Efficient Nash Equilibrium Under Adverse Selection

Abstract

This paper revisits the problem of adverse selection in the insurance market of Rothschild and Stiglitz [24]. We propose a simple extension of the game-theoretic structure in Hellwig [13] under which Nash-type strategic interaction between the informed customers and the uninformed firms results always in a particular separating equilibrium. The equilibrium allocation is unique and Pareto-efficient in the interim sense subject to incentive-compatibility and individual rationality. In fact, it is the unique neutral optimum in the sense of Myerson [20].

A Game Theoretic Foundation of Competitive Equilibria With Adverse Selection

“…Pursuing this line of research, Mimra and Wambach () have found that results similar to ours can be obtained in such a setting. Relatedly, Diasakos and Koufopoulos () have demonstrated that distinguishing between pre‐approved and standard applications for insurance and allowing firms to commit to not reject the applications for some contracts can yield similar results in a modified three stage game.…”

“…We rule out the withdrawal of individual contracts mainly because, otherwise, firms would withdraw any loss‐making contract in stage 2, as pointed out by Grossman (), precluding any cross‐subsidization in equilibrium. Other ways around this problem have been suggested in recent complementary studies, for instance, by allowing for multiple withdrawal rounds with endogenous termination in Mimra and Wambach () or endogenous commitment not to withdraw individual contracts from a menu in Diasakos and Koufopoulos () . We abstract from this issue by ruling out partial withdrawal, but these papers show that analogous results can be obtained when this restriction is relaxed.…”

“…Besides legal issues and claims, withdrawal may lead to adverse consequences such as consumer confusion or demand reductions in future periods or other markets in which the firm is active, effects that are not otherwise captured in our parsimonious three stage game. Alternatively, they could be motivated as a form of partial commitment not to withdraw contracts, an interpretation emphasized by Diasakos and Koufopoulos () in their distinction between pre‐approved versus standard applications. Methodologically, the only purpose of introducing withdrawal costs is to enable us later on to restrict attention to equilibria where firms remain active in case of indifference, so that there is no withdrawal along the equilibrium path.…”

“…This multiplicity result is much more encompassing than related results in the literature. For instance, Hellwig () argued that, if the Rothschild–Stiglitz equilibrium is inefficient, then the Rothschild–Stiglitz contracts as well as any pooling contract that Pareto dominates them can be the sequential equilibrium outcome of his three‐stage game (see also Diasakos and Koufopoulos, ). In contrast, our multiplicity result is not confined to the case where the Rothschild–Stiglitz contracts are inefficient, and it allows for outcomes that are Pareto worse or incomparable to them.…”

“…The most related is the work by Mimra and Wambach (), which shows that results similar to ours can be obtained in a model where, instead of withdrawing from the market, firms can withdraw individual contracts and there are multiple withdrawal stages that terminate endogenously. On the other hand, Diasakos and Koufopoulos () and Koufopoulos () have considered a three‐stage game that allows insurers to commit not to withdraw contracts, for example, by offering pre‐approved applications, or not to withdraw an individual contract unless all other contracts in the menu are also withdrawn. These papers show that the Miyazaki–Wilson allocation can result as the unique equilibrium outcome under some selection criterion: free entry during the withdrawal stages in Mimra and Wambach (), endogenous commitment in Diasakos and Koufopoulos ().…”