**by Aimé Lachapelle, Jean-Michel Lasry, Charles-Albert Lehalle & Pierre-Louis Lions**

This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogeneous traders. Offer and demand of liquidity drive price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: order book size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon is stabilized in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with lower Profit & Loss.