  We study the perfect information Nash equilibrium between a broker and her
clients -- an informed trader and an uniformed trader. In our model, the broker
trades in the lit exchange where trades have instantaneous and transient price
impact with exponential resilience, while both clients trade with the broker.
The informed trader and the broker maximise expected wealth subject to
inventory penalties, while the uninformed trader is not strategic and sends the
broker random buy and sell orders. We characterise the Nash equilibrium of the
trading strategies with the solution to a coupled system of forward-backward
stochastic differential equations (FBSDEs). We solve this system explicitly and
study the effect of information, profitability, and inventory control in the
trading strategies of the broker and the informed trader.
