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Probability and Cumulative Dice Sums

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Simplified Multinomial Kelly

Here's a simplified version for optimal Kelly bets when you have multiple outcomes (e.g. horse races).

The Smoczynski & Tomkins algorithm, which is explained here (or in the original paper):

https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_horses

Let's say there's a wager that, for every $1 you bet, will return a profit of $b if you win. Let the probability of winning be \(p\), and losing be \(q=1-p\).

The original Kelly criterion says to wager only if \(b\cdot p-q > 0\) (the expected value is positive), and in this case to wager a fraction \( \frac{b\cdot p-q}{b} \) of your bankroll.

But in a horse race, how do you decide which set of outcomes are favorable to bet on? It's tricky, because these wagers are mutually exclusive i.e. you can win at most one.

It turns out there's a simple and intuitive method to find which bets are favorable:

1) Look at \( b\cdot p-q\) for every horse.
2) Pick any horse for which \( b\cdot p-q > 0\) and mark "bet".
3) Adjust the probabilities for the remaining horses by dividing all win probabilities by \( \frac{1}{1-p} \) so they add up to 1 again ("renormalize").
4) Repeat!

That's it.

This should be substantially easier to understand than the exposition in Smoczynski & Tomkins.

The intuitive reasoning for why this should work is that you only need betting on a horse to be conditionally favorable assuming the other horses you've bet on don't win. That is, it must be a positive hedge.

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