Solver Methodology

Model: Nested logit demand (PPO nest = {BS, BC}, HMO nest = {Kaiser}) + Nash-in-Nash hospital bargaining with recapture.
Hospital A (community) serves BS and BC. Hospital B (integrated) serves Kaiser only.
Recapture: When Hospital A bargains with BS, it accounts for patients it would recapture through BC if BS disagreed. This raises BS's baseline hospital price (Hospital A has a better outside option). When BC is removed, recapture vanishes → hospital prices drop.
Price FOC: \(p = (1-\tau_j) \cdot (\varphi - \eta) / r + \tau_j \cdot (c + \rho)\), where \(\rho\) = recapture value per admission.