Phase transitions in optimal strategies for betting

TL;DR

This paper investigates optimal betting strategies balancing growth and risk, revealing a phase transition between risky and risk-free strategies in horse race models, and establishing bounds on growth fluctuations.

Contribution

It introduces a new class of optimal betting strategies that balance growth and risk, and uncovers a phase transition phenomenon in betting strategies.

Findings

Identifies a phase transition between risky and risk-free strategies.

Derives a bound on the average growth rate related to thermodynamic uncertainty.

Proves no other phase transitions exist between Kelly's strategy and risk-free betting.

Abstract

Kelly's criterion is a betting strategy that maximizes the long term growth rate, but which is known to be risky. Here, we find optimal betting strategies that gives the highest capital growth rate while keeping a certain low value of risky fluctuations. We then analyze the trade-off between the average and the fluctuations of the growth rate, in models of horse races, first for two horses then for an arbitrary number of horses, and for uncorrelated or correlated races. We find an analog of a phase transition with a coexistence between two optimal strategies, where one has risk and the other one does not. The above trade-off is also embodied in a general bound on the average growth rate, similar to thermodynamic uncertainty relations. We also prove mathematically the absence of other phase transitions between Kelly's point and the risk free strategy.