Designing Dynamic Reassignment Mechanisms: Evidence from GP Allocation

This paper studies the design of dynamic reassignment mechanisms—centralized systems that must not only provide good initial matches but also accommodate changes in agents’ preferences over time. The empirical setting is Norway’s system for allocating patients to general practitioners (GPs), where every individual is assigned a specific GP whose panel has a binding capacity cap. Since 2016, Norway has allowed patients to join waitlists for oversubscribed GPs while retaining their spot on their current GP’s panel, with reassignment proceeding strictly first-come, first-served (FCFS) as vacancies arise.

The paper makes three contributions. First, it provides direct evidence of unrealized gains from trade: in December 2019, 15 percent of the 133,332 patients then standing on waitlists could have been immediately reassigned via a single run of the Top-Trading Cycles (TTC) algorithm, which identifies not only bilateral swaps but arbitrary cycles. A mechanical simulation holding patient choices fixed shows that running TTC monthly from November 2016 through December 2019 would have left 23 percent fewer patients on waitlists by end-2019, with average waiting times among reassigned patients 29 percent shorter.

Second, the paper introduces a dynamic TTC mechanism and clarifies why static properties do not carry over. In the static case, TTC is both strategy-proof and Pareto-improving (Shapley and Scarf, 1974; Roth, 1982). In a dynamic setting, neither property holds. Repeated TTC is not strategy-proof because patients’ GP choices affect how long they wait. More importantly, TTC may leave some patients worse off: a panel slot that would have gone to the first person on a waitlist under FCFS may instead go to a later-arriving patient who can form a trading cycle, effectively de-prioritizing patients whose GPs are undersubscribed. In the mechanical simulation, 4.5 percent of patients face longer waiting times under TTC.

Third, the paper estimates a structural model of patient attention and GP choice using monthly Norwegian administrative data covering 4.78 million patients and 6,470 GP panels (2014–2019), restricting estimation to the Trondelag region (approximately 8 percent of the country). The model specifies: a Poisson attention process (patients consider switching only when an attention shock arrives); preferences over GPs as a function of travel time, GP fixed effects, and match characteristics; and a belief model mapping observed waitlist lengths into expected waiting times. Parameters are recovered via a Gibbs sampler with Metropolis-Hastings for the discount rate. Key estimates: the annual discount factor is approximately 0.91; a female patient under 45 would travel 7.3 minutes farther to see a female GP (6.3 minutes for a female patient over 45); GP fixed effects have a standard deviation of 31 minutes’ travel-time equivalent; idiosyncratic taste shocks have a standard deviation of 12.6 minutes.

The paper then simulates a stationary equilibrium for each counterfactual mechanism. Under the status quo in stationary equilibrium, 9.4 percent of patients are on a waitlist, 82.2 percent of GPs have a waitlist, and average expected waiting time is 16.7 months. Introducing TTC reduces average waiting time to 14.1 months and raises mean patient welfare by the equivalent of 0.75 minutes’ travel time (more than 13 percent of the gain achievable under a no-capacity-constraints benchmark). Over half of this gain (0.4 minutes) comes directly from patients obtaining geographically closer GPs. Benefits are concentrated among younger patients, female patients, and recent movers; rural patients gain 2.1 minutes. However, patients with undersubscribed GPs face waiting times that rise from 16.7 to 22.8 months and are worse off by the perpetuity equivalent of 0.8 minutes.

Two modified mechanisms are evaluated. Deferred Acceptance (DA), which strictly respects FCFS priority, achieves essentially no improvement over the status quo, illustrating a fundamental trade-off between eliminating envy and exploiting gains from trade. A “TTC with Priority” (TTCP) mechanism, which gives priority for panel vacancies to patients with undersubscribed GPs before running TTC, achieves 61 percent of TTC’s welfare gains (0.46 minutes flow payoff; 1.08 minutes NPV) while leaving patients with undersubscribed GPs no worse off than under the status quo. A benchmark simulation eliminating waitlists altogether raises mean welfare slightly (0.19 minutes) but lowers median welfare (−0.60 minutes), with gains concentrated among highly mismatched patients.

Q: What is the core market failure the paper documents? A: Norway’s waitlist mechanism assigns panel vacancies strictly first-come, first-served without allowing patients to trade. This creates a “double coincidence of wants” problem: patients can simultaneously be on each other’s waitlists but cannot swap. In December 2019, 15 percent of 133,332 waiting patients could have been immediately reassigned via a single TTC run. A mechanical simulation shows that monthly TTC would have left 23 percent fewer patients on waitlists by end-2019 and reduced average realized waiting times among reassigned patients by 29 percent.

Q: Why does TTC fail to be strategy-proof in a dynamic setting? A: In the static case, TTC gives every agent an assignment at least as good as their endowment, making truthful reporting a dominant strategy. In a dynamic setting, a patient’s choice of GP determines not only which GP they receive but also how long they wait — patients who choose less-demanded GPs reach the front of the waitlist faster. This creates incentives to misreport preferences strategically, breaking strategy-proofness. The paper shows this formally and builds it into the equilibrium model by requiring patients to optimize over both GP choice and expected waiting time.

Q: Why does dynamic TTC harm some patients relative to the status quo? A: Under FCFS, the first person on a waitlist is guaranteed the next available slot on the target GP’s panel. Under TTC, a patient who arrived later but whose current GP is oversubscribed can form a trading cycle that redirects that slot, effectively jumping the queue. Patients with undersubscribed GPs — whose panel endowment is not a scarce resource that others want — cannot form cycles and are systematically de-prioritized. In the stationary equilibrium, their expected waiting time rises from 16.7 to 22.8 months, and they are worse off by the perpetuity equivalent of 0.8 minutes’ travel time.

Q: What are the main parameter estimates and what do they imply? A: The annual discount factor is estimated at approximately 0.91 once GP fixed effects are included (rising to near 0.95 without them, because more desirable GPs have longer waitlists). Gender homophily is worth 6.3–7.3 minutes of travel time for female patients under 45. Age homophily is worth approximately 1 minute. The standard deviation of GP fixed effects is 31 minutes and idiosyncratic shocks are 12.6 minutes, both in travel-time equivalents, indicating substantial horizontal differentiation across GPs and across patients’ idiosyncratic tastes.

Q: How important are moves as a driver of GP switching? A: Moves are the dominant driver. Among non-movers, older men consider switching just once every 25 years; temporary residents consider switching approximately once every 7.5 years (1.084 percent per month). Among patients who moved more than 30 minutes, a temporary resident has an 18.59 percent monthly probability of considering switching in the month of or month after the move. For a permanent resident making a long-distance move, the cumulative attention probability over the 8 months surrounding the move rises to 34 percent (versus 22 percent for a short-distance move). In the data, 26 percent of waitlist users moved municipality during 2017–2019, versus 6 percent of non-switchers.

Q: What does the stationary equilibrium under the status quo look like? A: In the long-run stationary equilibrium, 9.4 percent of patients are on a waitlist, 82.2 percent of GPs have a waitlist, and the average expected waiting time to switch GPs is 16.7 months. Each month, 2,299 patients on average draw attention shocks; 85.2 percent of these choose to join a waitlist, while the remainder either switch to an open GP or stay with their current GP. The average attentive patient expects to successfully obtain their chosen GP after 16.8 months.

Q: What are the distributional consequences of TTC across patient subgroups? A: Female patients benefit especially because they are more likely to be attentive (and thus use waitlists) than males. Recent movers gain 2.3 minutes’ travel-time equivalent. Patients who have never moved still gain 1.0 minutes. Rural patients gain 2.1 minutes (larger than average), reflecting their longer baseline travel times and greater geographic mismatch potential. Urban patients also benefit but less so. The one group that is harmed is patients with undersubscribed GPs, who face longer waits and a welfare loss of 0.8 minutes perpetuity equivalent.

Q: Why does the Deferred Acceptance mechanism fail to improve on the status quo? A: DA strictly respects FCFS waiting-time priority: no patient may be reassigned to a GP for whom another patient has been waiting longer. This means DA can only execute swaps in which all patients ahead of each participant on their respective waitlists are also reassigned in the same month. In practice, this virtually never occurs, so DA reassigns almost no patients earlier than the status quo Waitlists mechanism. The result illustrates a fundamental trade-off: fully respecting FCFS priority eliminates nearly all gains from trade.

Q: How does TTCP restore fairness while preserving most of the efficiency gains? A: TTCP modifies TTC by prioritizing patients with undersubscribed GPs over those with oversubscribed GPs when assigning panel vacancies, while still respecting the constraint that patients cannot be assigned a GP they prefer less than their current one. This gives patients with undersubscribed GPs a compensating advantage in the queue that offsets their inability to trade via cycles. TTCP achieves 0.46 minutes’ mean flow payoff improvement versus 0.75 for TTC (61 percent of TTC’s gains), and an NPV measure of 1.08 minutes versus 1.25 for TTC. Patients with undersubscribed GPs are left no worse off than under the status quo.

Q: What happens when waitlists are eliminated entirely? A: Under No Waitlists, attentive patients may only choose among GPs with open panels at the moment of attention. Mean welfare rises slightly (0.19 minutes) because patients spend less time mismatched while waiting, but median welfare falls by 0.60 minutes. The gains are concentrated among a minority of highly mismatched patients who prefer limited choice with no waiting over broader choice with long waits, while most patients prefer the option to wait for a more preferred GP. The authors note this may partly explain why formal waitlists are rare in other primary care systems.

Q: What is the welfare benchmark and how large are the gains? A: The benchmark is a “No Caps” scenario in which all panel caps are removed, representing the maximum achievable improvement. The mean welfare gain from TTC (0.75 minutes) represents more than 13 percent of this upper bound. The “Truthful TTC” benchmark, where patients submit full preference lists, yields 1.04 minutes, but its gains are also concentrated: the median patient is no better off than under the status quo Waitlists mechanism.

Q: What are the scope conditions for these findings? A: The demand model is estimated on the Trondelag region of Norway (approximately 8 percent of the national population) over 2017–2019, a period when waitlists were growing rapidly rather than in steady state. Counterfactual comparisons are made in a stationary equilibrium calibrated to Trondelag. The model excludes patients under 16 (whose enrollment is managed by parents). The partially capitated payment structure and fixed panel caps are institutional features specific to Norway, though similar systems exist in Canada, the UK, Italy, and Sweden. GP characteristics are held fixed in the model. The analysis abstracts from health outcomes, focusing on preference-based welfare from GP assignment.

Top-Trading Cycles (TTC) algorithm: A centralized reassignment algorithm that takes agents’ preference lists and objects’ priority lists as inputs, has each agent “point to” their preferred object and each object “point to” their highest-priority current or waiting agent, identifies cycles of mutual pointing, and executes the trades in those cycles simultaneously. In the paper’s static application, TTC is both Pareto-improving (every participant receives an assignment at least as good as their endowment) and strategy-proof. In the dynamic setting studied here, neither property holds.

Dynamic TTC mechanism: A mechanism that runs the TTC algorithm repeatedly at the end of each period after naturally arising vacancies have been filled from waitlists. Because patients’ GP choices affect how long they wait — not only which GP they receive — this mechanism is not strategy-proof and may leave patients with undersubscribed GPs worse off than under strictly FCFS waitlists.

TTC with Priority (TTCP): A modified version of dynamic TTC that changes the priority ordering so that patients with undersubscribed current GPs are prioritized above patients with oversubscribed GPs when panel vacancies are allocated. This modification preserves patients’ endowment rights but compensates the group harmed by standard TTC. In the paper’s simulations, TTCP achieves 61 percent of TTC’s mean welfare gains while leaving patients with undersubscribed GPs no worse off than under the status quo.

Patient attention model: A model in which patients consider switching GPs only when they receive a Poisson-distributed attention shock. Attention rates vary by observable characteristics (age, gender, temporary vs. permanent residency, whether and how far the patient recently moved). The model interprets any switch request as evidence of both an attention shock and a preference for the requested GP over the current one. Patients who do not request switches may be either inattentive or attentive but satisfied.

Horizontal differentiation (GP preference heterogeneity): The extent to which different patients prefer different GPs for reasons unrelated to overall GP quality — primarily driven by geographic proximity, gender homophily (worth 6.3–7.3 travel-time-equivalent minutes for young female patients), and age similarity (approximately 1 minute). Horizontal differentiation is the fundamental source of gains from trade: if all patients preferred the same GP, there would be no mutual-benefit swaps to find.

Deferred Acceptance (DA) algorithm: The patient-proposing DA algorithm, which strictly respects FCFS waiting-time priority: no patient may be reassigned ahead of another patient who has been waiting longer for the same GP. In the dynamic context, DA achieves essentially no welfare improvement over the status quo because its strict respect for priority eliminates nearly all trading opportunities, illustrating the trade-off between envy-freeness and efficiency.

Double coincidence of wants: The situation in which two (or more) patients are simultaneously on each other’s waitlists and would mutually benefit from trading GP assignments, but cannot do so under the current mechanism because there is no vacancy on either panel. The paper’s direct evidence of this phenomenon — 15 percent of waiters could be immediately reassigned via one TTC run — motivates the counterfactual analysis.