We study the efficiency of the equilibrium price in a centralized, orderdriven market where many asymmetrically informed traders are active for many periods. We show that asymmetries of information can lead to suboptimal information revelation with respect to the symmetric case. In particular, we assess that the more precise the information the higher the incentive to reveal it, and that the value of private information is related to the volume of exogenous trade present on the market. Moreover, we prove that any informed trader, whatever his information, reveals its private signal during an active phase of the market, concluding that long pre-opening phases are not effective as an information discovering device in the presence of strategic players.