Market Making and Incentives Design in the Presence of a Dark Pool: A Stackelberg Actor–Critic Approach

A Stackelberg actor–critic approach to optimal market making and incentives design with dark pools. We consider the issue of a market maker acting at the same time in the lit and dark pools of an exchange. The exchange wishes to establish a suitable make–take fee policy to attract transactions on its venues. We first solve the stochastic control problem of the market maker without the intervention of the exchange. Then, we derive the equations defining the optimal contract to be set between the market maker and the exchange. This contract depends on the trading flows generated by the market maker’s activity on the two venues. In both cases, we show existence and uniqueness, in the viscosity sense, of the solutions of the Hamilton–Jacobi–Bellman equations associated to the market maker and exchange’s problems. We finally design an actor–critic algorithm inspired by deep reinforcement learning methods, enabling us to approximate efficiently the optimal controls of the market maker and the optimal incentives to be provided by the exchange.