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If the Total Overround is greater than 6.18% and the Bookmaker Sigma is less than 5.27%, then the best strategy is not to bet, otherwise
If the Total Overround is less than 5.44%, then the best strategy is to Kelly stake
It also suggests that, within the range of parameter values explored, the variability to Kelly staking grows with Bookmaker imprecision and shrinks with the product of Punter Bias and Bookmaker Overround.
Using Eureqa's Formulize to build a model of the ROI to Kelly staking, we find that one of the best fitting models, with an R squared in excess of 85%, is:
Expected Kelly ROI = 2.143Bookie Sigma Bookie Total Overround Bookie SigmaPunter Bias 0.463Punter Sigma
the TAB Bookmaker can be thought of as Converse Shoes Uk Womens a Bookmaker with zero bias and a 5 5.5% sigma, and the Head to Head Probability Predictor can be thought of as a Punter with a Converse Gray
We also find that, for every set of parameters in the simulation, the expected return to Kelly staking exceeds that for Level staking, which is broadly consistent with what we found when exploring the larger parameter space, but the correlation between the average Log Probability Score and the ROI to Kelly staking is, in absolute terms, always lower than the correlation between the average Brier Score and the ROI to Kelly staking, which is contrary to what we found when exploring the larger scenario space.
Using RWeka to create simple rules for when to Kelly stake and when to beat a considered retreat from wagering, the first few rules we're offered are:
(Note that these images can be clicked for larger versions.)
What's also interesting is that the variability of the Converse Black Shoes For Men
+1 2% bias and a 10 12% sigma
Expected SD Kelly ROI = 0.067 + 0.415Bookie Sigma 3.600Punter BiasBookie Total Overround
If the Total Overround is less than 5.63% and the Punter Sigma is less than 11.00%, then the best strategy is, again, to Kelly stake, otherwise
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returns to Kelly staking is positively related to Bookmaker Sigma, broadly unrelated to Punter Sigma, and negatively related to Punter Bias.
For about 70% of the remaining scenarios the recommendation is not to bet.
So, not only does the expected return to Kelly staking rise with the Bookmaker's Sigma, so too does the variability of that return.
If the Total Overround is greater than 5.34%, the Punter Sigma is less than 10.73%, and the Bookmaker Sigma exceeds 5.14%, then the best strategy is to Kelly stake, otherwise
If the Total Overround is less than 5.88% and the Bookmaker Sigma exceeds 5.24%, then the best strategy is to Kelly stake, otherwise
Simulating these parameter ranges, with an LPSO like Bookmaker and with his overround varying in the range 5% to 6.5%, reveals that the return to Kelly staking for a Punter with Bias and Sigma in these ranges (who, unlike MAFL's Head to Head Fund, wagers on Away as well as Home teams) is positively related to Bookmaker Sigma .
vs TAB Bookmaker
And, finally, using Eureqa's Formulize to build a model of the standard deviation of the ROI to Kelly staking, we find that one of the best fitting models, but with an R squared of only around 16%, is:
This suggests that, as we've found previously, the ROI to Kelly staking is heavily dependent on the Bookmaker's precision and, to a lesser extent, on the Punter's. As well, the ROI to Kelly staking drops percent for percent with the Bookmaker's Total Overround.
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