We study imperfect competition between insurers in a multiple-risk environm
ent. In the absence of asymmetric information. equilibria are efficient, an
d we determine the degrees of specialization under which the specialized in
surers are able or unable to capture the surplus. We show in contrast that
under adverse selection, specialization systematically prevents second-best
efficiency. Concluding on the role of our notions of strategic complementa
rity/substituability on the tradeoff between efficiency and fairness of the
allocation, we give indications on the desirable structure of the insuranc
e industry.