selectivity unobservable via (eq. 5.2, p. 414):

$$E\!\left[Y_{i,0} \times 1\{T_i = 0\} \mid q_j^k, z_i\right] = \int_0^1 E\!\left[Y_{i,0} \mid \nu_D = v(\varepsilon_D;\, z_i, q_j)\right] \varepsilon_D^k \,\mathrm{d}v(\varepsilon_D;\, z_i, q_j). \tag{5.2}$$

Estimation uses a parameterized Box-Cox version of equations (3.1)-(3.3), estimated by Gibbs sampling (McCulloch and Rossi (1994)) (eqs. 5.3-5.7, pp. 415-416). Let $B(Y; \rho) = (Y^\rho - 1)/\rho$ denote the Box-Cox transformation (Box and Cox (1964)):

$$y_{i,0} = B(Y_{i,0};\, \rho_0) = x_i \beta_x + \nu_{i,0}, \tag{5.3}$$ $$y_{i,j} = B(Y_{i,j};\, \rho_1) = \chi(x_i, q_j)\alpha_{x,q} + \alpha_\eta \eta_j + \nu_{i,1} + \varepsilon_{i,j,1}, \tag{5.4}$$ $$D_{i,j} = 1\!\left\{\chi(x_i, q_j)\gamma_{x,q} + z_i \gamma_z + \eta_j - \nu_{i,D} + \varepsilon_{i,j,D} > 0\right\}, \tag{5.5}$$

where $\eta_j \sim N(0, \sigma_\eta^2)$ captures unobserved organ-level quality and $\chi(x_i, q_j)$ is a flexible function of patient and donor characteristics. The unobservables follow the factor structure (eqs. 5.6-5.7, p. 415):

$$\nu_{i,1} = \delta_{1,D}\, \nu_{i,D} + \nu_{i,f}, \quad \nu_{i,0} = \delta_{0,D}\, \nu_{i,D} + \delta_{0,f}\, \nu_{i,f} + \tilde{\nu}_{i,0}, \tag{5.6-5.7}$$

where $\nu_{i,D}$, $\nu_{i,f}$, and $\tilde{\nu}_{i,0}$ are independently distributed mean-zero normals. This factor structure allows selectivity into transplantation ($\nu_{i,D}$) to be correlated with both post-transplant and untransplanted survival. The Gibbs sampler (Geweke, Gowrisankaran, and Town (2003)) draws sequentially from conditional posteriors; by the Bernstein-von Mises theorem this is interpreted as maximum likelihood. Monte Carlo simulations on 100 data sets with 10,000 patients and 2,500 donors confirm good coverage and convergence (footnote 23, p. 416).

Empirical specifications

Survival and choice estimates (Table V, pp. 417-419). The structural model is estimated across three specifications: (1) observational (no scarcity instrument, with selection-related patient and match unobservables assumed mutually independent), (2) preferred (scarcity instrument = number of past donors), and (3) robustness (past offers instrument). Marginal half-life effects are reported for a 1-SD increase in continuous characteristics. Diabetic patients have a shorter half-life by 3.58 years with a transplant (Panel B) and 1.45 years without a transplant (Panel A; PDF p. 419). Positive tissue-type matching raises post-transplant half-life substantially (Panel B). A 1-SD increase in $\nu_{i,D}$ is associated with a 0.316-year increase in untransplanted survival (Table VI, Panel A, p. 420).

Offer-instrument first stage (Table III, p. 409). The outcomes are any transplant, a transplant from a KDPI <= 50% donor, and a transplant from a KDPI > 50% or missing donor. For any transplant, columns 1-2 add progressively richer patient controls; columns 3-4 report the donor-quality outcomes:

$$Y_i^{(m)} = \beta_{m,L}\log(1+O_{i,L}) + \beta_{m,H}\log(1+O_{i,H}) + X_i'\gamma_m + \alpha_{d(i)} + \lambda_{y(i)} + \mu_{b(i)} + e_i^{(m)}.$$

Here $O_{i,L}$ and $O_{i,H}$ count potential top-10 offers from the two donor-quality groups; $X_i$ includes priority and (in columns 2-4) patient controls. All columns include DSA, registration-year, and blood-type fixed effects. Standard errors cluster by DSA, registration year, and blood type. The sample is non-pediatric patients registered 2000-2008; N = 132,507 (columns 1, 3, 4) and 130,923 (column 2).

Scarcity first stage (Table IV, p. 410). Acceptance on the first 100 offers per donor is estimated with both scarcity measures and progressively expanded controls:

$$\text{Accept}_{ij} = \beta_D\log(1+\text{Donors}_{it}) + \beta_O\log(1+\text{Offers}_{it}) + X_i'\gamma + W_j'\psi + M_{ij}'\delta + \alpha_{\text{offer-year}} + \alpha_{\text{priority-type}} + \alpha_{\text{DSA}} + \alpha_{\text{blood-type}} + \alpha_{\text{years-waited}} + e_{ij}.$$

Columns 1-4 omit some donor/match covariates and use N = 863,073; columns 5-8 include donor and match controls and use N = 851,753. Standard errors cluster by DSA, offer year, years waited at offer, and blood type. Priority controls include pediatric status, CPRA categories, and time on dialysis; patient, donor, and match control definitions are reported in the Table IV note.

Structural survival and choice estimates (Table V, pp. 417-419). Equations (5.3)-(5.7) above define the estimated Box-Cox survival outcomes and latent-index acceptance model. Table V reports the model-implied half-life and acceptance marginal effects at median observable covariates, integrating over unobservables, for no-instrument, past-donor, and past-offer specifications. Each specification includes DSA, blood-type, and registration-year fixed effects, patient/donor/offer covariates, and standard errors in parentheses. The MCMC summary draws 250,000 samples, burns in 50,000, and retains every tenth draw. The estimation cohort is the paper’s non-pediatric 2000-2010 waitlist sample.

Match-value correlation diagnostic (Table VI, p. 420; text p. 423, footnote 28). Table VI reports model-implied marginal changes from 1-SD shifts in patient selectivity and match value, rather than a separate causal regression. For the additional acceptance-sorting check, the paper regresses conditional expected LYFT on predicted acceptance probability, controlling for patient and donor fixed effects; the text reports the 0.24% increase for a 1-SD increase in match-specific LYFT. No additional fixed effects beyond those stated in the footnote are specified there.

LYFT and assignment benchmarks (Table VII, p. 421; Figure 4, p. 425; text pp. 426, 429). Table VII averages model-implied median-survival differences across realized transplants, using the three instrument specifications and an additional donor-characteristics specification; standard errors are in parentheses. Figure 4 benchmarks the 2005 registrant cohort. The full-information optimal assignment solves the simulated allocation problem (footnote 29, p. 424):

$$\max_{a_{ij}\in\{0,1\}} \sum_{i,j} a_{ij}\,\text{LYFT}^{s}_{ij} \quad\text{subject to}\quad a_{ij}(1-c_{ij})=0,\quad \sum_i a_{ij}\leq k_j,\quad \sum_j a_{ij}\leq 1.$$

Here $c_{ij}$ marks a feasible patient-kidney pair and $k_j$ is the number of kidneys from donor $j$. Random and no-choice benchmarks use the feasibility and priority rules described in Section 8. Table VIII (p. 427) reports descriptive distributions by assignment; it has no estimating regression or standard-error treatment. The sickest-first counterfactual is reported in the conclusion (p. 429).

Datasets used

Dataset	Role in paper	Wiki page
OPTN Potential Transplant Recipient (PTR) dataset	All organ offers made to each waitlisted patient and each patient’s accept/reject decision; match characteristics including HLA mismatches	no page yet
OPTN Standard Transplantation Analysis and Research (STAR) dataset	Deceased donor characteristics, patient histories, transplant outcomes, annual follow-up data, patient survival (death dates merged from Social Security records)	no page yet

Sample: 175,640 patients registered January 1, 2000 through December 31, 2010, excluding pediatric patients and those needing multiple organs or a living donor. Data on approximately 6,195 donors per year and 15,967 new patients per year. Survival tracked through February 29, 2020 (up to 20 years and 2 months from registration).

Both datasets are supplied by UNOS as contractor for OPTN. Access requires a data use agreement with OPTN (https://optn.transplant.hrsa.gov/data/request-data/); data are proprietary-confidential.

When to read the full paper

Use the original if you are:

• Extending the identification framework to other assignment settings (public housing, school choice, gig-economy jobs) where agents face sequentially arriving heterogeneous objects and outcomes depend on the match;
• Evaluating the distributional consequences of alternative kidney allocation policies, particularly the trade-off between LYFT maximization and prioritizing the sickest (Table VIII, p. 427);
• Replicating the Gibbs sampler estimation of the joint model (replication code available; footnote 23, p. 416);
• Building on the identification results (Lemmas 1-2, Theorem 1) for settings with multiple unobserved dimensions of heterogeneity in both choices and outcomes.

The variance decomposition (patient vs donor vs match, p. 423) and the planner’s dilemma (Figure 4, p. 425; Table VIII, p. 427) are the most directly policy-relevant sections.

Attribution and rights

Source: peer-reviewed, Econometrica 93(2), March 2025. This distillation was extracted and model-verified on 2026-10-04; it is not independently reproduced. The paper is paywalled; only text extraction is permitted here.

Agarwal, Nikhil, Charles Hodgson, and Paulo Somaini. “Choices and Outcomes in Assignment Mechanisms: The Allocation of Deceased Donor Kidneys.” Econometrica 93, no. 2 (March 2025): 395-438. DOI: 10.3982/ECTA20203. (c) 2025 The Econometric Society. Extract-only; full text at the Econometric Society.