Set the Derivative of the Profit Function Equal to Zero

Apparently the idea explored in my prior post, which I thought was a novel one, is not so new. Mark Cramer long ago had asserted a critical corollary in handicapping; that the wager value of handicapping data is inversely proportional to its degree of use. Information that allows a handicapper to duplicate the public's laudable 33% win rate will also duplicate the public's 9% flat bet loss on favorites. How did I learn of Mark Cramer? I learned of Cramer from James Quinn's masterpiece, The Best of Thoroughbred Handicapping. This is by far the best book for those new to Handicapping as it provides a survey of the literature. That notwithstanding, however, Chapter 14 of said book is one of the worst I have read. After advocating for Optimal Betting via the Kelly Criterion in Chapter 2, Quinn seems to indicate that longer odds should have higher bets. While there is a more devastating reason that this is absolutely false, even the criterion he advocates teaches the exact opposite. Assuming a flat bet ROI of 10%, the percent of capital wagered via the Kelly Criterion is, for example, 10% on a 1:1 shot while it is 0.5% on a 20:1 shot.

Unfortunately for bettors, the Kelley Criterion is not usually the limiting factor for determining bet size, at least not for bettors with any appreciable bankroll. The tricky thing about pari-mutuel betting is that it allows the greedy gambler to destroy the overlay he meant to exploit. Placing a large bet on a long shot will significantly alter the odds.

The math is somewhat complicated and it is out of scope for me to present it here. The profit function is the bet size times the expectation value. The expectation value is the probability of winning times the odds being offered. The odds being offered are a function of the total wager pool, the wager pool on the horse in question, and the house cut. Two of these are a function of the bet size. Only by setting the derivative of the profit function to zero can the maxima of the profit function be determined.

Keep in mind too, that this can be done with the best of intentions but that the odds are still subject to change after you place your wager. It is also the case that the expectation values that feed into the Kelly Criterion optimal bet size are also a function of wager size. These two functions (profit function and optimum bet) can be plotted against bet size. There are generally four possibilities. There is no profitable bet, the Kelly Criterion dictates, the Profit Function Dictates, or the two functions intersect. In the event where a profitable bet is possible, I advocate 50 - 100% of the lesser of the three (the Kelly Optimum, the Intersection, or the Profit Function Maximum).

From what I have read in the handicapping literature, this is my new contribution (I think, I haven't stumbled across it yet). Note that the profit function derivative method can also be applied to Dr. Z's system for place and show betting on favorites.

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