Moral hazard, insurance, and some collusion

A risk-averse consumer purchases an insurance policy; if she suffers a loss, she may receive services from a provider to recover some of the loss. Only the consumer and the provider know if the loss has actually occurred. The provider's behavior is uncertain. With some positive probability, the provider is honest, reporting the loss information truthfully to the insurer; with the complementary probability, the provider reports the information strategically, by writing a side-contract with the consumer to maximize the joint surplus of the provider-consumer coalition. We show that there is a loss of generality in considering only collusion-proof contracts, and characterize equilibria implemented by collusion-proof and noncollusion-proof contracts. When the probability of a provider acting collusively is small, the equilibrium contract is not collusion-proof but approximately first-best. When the probability of a provider acting collusively is large, the equilibrium contract is independent of this probability and identical to the equilibrium collusion-proof contract when the provider is collusive with probability 1.