We characterize and prove the existence of Nash equilibrium in a limit order market with a finite number of risk-neutral liquidity providers. We show that if there is sufficient adverse selection, then pointwise optimization (maximizing in p for each q) in a certain nonlinear pricing game produces a Nash equilibrium in the limit order market. The need for a sufficient degree of adverse selection does not vanish as the number of liquidity providers increases. Our formulation of the nonlinear pricing game encompasses various specifications of informed and liquidity trading, including the case in which nature chooses whether the market-order trader is informed or a liquidity trader. We solve for an equilibrium analytically in various examples and also present examples in which the first-order condition for pointwise optimization does not define an equilibrium, because the amount of adverse selection is insufficient.

This paper analyzes models of securities markets with a single strategic informed trader and competitive market makers. In one version, uninformed trades arrive as a Brownian motion and market makers see only the order imbalance, as in Kyle (1985). In the other version, uninformed trades arrive as a Poisson process and market makers see individual trades. This is similar to the Glosten–Milgrom (1985) model, except that we allow the informed trader to optimize his times of trading. We show there is an equilibrium in the Glosten–Milgrom-type model in which the informed trader plays a mixed strategy (a point process with stochastic intensity). In this equilibrium, informed and uninformed trades arrive probabilistically, as Glosten and Milgrom assume. We study a sequence of such markets in which uninformed trades become smaller and arrive more frequently, approximating a Brownian motion. We show that the equilibria of the Glosten–Milgrom model converge to the equilibrium of the Kyle model.

We analyze dynamic trading by an activist investor who can expend costly effort to affect firm value. We obtain the equilibrium in closed form for a general activism technology, including both binary and continuous outcomes. Variation in parameters can produce either positive or negative relations between market liquidity and economic efficiency, depending on the activism technology and model parameters. Two results that contrast with the previous literature are that (a) the relationship between market liquidity and economic efficiency is independent of the activist's initial stake for a broad set of activism technologies, and (b) an increase in noise trading can reduce market liquidity because it increases uncertainty about the activist's trades (the activist trades in the opposite direction of noise traders) and thereby increases information asymmetry about the activist's intentions.