On the Analysis of Kelly Criterion and Its Application

  • Mu-En Wu
  • Wei-Ho Chung
  • Chia-Jung LeeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11432)


We analyze the return of a game for a gambler after bidding \( T \) time steps. Consider a gamble with known odds and win rate, the optimal solution is to use Kelly criterion which determines the optimal fraction in each bidding step. In this paper we show that the logarithm of return when bidding optimal fraction is \( {\text{KL}}\left( {{\text{R}}||{\text{P}}(b)} \right) - {\text{KL}}\left( {{\text{R}}||{\text{P}}} \right) \), where \( {\text{R}} \) is the proportion of winning\losing outcome in \( T \) time steps, \( {\text{P}}\left( b \right) \) is the risk-neutral probability corresponding to odds \( b \), and \( {\text{P}} \) is the gambler’s individual belief about the win probability of the game. This argument shows that, in a gamble with fixed odds, the KL divergence of the win\lose proportion, say \( {\text{R}} \), and the win rate, say \( {\text{P}} \), determines the portion of the losing amount. On the other hand, the profit is determined by the proportion \( {\text{R}} \) and the odds \( b \), irrelevant to win probability \( {\text{P}} \). Any improvement is not obtainable even when the win probability is estimated precisely in advance.


Kelly criterion Optimal fraction KL-divergence 


  1. 1.
    Kelly, J.L.: A new interpretation of information rate. Bell Labs Tech. J. 35(4), 917–926 (1956)MathSciNetCrossRefGoogle Scholar
  2. 2.
    William, L.R.: How I Made One Million Dollars Last Year Trading Commodities. Windsor Books, Publisher (1979)Google Scholar
  3. 3.
    Thorp, E.: Beat the Dealer: A Winning Strategy for the Game of Twenty-OneGoogle Scholar
  4. 4.
    Vince, R.: The Mathematics of Money Management, Risk Analysis Techniques for Traders, A Wiley Finance Edition. Wiley (1992)Google Scholar
  5. 5.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, Wiley Series in Telecommunications and Signal Processing (2006)Google Scholar
  6. 6.
    Chou, J.-H., Lu, C.-J., Wu, M.-E.: Making profit in a prediction market. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 556–567. Springer, Heidelberg (2012). Scholar
  7. 7.
    Chen, C.-H., Chen, Y.-H., Wu, M.-E., Hong, T.-P.: A sophisticated optimization algorithm for obtaining a group trading strategy portfolio and its stop-loss and take-profit points. In: Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, IEEE SMC 2018, 7–10 October 2018, Miyazaki, Japan (2018)Google Scholar
  8. 8.
    Hsieh, C.H., Barmish, B.R.: On Kelly betting: some limitations. In: Proceedings of Annual Allerton Conference on Communication, Control, and Computing, Monticello, pp. 165–172 (2015)Google Scholar
  9. 9.
    Hsieh, C.H., Barmish, B.R., Gubner, J.A.: Kelly betting can be too conservative. In: Proceedings IEEE 55th Conference Decision Control, December 2016, vol. 476, pp. 3695–3701 (2016)Google Scholar
  10. 10.
    Hsieh, C.H., Barmish, B.R.: On drawdown-modulated feedback in stock trading. In: Proceedings of the IFAC World Congress, Toulouse, France, pp. 975–981 (2017)Google Scholar
  11. 11.
    Chen, C.-H., Chen, Y.-H., Wu, M.-E.: A GGA-based algorithm for group trading strategy portfolio optimization. In: Proceedings of the 4th Multidisciplinary International Social Networks Conference (MISNC 2017), 17–19 July 2017, Bangkok, Thailand (2017)Google Scholar
  12. 12.
    Wu, M.-E., Tsai, H.-H., Tso, R., Weng, C.-Y.: An adaptive Kelly betting strategy for finite repeated games. In: Proceedings of the 9th International Conference on Genetic and Evolutionary Computing (ICGEC 2015), 26–28 August 2015, Yangon, Myanmar (2015)Google Scholar
  13. 13.
    Wu, M.-E., Chung, W.-H.: A novel approach of option portfolio construction using the Kelly criterion. Accepted and to appear in IEEE Access (SCI)Google Scholar
  14. 14.
    Wu, M.-E., Wang, C.-H., Chung, W.-H.: Using trading mechanisms to investigate large futures data and its implications to market trends. Soft. Comput. 21(11), 2821–2834 (2017). (SCI)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Information and Finance ManagementNational Taipei University of TechnologyTaipeiTaiwan
  2. 2.Department of Electrical EngineeringNational Tsing Hua UniversityHsinchuTaiwan
  3. 3.School of Big Data ManagementSoochow UniversityTaipeiTaiwan

Personalised recommendations