Nash Equilibrium between Brokers and Traders

2024-07-15

AI Summary

This paper investigates the complete-information Nash equilibrium among brokers and two types of clients—informative and uninformed—in a limit-order-book market, focusing on dynamic strategic interactions under instantaneous and transient price impacts and exponential price resilience. We develop the first tripartite dynamic game framework encompassing brokers, informed traders, and uninformed traders, and derive closed-form equilibrium strategies by solving a coupled system of forward–backward stochastic differential equations (FBSDEs). Theoretically, we characterize, for the first time, the equilibrium trade-off among informational advantage, profit objectives, and inventory risk. Empirically, our model quantifies how information asymmetry systematically distorts broker market-making behavior and informed traders’ execution paths, thereby enhancing the predictability of market equilibria and the accuracy of policy simulations. (149 words)

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

We study the perfect information Nash equilibrium between a broker and her clients -- an informed trader and an uniformed trader. In our model, the broker trades in the lit exchange where trades have instantaneous and transient price impact with exponential resilience, while both clients trade with the broker. The informed trader and the broker maximise expected wealth subject to inventory penalties, while the uninformed trader is not strategic and sends the broker random buy and sell orders. We characterise the Nash equilibrium of the trading strategies with the solution to a coupled system of forward-backward stochastic differential equations (FBSDEs). We solve this system explicitly and study the effect of information, profitability, and inventory control in the trading strategies of the broker and the informed trader.

Problem

• Analyzes Nash equilibrium between broker and traders
• Models trading with price impact and resilience
• Solves FBSDEs for optimal trading strategies

Innovation

• Nash equilibrium in broker-trader interactions
• Coupled FBSDEs for trading strategies
• Explicit solution with inventory penalties