# Relative growth optimal strategies in an asset market game

## TL;DR

This paper introduces a unique relative growth optimal strategy in a market game, proving its existence, uniqueness, and asymptotic optimality, and analyzes the market's long-term behavior under this strategy.

## Contribution

It establishes the existence and uniqueness of a relative growth optimal strategy and demonstrates its asymptotic optimality in a competitive market setting.

## Key findings

- Proved the existence and uniqueness of the relative growth optimal strategy.
- Showed that the strategy is asymptotically optimal for capital growth.
- Analyzed the market's asymptotic structure when all investors adopt this strategy.

## Abstract

We consider a game-theoretic model of a market where investors compete for payoffs yielded by several assets. The main result consists in a proof of the existence and uniqueness of a strategy, called relative growth optimal, such that the logarithm of the share of its wealth in the total wealth of the market is a submartingale for any strategies of the other investors. It is also shown that this strategy is asymptotically optimal in the sense that it achieves the maximal capital growth rate when compared to competing strategies. Based on the results obtained, we study the asymptotic structure of the market when all the investors use the relative growth optimal strategy.