Liquidity Provision with Adverse Selection and Inventory Costs.

The article analyzes one-shot Nash competition among multiple identical dealers who quote price schedules to attract order flow from a client trading due to private information, idiosyncratic risk, or both. Dealers do not observe the client's exact type but know its distribution and balance adverse selection against inventory costs when setting prices. Under minimal assumptions, the authors prove the existence and uniqueness of a symmetric Nash equilibrium characterized by a nonlinear ordinary differential equation (ODE) for the dealers' marginal prices. They show that admissible price schedules must be strictly convex in competitive settings, and that sufficient adverse selection combined with dealer inventory costs ensures the ODE solution corresponds to an equilibrium. The paper extends prior models by allowing general type distributions without compact support and by endogenizing convexity of price schedules, providing constructive methods for numerical solutions and highlighting differences between monopolistic and oligopolistic dealer markets.