Journal of Investment Strategies

Risk.net

Optimal betting sizes for the game of blackjack

Ralph Vince and Qiji Zhu

  • The authors develop the theory of Kelly and Thorp for determining optimal bet sizes for blackjack.
  • The ratio of return to bet size is approximately proportional to the ratio of return to drawdown, and is a reasonable proxy for risk-adjusted returns.
  • Bet sizes that maximize this measure or its marginal increase are reasonable choices for maximizing risk-adjusted returns in the case of a finite number of plays.
  • Theoretical analysis and computer simulation shows that these alternative choices presented for bet sizes are much more conservative than what the Kelly-Thorp theory suggests.

ABSTRACT

We develop the theory of Kelly and Thorp for determining optimal bet sizes for blackjack by incorporating two vital, practical considerations. We first address the necessary requirement that only a finite number of plays may transpire. Further, we seek to maximize risk-adjusted returns, which are often a more likely criterion that players subscribe to. We show that the ratio of return to bet size is approximately proportional to the ratio of return to drawdown, and is a reasonable proxy for risk-adjusted returns. Thus, bet sizes that maximize this measure or its marginal increase are reasonable choices for maximizing risk-adjusted returns in the case of a finite number of plays. Theoretical analysis and computer simulation show that the alternative choices presented for bet sizes are much more conservative than the Kelly-Thorp theory suggests, and hence make sense in practice. In principle, the analysis and results here also apply to money management problems for investment as well as more generalized cumulative resource allocation considerations involving risk.

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