Abstract
The concept of shrinking bet size in Kelly betting to minimize estimated frequentist risk has recently been mooted. This rescaling appears to conflict with Bayesian decision theory through the likelihood principle and the complete class theorem; the Bayesian solution should already be optimal. We show theoretically and through examples that when the modeldetermining the likelihood function is correct, the prior distribution (if not dominated by data) is `correct' in a frequentist sense, and the posterior distribution is proper, then no further rescaling is required. However, if the model or the prior distribution is incorrect, or the posterior distribution improper, frequentist risk minimization can be a useful technique. We discuss how it might best be exploited. Another example, from maintenance, is used to show the wider applicability of the methodology; these conclusionsapply generally to decision-making.
Highlights
Bet ShrinkageBaker and McHale (2013) introduced the concept of bet shrinkage in Kelly betting, for the case where the probability of winning is not accurately known
They derived approximations to the amount of shrinkage required, and showed that shrinking bet size increased expected utility both in a simulated gambling situation, and in tennis betting. Their methodology is frequentist, and can be applied in contexts other than Kelly betting. It will doubtless raise Bayesian hackles, because Bayesian decision theory would seem to state that no such rescaling is required, and it is underpinned by two important theorems, the likelihood principle and the complete class theorem (see e.g. Robert (2007))
We examine what is implied in shrinking bet sizes from the viewpoint of Bayesian decision theory
Summary
Baker and McHale (2013) introduced the concept of bet shrinkage in Kelly betting, for the case where the probability of winning is not accurately known They derived approximations to the amount of shrinkage required, and showed that shrinking bet size increased expected utility both in a simulated gambling situation, and in tennis betting. This would give some information about θ, but it would not be the very accurate knowledge we could gain from simple probability theory, where the only inexactness in our knowledge of θ arises from tiny inaccuracies in the manufacture of the dice We focus on this example, Baker and McHale (2013) consider the general problem of deciding how much to bet, if the bettor has only an inexact estimate (maybe a guess) of the probability of winning.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Statistics and Probability
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
