Relative performance criteria of multiplicative form in complete markets

Paper: arXiv 2303.07941

Authors: Anastasiya Tanana

Abstract

We consider existence and uniqueness of Nash equilibria in an $N$-player game of utility maximization under relative performance criteria of multiplicative form in complete semimartingale markets. For a large class of players’ utility functions, a general characterization of Nash equilibria for a given initial wealth vector is provided in terms of invertibility of a map from $\mathbb{R}^N$ to $\mathbb{R}^N$. As a consequence of the general theorem, we derive existence and uniqueness of Nash equilibria for an arbitrary initial wealth vector, as well as their convergence, if either (i) players’ utility functions are close to CRRA, or (ii) players’ competition weights are small and relative risk aversions are bounded away from infinity.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical analysis of N-player games with relative performance criteria in complete semimartingale markets using advanced stochastic calculus and game theory. However, it is purely mathematical without any empirical backtesting, data analysis, or numerical simulations.

Research Flowchart

  flowchart TD
    A["Research Question: Existence & Uniqueness of Nash Equilibria in Relative Performance Games"] --> B["Inputs: Utility Functions, Semimartingale Markets, Wealth & Competition Weights"]
    B --> C["Formulate N-Player Utility Maximization Model"]
    C --> D["Characterize Equilibrium via Invertibility of N-Dimensional Map"]
    D --> E{"Special Cases & Conditions"}
    E -- "Case 1: Near CRRA Utilities" --> F["Outcome: Proven Nash Equilibrium Existence, Uniqueness & Convergence"]
    E -- "Case 2: Small Competition Weights & Bounded Risk Aversion" --> F