On competitive nonlinear pricing

We study a discriminatory limit-order book in which market makers compete in nonlinear tariffs to serve a privately informed insider. Our model allows for general nonparametric specifications of preferences and arbitrary discrete distributions for the insider's private information. Adverse selection severely restricts equilibrium outcomes: in any pure-strategy equilibrium with convex tariffs, pricing must be linear and at most one type can trade, leading to an extreme form of market breakdown. As a result, such equilibria exist only under exceptional circumstances that we fully characterize. These results are strikingly different from those of existing analyses that postulate a continuum of types. The two approaches can be reconciled when we consider epsilon-equilibria of games with a large number of market makers or a large number of types.