Nash equilibrium in competitive insurance

•I study an insurance market with any finite number of types as a standard duopoly.•I formally specify demand functions and profits.•I provide an easy proof for the (non) existence of (pure strategy) Nash equilibrium.

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