Bid-ask price competition with asymmetric information between market makers

We consider the effect of asymmetric information on price formation process in a financial market where private information is held by a market maker. A Bayesian game is proposed in which there is price competition between two market makers with two different information partitions. At each stage players set bid and ask prices simultaneously and then trade occurs between market maker who proposes the most profitable price and liquidity traders. We characterize a set of partially revealing equilibria where the informed market maker's prices do not convey his private information. Informed player's equilibrium payoffs are proportional to prior beliefs of the market