I formalise a rather stylised insurance market with adverse selection as a standard duopoly. I formally specify demand functions and profits and I prove that a Nash equilibrium in pure strategies exists if and only if the well-known Rothschild-Stiglitz allocation is efficient

I construct an efficient game of competition for insurance markets with adverse selection. In the game, each company offers two menus of contracts: a public menu and a private menu. The union of all the public menus needs to be offered by every active company in the market. A private menu...