Nash equilibria for relative investors via no-arbitrage arguments

TL;DR

This paper characterizes Nash equilibrium investment strategies for multiple agents in arbitrage-free markets, linking them to classical utility maximization and providing applications and comparisons with existing literature.

Contribution

It introduces a novel approach to derive all Nash equilibria in multi-agent utility maximization using no-arbitrage arguments and classical pricing theory.

Findings

Explicit formulas for Nash equilibria in terms of single-agent solutions

Application to four specific financial market models

Comparison showing advantages over previous methods

Abstract

Within a common arbitrage-free semimartingale financial market we consider the problem of determining all Nash equilibrium investment strategies for $n$ agents who try to maximize the expected utility of their relative wealth. The utility function can be rather general here. Exploiting the linearity of the stochastic integral and making use of the classical pricing theory we are able to express all Nash equilibrium investment strategies in terms of the optimal strategies for the classical one agent expected utility problems. The corresponding mean field problem is solved in the same way. We give four applications of specific financial markets and compare our results with those given in the literature.