Efficient Nash equilibrium in insurance markets with adverse selection

We revisit the seminal analysis of adverse selection by Rothschild and Stiglitz (1976) and propose a game-theoretic structure under which Nash-type strategic interaction between the informed and uninformed agents has always a unique, Pareto-optimal, incentive-compatible, separating, competitive equilibrium. This obtains under a simple extension of the mechanism in Hellwig (1987): expanding the insurance sellers' strategy space to allow them to offer menus of contracts as well as to publicly commit, if they so wish, upon not withdrawing a given contract. Under this structural enhancement, a separating, efficient, Nash equilibrium allocation exists even when an equilibrium does not exist in the Rothschild-Stiglitz setting. It is always unique, coinciding with the Rothschild-Stiglitz unsubsidized separating arrangement when the latter is an equilibrium in their model. Otherwise, it involves cross-subsidization across risk-classes and contracts.

Our contribution adds to but also stems from the large literature that investigates one of the most fundamental issues in economics: whether, if at all, and how efficiently competitive markets may allocate resources in the presence of adverse selection. Yet, a crucial question had been left open: whether, if at all, and under which game-theoretic structure a competitive market whose participants engage in Nash-type strategic behavior may allocate resources efficiently. We answer in the affirmative, and in the strongest sense, using structural elements introduced in the literature a long time ago. Hellwig's game-structure has been used extensively in applications of contract theory, its popularity due mainly to its simplicity and realism. Within this structure, Miyasaki (1977) suggested that each supplier may offer a menu of contracts rather than a single contract. His focus was on the labor market and, identifying a firm with a menu of wage-output contracts, viewed free entry as dictating that a firm may withdraw its entire menu but not single contracts from a menu. In the realm of insurance markets, the latter restriction was heavily criticized by Grossman (1979) who pointed out that insurers may indeed offer menus of contracts but are also able to withdraw specific contracts from these menus by simply rejecting the corresponding applications from customers.

We allow an insurance supplier not only to withdraw but also to commit to not withdraw a particular contract from a menu. As Grossman did, we take into account an important element of insurance suppliers' behavior in real markets: they do make contractual offers by sending out applications to potential customers. Yet, we also make use of another, equally realistic, element: they often choose to send certain customers ``pre-approved'' applications. By showing that an efficient  allocation obtains uniquely as a Nash equilibrium through this very simple mechanism, effectively we return the discussion to the original issue raised by Rothschild and Stiglitz in their seminal paper, albeit under a substantially different light. Our analysis suggests that the lack of efficient outcomes in real world competitive markets under adverse selection may not be due to the presence of private but rather due to the absence of public information.