Matching with Choice: Menu Design for Patient--Provider Assignment
High-quality matches between patients and primary care providers improve downstream health outcomes, yet provider turnover breaks these matches and forces patients to be reassigned. Health systems facing this problem must decide not only which providers to recommend, but how much choice to offer. Direct assignment avoids competition for scarce provider capacity but relies on noisy estimates of patient preferences; broader choice allows patients to correct these estimates using private information but can create costly coordination failures. We study this tradeoff through the design of fixed, personalized menus from which patients choose providers. We show that constructing the optimal menu is NP-hard and establish an information-theoretic lower bound. We find that offering each patient a single provider matches this bound up to a constant factor, while unrestricted choice can perform arbitrarily poorly. However, we show that additional choice improves on single-provider assignment when the system cannot reliably distinguish among a patient's alternatives. We formalize this tradeoff through two forms of loss: menu loss, which arises when menus offer too few options, and coordination loss, which arises when scarce capacity is exposed to too much choice. Motivated by this decomposition, we develop the Scenario-Averaged Marginals (SAM) algorithm, which tailors menus based on provider shadow prices so choice is offered only when utility exceeds the capacity cost. We identify sufficient conditions which SAM's regret matches the lower bound, namely when patients have valuable alternatives and offered menus don't create competition for scarce capacity. In a semi-synthetic environment calibrated to a provider network using Medicare data, SAM improves total utility by up to 19% over single-provider assignment as preference uncertainty grows, while improving utility for the least-served ZIP codes.