Phase transitions in optimal betting strategies

¹ GISC - Grupo Interdisciplinar de Sistemas Complejos and Dpto. de Estructura de la Materia, Física Térmica y Electrónica, Universidad Complutense de Madrid - E-28040 Madrid, Spain
² Institut Elie Cartan, UMR CNRS 7502, Université de Lorraine - BP 239, F-54506 Vandoeuvre- lès-Nancy Cedex, France
³ Gulliver Laboratory, UMR CNRS 7083, PSL Research University, ESPCI - 10 rue Vauquelin, F-75231 Paris Cedex 05, France

Received: 18 May 2020
Accepted: 23 September 2020

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.