Optimal Insurance with Information Asymmetry: Nonlinear and Linear Pricing

We propose a new framework for studying optimal insurance under information asymmetry within the Stackelberg game framework. In this setting, a monopolistic insurer faces uncertainty regarding a customer's loss distribution or risk attitude. The customer is assumed to follow a mean–variance preference in continuous time, while the insurer sets premiums through a risk loading based on the expected loss. An optimal menu is explicitly derived for a general class of aggregate loss models. Our approach connects with the extensive literature on optimal insurance demand, stemming from the seminal work of Arrow (1963), and leads to an interesting finding: a nonlinear pricing structure for risk-type uncertainty versus a linear pricing structure for risk-attitude uncertainty. Specifically, if an insurer is uncertain about a customer's risk type and seeks to elicit this information, the risk loading (premium minus expected loss) is set lower for high-risk individuals to encourage them to select the corresponding contract. In contrast, if the insurer is only uncertain about the customer's risk attitude, no such discounts---in terms of risk loading---are provided. This reveals that information about customers' risk types is more valuable than information about their risk attitudes. Additionally, we compare our optimal menu with the worst-case contract derived from the maxmin expected utility, we find that our optimal menu increases the insurer's expected profit and enhances the likelihood of trading.