Probability and Intuition by Vijay Sujith

Author:Vijay, Sujith [Vijay, Sujith]
Language: eng
Format: epub
Published: 2020-09-29T00:00:00+00:00

Chapter 7: The Kelly Criterion

Suppose we are asked to make bets on the outcome of twenty successive tosses of a biased coin, where the probability of the coin showing heads on each toss equals 0.6. (Never mind why someone would offer such a bet, although this is certainly something that should be asked in real life.) Obviously it makes sense to always bet on heads, but if the initial capital is $1000, how much money should be bet on each toss to maximize the expected winnings? Clearly, since we are more likely to win than lose, we can afford to be somewhat brave. But there is a fine line between brave and foolhardy, and that is what we are trying to figure out.

Suppose we bet half of our current capital on each bet. So in a typical scenario, our capital gets multiplied by 1.5 twelve times, and gets halved eight times. Since (1.5)¹² (0.5)⁸ = 0.5068, we surprisingly end up losing nearly half our capital in spite of the odds being in our favour. Clearly, betting 50% of our bankroll is way too aggressive.

What if we only bet 5% of our current capital on each toss? Then our final earnings get magnified by (1.05)¹² (0.95)⁸ = 1.1914 in a typical scenario. In other words, we make a return of 19% on our original investment. Not bad, but can we do better?

Let us work in a general setting. Suppose we have N bets and our probability of winning each bet is p . We will assume that p > 0.5, as we should not be betting at all if this is not the case. (Note that this only applies when the potential gain and potential loss on each bet are equal; otherwise betting on the less likely outcome could very well yield a positive expectation.) Let x be the fraction of our capital that we bet each time. Then the typical value W of our final wealth, starting from an initial capital W₀ is

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