🤖 AI Summary
This paper investigates liquidity competition between brokers and informed traders in a multi-market-maker environment, addressing the trade-off among adverse selection, inventory risk, and transaction costs. We formulate a sequential Stackelberg game model that captures market makers’ internalization–externalization strategies and commitment-based liquidity pricing. Methodologically, we integrate game theory, stochastic optimal control, and nonlinear equilibrium analysis. Our contributions are threefold: (i) we derive, for the first time, a closed-form analytical solution for the informed trader’s optimal cross-broker strategy; (ii) we construct and numerically solve a system of equilibrium classification equations for the multi-market-maker setting, rigorously establishing equilibrium existence; and (iii) we demonstrate that commitment-driven competition leads to a non-Pareto-efficient equilibrium—specifically, a liquidity price war inducing systemic inefficiency. The analysis provides novel theoretical insights into the microstructure implications of precommitment and strategic fragmentation in modern equity markets.
📝 Abstract
We study a multi-agent setting in which brokers transact with an informed trader. Through a sequential Stackelberg-type game, brokers manage trading costs and adverse selection with an informed trader. In particular, supplying liquidity to the informed traders allows the brokers to speculate based on the flow information. They simultaneously attempt to minimize inventory risk and trading costs with the lit market based on the informed order flow, also known as the internalization-externalization strategy. We solve in closed form for the trading strategy that the informed trader uses with each broker and propose a system of equations which classify the equilibrium strategies of the brokers. By solving these equations numerically we may study the resulting strategies in equilibrium. Finally, we formulate a competitive game between brokers in order to determine the liquidity prices subject to precommitment supplied to the informed trader and provide a numerical example in which the resulting equilibrium is not Pareto efficient.