Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion

The Kelly betting criterion ignores uncertainty in the probability of winning the bet, and uses an estimated probability. In general, such replacement of population parameters by sample estimates gives poorer out-of-sample than in-sample performance. We show that to improve out-of-sample performance the size of the bet should be shrunk in the presence of this parameter uncertainty, and compare some estimates of the shrinkage factor. From a simulation study and from an analysis of some tennis betting data we show that the shrunken Kelly approaches developed here offer an improvement over the ‘raw’ Kelly criterion. One approximate estimate of the shrinkage factor gives a ‘back of envelope’ correction to the Kelly criterion that could easily be used by bettors. We also study bet shrinkage and swelling for general risk-averse utility functions, and discuss the general implications of such results for decision theory.