  We propose a general non-linear order book model that is built from the
individual behaviours of the agents. Our framework encompasses Markovian and
Hawkes based models. Under mild assumptions, we prove original results on the
ergodicity and diffusivity of such system. Then we provide closed form formulas
for various quantities of interest: stationary distribution of the best bid and
ask quantities, spread, liquidity fluctuations and price volatility. These
formulas are expressed in terms of individual order flows of market
participants. Our approach enables us to establish a ranking methodology for
the market makers with respect to the quality of their trading.
