Fortune's Formula or the Road to Ruin? The Generalized Kelly Criterion With Multiple Outcomes

Whelan, Karl (2023): Fortune's Formula or the Road to Ruin? The Generalized Kelly Criterion With Multiple Outcomes.

Abstract

You can bet on an event where there are multiple possible winners but only one will actually win. At the odds offered, you think there may be multiple bets worth taking. How much do you place on each bet to maximize your expected utility? We describe how this problem can be solved for concave utility functions and illustrate the properties of the solution. The optimal betting strategy is more aggressive than strategies derived from considering each outcome separately such as the Kelly criterion. The strategy also recommends sometimes placing bets with negative expected returns because they act as hedges against losses on other bets. While this strategy maximizes the bettor's subjective expected utility, if betting odds incorporate a profit margin and reflect underlying probabilities correctly, then this more aggressive approach loses more money and results in lower realized utility.

• Item Type: MPRA Paper
• Language: English
• Keywords: Decision-making under uncertainty; optimal betting; Kelly criterion
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Hegarty, Tadgh and Karl Whelan (2023). Forecasting Soccer Matches With Betting Odds: A Tale of Two Markets, CEPR Discussion Paper No.\ 17949.

Kelly, John L. (1956). ``A New Interpretation of Information Rate.'' \textit{Bell System Technical Journal}, Volume 35, pages 917–926.

Maclean, Leonard, William T. Ziemba and Yuming Li (2005). ``Time to Wealth Goals in Capital Accumulation,'' \textit{Quantitative Finance}, Volume 5, pages 343–355.

Maclean, Leonard, Edward Thorp, Yonggan Zhao and William Ziemba (2011). ``How Does the Fortune's Formula Kelly Capital Growth Model Perform?'' \textit{Journal of Portfolio Management}, Volume 37, pages 96-111.

Moscowitz, Tobias and Kaushik Vasudevan (2022). Betting without Beta. Yale University Working paper.

Poundstone, William (2006). \textit{Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street}, Hill \& Wang.

Samuelson, Paul (1971). ``The Fallacy of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling,'' \texit{Proceedings of the National Academy of Sciences}, Volume 68, pages 2493-2496.

Samuelson, Paul (1979). ``Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long,'' \textit{Journal of Banking and Finance}, Volume 3, pages 305-307.

Smoczynski, Peter and Dave Tomkins (2010). ``An Explicit Solution to the Problem of Optimizing the Allocations of a Bettor’s Wealth When Wagering on Horse Races'', \textit{Mathematical Scientist}, Volume 35, pages 10-17.

Snowberg, Erik and Justin Wolfers, (2008). Examining Explanations of a Market Anomaly: Preferences or Perceptions? in \textit{Handbook of Sports and Lottery Markets}, pages 103–136. Elsevier.

Sung, Ming-Chien and Johnnie Johnson (2010). ``Revealing Weak-Form Inefficiency in a Market for State Contingent Claims: The Importance of Market Ecology, Modelling Procedures and Investment Strategies,'' \textit{Economica}, Volume 305, pages 128-147.

Whelan, Karl (2023). ``US Taxation of Gambling Winnings and Incentives to Bet,'' \textit{Journal of Gambling Studies} https://doi.org/10.1007/s10899-023-10189-z.

Whitrow, Chris (2007). ``Algorithms for Optimal Allocation of Bets on Many Simultaneous Events,'' \textit{Journal of the Royal Statistical Society. Series C. (Applied Statistics)}, Volume 56, pages 607-623.

Wright, Margaret (1992). ``Interior Methods for Constrained Optimization,'' \textit{Acta Numerica}, Volume 1, pages 341–407