Nash equilibria for relative investors via no-arbitrage arguments
Nicole Bäuerle, Tamara Göll
TL;DR
The paper analyzes Nash equilibria for $n$ agents maximizing utilities of relative wealth in an arbitrage-free semimartingale market, allowing general utility functions including CPT. It solves the multi-agent problem by pricing the fixed wealth of competitors, reducing each agent to a standard single-agent optimization with reduced initial wealth, and then reconstructs the Nash strategies via a linear transformation of the single-agent solutions; a mean-field limit is derived that preserves the same linear structure. The main contributions are (i) an explicit Nash-equilibrium representation in terms of unique single-agent optimizers, (ii) applicability to discrete time and CPT, and (iii) a clear connection between Nash equilibria and Pareto-optimal allocations. The framework is illustrated through four explicit market settings (Lévy, stochastic volatility, CPT, and CRR), showing how jumps and nonstandard preferences shape strategic investment, and the mean-field results provide scalable insight for large populations of competing investors.
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.