In a two-stage model insurance companies first decide upon risk classification and then compete in prices. I show that the observed heterogeneous behavior of similar firms is compatible with rational behavior. On the deregulated German insurance market individual application of classification schemes induces welfare losses due to cream skimming. Classification costs and pricing above marginal cost can be prevented by common industry-wide loss statistics which already exist to a rudimentary extent. They allow competition to approach Bertrand type. The computation of a mixed-strategy equilibrium for Bertrand competition allows to explain the decrease of industry profit after deregulation.