Extending Kelly Staking Strategies to Peer-to-Peer Betting Exchanges

Abstract

In his seminal paper J. L. Kelly Jr. linked information theory with a staking system for calculating the stake of a favourable bet. Kelly’s system is elegant, and under certain conditions can be said to be optimal. In the fifty years since, his ideas have been extended to financial markets, with much success. There have been fewer such extensions to betting. In many countries gambling has been liberalised and the range of bets available has increased markedly. One recent innovation has made a particular impact, i.e. the introduction of peer-to-peer betting exchanges. Betting exchanges allow their members to place bets with each other, behaving as bookmakers, something Kelly did not consider. This and the Internet have increased the number of bets available on a given event. These are often highly correlated. We extend Kelly’s method to incorporate lay bets; and we do so analytically. We extend this numerically to incorporate highly correlated bets and commission, another feature unique to exchanges. We further extend this solution to include bets previously placed, making possible a profit before the match has started. Whilst extending the optimal solution to cover new situations we make use of the increase in data available to examine the applicability of the original assumptions. We run Monte-Carlo simulations to show when these assumptions are likely to fail. One of the features of betting exchanges is the ability to increase or reduce the bet size as the odds change over time. We examine the optimal bets to add to an existing portfolio of bets if the odds have changed. We briefly consider changes in the modelled probabilities. Finally, we suggest a novel way of reversing Kelly’s strategy to provide a market making strategy, even when there is some uncertainty in where the market prices should be. However we also show that if that uncertainty is too great such a strategy is unlikely to be profitable.Open Acces