  We consider a stochastic game between three types of players: an inside
trader, noise traders and a market maker. In a similar fashion to Kyle's model,
we assume that the insider first chooses the size of her market-order and then
the market maker determines the price by observing the total order-flow
resulting from the insider and the noise traders transactions. In addition to
the classical framework, a revenue term is added to the market maker's
performance function, which is proportional to the order flow and to the size
of the bid-ask spread. We derive the maximizer for the insider's revenue
function and prove sufficient conditions for an equilibrium in the game. Then,
we use neural networks methods to verify that this equilibrium holds. We show
that the equilibrium state in this model experience interesting phase
transitions, as the weight of the revenue term in the market maker's
performance function changes. Specifically, the asset price in equilibrium
experience three different phases: a linear pricing rule without a spread, a
pricing rule that includes a linear mid-price and a bid-ask spread, and a
metastable state with a zero mid-price and a large spread.
