“The Kelly Criterion in Blackjack Sports Betting, and the Stock Market”, Edward O. Thorp2006 (decision theory; similar)⁠:

[By Edward O. Thorp] The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, ie. how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an “investor”, looks for “excess risk adjusted return”.

In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth (“maximize expected logarithmic utility”). The criterion is known to economists and financial theorists by names such as the “geometric mean maximizing portfolio strategy”, maximizing logarithmic utility, the growth-optimal strategy, the capital growth criterion, etc.

The author initiated the practical application of the Kelly criterion by using it for card counting in blackjack. We will present some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. Then we illustrate its recent use in a successful casino sports betting system. Finally, we discuss its application to the securities markets where it has helped the author to make a 30 year total of 80 billion dollars worth of “bets”.

[Keywords: Kelly criterion, betting, long run investing, portfolio allocation, logarithmic utility, capital growth]

0. Abstract

1. Introduction

2. Coin tossing

3. Optimal growth: Kelly criterion formulas for practitioners

   1. The probability of reaching a fixed goal on or before n trials

   2. The probability of ever being reduced to a fraction x of this initial bankroll

   3. The probability of being at or above a specified value at the end of a specified number of trials

   4. Continuous approximation of expected time to reach a goal

   5. Comparing fixed fraction strategies: the probability that one strategy leads another after n trials

4. The long run: when will the Kelly strategy “dominate”?

5. Blackjack

6. Sports betting

7. Wall Street: the biggest game

   1. Continuous approximation

   2. The (almost) real world

   3. The case for “fractional Kelly”

   4. A remarkable formula

8. A case study

   1. The constraints

   2. The analysis and results

   3. The recommendation and the result

   4. The theory for a portfolio of securities

9. My experience with the Kelly approach

10. Conclusion

11. Acknowledgments

12. Appendix A: Integrals for deriving moments of E_∞

13. Appendix B: Derivation of formula (3.1)

14. Appendix C: Expected time to reach goal

15. References

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