Tail expectation and imperfect competition in limit order book markets

Perfect competition in liquidity provision in limit order markets is characterized by a tail expectation condition (Glosten 1994). In this paper, we model imperfect competition in schedules by infinitely many liquidity suppliers, quoting on a limit order book. We show that there are zero-rent mixed-strategy equi- libria featuring finite numbers of active liquidity suppliers. None of the equilibria satisfies the competitive outcome, not even on average. Considering a sequence of equilibria with the number of active liquidity suppliers becoming large, we show that the aggregate stochastic marginal price schedule converges to the deterministic competitive marginal price schedule.