Financial support by a Marie Curie Fellowship under contract no. HPMF-CT-01206. Olivella acknowledges financial support from projects SEJ2006-00538, ECO2009-7616, Consolider-Ingenio CSD2006-16, 2009SGR-169, and Barcelona Economics-Xarxa CREA. Olivella is a Research Fellow of MOVE (Markets, Organizations and Votes in Economics)

When health plans compete under adverse selection, the competitive equilibrium set of contracts is unique. However, the allocation of these contracts among health plans is undetermined. We show that three health plans suffice to sustain an equilibrium where each health plan offers a single contract and attracts a single type of agent (full specialization). We also show that this equilibrium can be ruled out by introducing any horizontal differentiation, and that if in equilibrium each health plan attracts all types of agents, at least one of the health plans must do so through a menu of contracts.