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 equilibria 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. (C) 2019 Elsevier Inc. All rights reserved.