Nash equilibria for relative investors with (non)linear price impact

Paper: arXiv 2303.18161

Authors: Nicole Bäuerle, Tamara Göll

Abstract

We consider the strategic interaction of $n$ investors who are able to influence a stock price process and at the same time measure their utilities relative to the other investors. Our main aim is to find Nash equilibrium investment strategies in this setting in a financial market driven by a Brownian motion and investigate the influence the price impact has on the equilibrium. We consider both CRRA and CARA utility functions. Our findings show that the problem is well-posed as long as the price impact is at most linear. Moreover, numerical results reveal that the investors behave very aggressively when the price impact is beyond a critical parameter.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of n-player Nash equilibria incorporating both relative performance concerns and non-linear market impact under CARA/CRRA utilities. However, it is predominantly a mathematical and numerical study lacking empirical market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Find Nash Equilibria for Relative Investors with Price Impact] --> B[Model Inputs: n Strategic Investors, Brownian Motion Market, CARA/CRRA Utilities]
    B --> C[Formulate Price Impact Dynamics: Linear vs Nonlinear]
    C --> D[Solve Stochastic Games & Derive Equilibrium Strategies]
    D --> E[Key Finding 1: Well-posedness Holds Only for Sub-Linear to Linear Impact]
    D --> F[Key Finding 2: Aggressive Trading Behavior Beyond Critical Impact Parameter]