Chapter 4: Further Reading — Bankroll Management Fundamentals

Foundational Papers

1. Kelly, J. L. Jr. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal, 35(4), 917-926.

The original paper that started it all. Kelly formulated the criterion in the context of a gambler receiving noisy information over a communication channel. While the paper is technical and rooted in information theory, the core insight -- that maximizing the expected logarithm of wealth leads to long-run optimal growth -- is elegantly simple. Every serious student of bankroll management should read this paper at least once.

2. Thorp, E. O. (1969). "Optimal Gambling Systems for Favorable Games." Review of the International Statistical Institute, 37(3), 273-293.

Edward Thorp, who famously applied the Kelly Criterion to blackjack and later to financial markets, provides a rigorous treatment of optimal staking for games with positive expected value. Thorp addresses practical complications including simultaneous bets, variable odds, and finite time horizons. This paper bridges the gap between Kelly's theoretical framework and real-world gambling applications.

3. Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." In S. A. Zenios & W. T. Ziemba (Eds.), Handbook of Asset and Liability Management, Volume 1 (pp. 385-428). North-Holland.

A comprehensive overview by Thorp that traces the history, theory, and application of the Kelly Criterion across multiple domains. Particularly valuable for the sports betting sections, where Thorp discusses practical adjustments to the pure Kelly approach. Highly accessible and recommended as a starting point for advanced study.

4. Breiman, L. (1961). "Optimal Gambling Systems Which Minimize the Probability of Ruin." Journal of the Society for Industrial and Applied Mathematics, 9(3), 392-407.

Breiman provides the mathematical foundation for understanding when and why Kelly staking minimizes the probability of ruin and maximizes the growth rate. His proof that the Kelly strategy asymptotically outperforms any other strategy is a cornerstone result. Requires comfort with probability theory and stochastic processes.

5. Latane, H. A. (1959). "Criteria for Choice Among Risky Ventures." Journal of Political Economy, 67(2), 144-155.

Latane independently discovered the connection between maximizing geometric mean returns and optimal long-term growth, contemporaneous with Kelly's work. This paper approaches the problem from an economics/finance perspective rather than information theory, providing a complementary viewpoint that many readers find more intuitive.

Books

6. Poundstone, W. (2005). Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Hill and Wang.

An engaging narrative history of the Kelly Criterion, from Claude Shannon's information theory lab at MIT to Ed Thorp's blackjack exploits to the trading floors of Wall Street. Poundstone makes the mathematics accessible through storytelling and explains the famous debate between Kelly advocates and Paul Samuelson's efficient market camp. Essential reading for context and motivation.

7. Thorp, E. O. (2017). A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market. Random House.

Thorp's autobiography details his journey from academic mathematician to casino-beating blackjack player to hedge fund pioneer, with the Kelly Criterion as a recurring theme throughout. The book provides invaluable practical insights into how a master practitioner actually implements bankroll management principles under real-world constraints and uncertainty.

8. Epstein, R. A. (2009). The Theory of Gambling and Statistical Logic. 2nd edition. Academic Press.

A mathematically rigorous textbook covering the theory of gambling, including extensive treatment of optimal staking, risk of ruin, and bankroll management. Chapter coverage of the Kelly Criterion is thorough and precise. Suitable for readers with a strong mathematical background who want formal proofs and derivations.

9. MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (Eds.) (2011). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific.

The definitive academic anthology on the Kelly Criterion, collecting the most important papers in the field from 1956 to 2010. Includes Kelly's original paper, Thorp's key contributions, critiques by Samuelson, and modern applications to portfolio management. At over 800 pages, it is a comprehensive reference rather than a casual read, but it is the single most complete source on the topic.

10. Ziemba, W. T., & Hausch, D. B. (1986). Betting at the Racetrack. Dr. Z Investments.

While focused on horse racing, this book provides an excellent practical treatment of Kelly staking in the context of parimutuel betting. Ziemba and Hausch address the real-world complications of implementing Kelly, including transaction costs (track take), estimation error, and multiple simultaneous bets. Many of the insights transfer directly to sports betting.

Sports Betting Specific

11. Miller, E. (2015). "Defining and Measuring Edge in Sports Betting." Unpublished manuscript / online resource.

Miller provides a clear framework for understanding and quantifying edge in sports betting, which is the essential input to the Kelly Criterion. His treatment of closing line value (CLV) as a proxy for edge estimation is particularly relevant for bettors who struggle with the "how do I know my true probability?" problem that makes Kelly hard to apply in practice.

12. Cortis, D. (2015). "Expected Values and Variances in Bookmaker Payouts: A Theoretical Approach Towards Setting Limits on Odds." Journal of Prediction Markets, 9(1), 1-14.

An academic treatment of how bookmaker odds relate to true probabilities, with implications for edge estimation and bankroll management. Understanding the structure of bookmaker margins is essential for estimating the "p" that goes into the Kelly formula.

13. Kaunitz, L., Zhong, S., & Kreber, J. (2017). "Beating the Bookies with Their Own Numbers -- and How the Online Sports Betting Market is Rigged." arXiv preprint arXiv:1710.02824.

This paper demonstrates a betting strategy based on odds discrepancies across bookmakers and addresses the bankroll management challenges of implementing such a strategy at scale. The authors' experience with bet size optimization and account restrictions provides a real-world case study in the practical limits of Kelly-based approaches.

Risk Management and Advanced Topics

14. Vince, R. (1992). The Mathematics of Money Management: Risk Analysis Techniques for Traders. John Wiley & Sons.

While written for financial traders rather than sports bettors, Vince's treatment of optimal f (his term for the Kelly-analogous concept), risk of ruin, and drawdown analysis is directly applicable to bankroll management. His concept of "leverage space" provides a geometric intuition for understanding multi-bet portfolios that many bettors find enlightening.

15. MacLean, L. C., Thorp, E. O., Zhao, Y., & Ziemba, W. T. (2011). "How Does the Fortune's Formula Kelly Capital Growth Model Perform?" Journal of Portfolio Management, 37(4), 96-111.

An empirical examination of Kelly-based strategies using historical financial data. The authors compare full Kelly, fractional Kelly, and other strategies on metrics including terminal wealth, drawdowns, and Sharpe ratios. Their findings strongly support fractional Kelly approaches and provide quantitative evidence for the practical superiority of half Kelly over full Kelly.

16. Samuelson, P. A. (1971). "The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling." Proceedings of the National Academy of Sciences, 68(10), 2493-2496.

The most famous critique of the Kelly Criterion, by Nobel laureate Paul Samuelson. Samuelson argues that maximizing geometric mean is not universally optimal and depends on the investor's utility function. This short paper (intentionally written with monosyllabic words to emphasize simplicity) presents the strongest intellectual counterargument to Kelly. Understanding this critique is essential for a balanced perspective on bankroll management.

17. Haghani, V., & Dewey, R. (2016). "Rational Decision-Making Under Uncertainty: Observed Betting Behavior on the Outcome of a Coin Flip Experiment." Working paper.

A fascinating experimental study where finance professionals were given a biased coin (60% heads) and $25, then allowed to bet repeatedly. Despite the large edge, most participants used suboptimal strategies (many went broke, and very few used Kelly). This paper vividly demonstrates why bankroll management education matters -- even financially sophisticated people make catastrophic staking errors.

18. Benter, W. (2008). "Computer Based Horse Race Handicapping and Wagering Systems: A Report." In D. B. Hausch, V. S. Y. Lo, & W. T. Ziemba (Eds.), Efficiency of Racetrack Betting Markets (2008 edition, pp. 183-198). World Scientific.

Bill Benter's landmark paper on his horse racing model, which generated hundreds of millions of dollars in profits. While the modeling aspects are specific to horse racing, Benter's discussion of bankroll management, Kelly implementation with simultaneous bets, and the practical challenges of scaling a betting operation is directly relevant to serious sports bettors.

Online Resources

19. Pinnacle Sports Betting Resources: "Staking Methods" series.

Pinnacle, one of the sharpest bookmakers, publishes educational articles on staking and bankroll management. Their series covers flat staking, proportional staking, Kelly, and fractional Kelly with practical examples from sports betting. Freely available and written for a practitioner audience rather than an academic one.

20. "The Kelly Criterion" -- Wizard of Odds (wizardofodds.com).

Michael Shackleford's treatment of the Kelly Criterion includes clear derivations, worked examples, and an interactive calculator. The site also provides risk-of-ruin calculators for various scenarios. A reliable free resource for checking calculations and building intuition.