A Dynamic Kelly Criterion for Games with a Maximum Payout

Abstract

In 1956, J.L. Kelly provided the formula for the optimal portion f of one’s capital to bet on a coin toss with favorable odds. We show that when the bet can be repeated a fixed number of times, but the game has a maximum payout M, the optimal portion f becomes a function of both k, the number of games remaining, and B, the current bankroll. We describe this function $f_k(B)$ and quantify the advantages gained over the fixed Kelly criterion. We also discuss the impact of changing the underlying utility function from a logarithmic utility to a more linear utility, and thereby optimizing a risk-reward ratio.

MSC:

• 91
• 60

Acknowledgment

The authors are grateful for helpful conversations with Jesse Frey, Tasha Boland, Ryan Harkin, and Joachim Rebholz. Furthermore, the authors are grateful to the anonymous reviewers for their careful reading and insightful advice.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Notes

1 A difference in the design of the original experiment was that the amount of the maximum payout was only revealed to players when they were close to reaching the maximum. Also the subjects were allowed to play as many games as they could in 30 minutes. On average the players completed just over 100 games.

2 Kelly’s criterion assumes favorable odds. While we humans often bet on games with negative expected value (as in casinos and lotteries), Kelly’s question about the optimal portion of one’s capital to bet is rational only for games with positive expected value, that is, $\mathit{\text{bp}}-(1-p)c>0$. In the case (b = 1, c = 1), this means p > 1∕2.

3 For readability we assume in this paper that a $1 bet wins $1 on a successful trial and loses $1 otherwise. That is, we set $b=c=1$.

4 A logarithmic utility function models the assumption that players measure their success (‘utility’) using the percentage increase of their capital rather than an absolute increase. A linear utility function, in contrast, would assign the same utility to increasing one’s capital by, say, $1000, whether one’s current capital is $1000 or $1 million. This differentiation in the utility of money was first introduced by G. Cramer in 1728 in a letter to N. Bernoulli in an attempt to solve the St. Petersburg paradox, and has since become fundamental to economic models [4].

5 All our results are obtained via Monte Carlo Simulations, implemented in RStudio, with the number of simulations being $900,000$.

6 i.e., choosing f for each trial to a be a uniformly distributed random variable on the unit interval.

7 John Maynard Keynes, 1923.

8 The notion of a fractional, but fixed, Kelly proportion has been discussed by other authors as a way to reduce risk in the presence of uncertainties, such as not knowing the precise magnitude of the winning probability [2, 14], or in the presence of a limited time horizon [10]. The notion of a fractional Kelly as a function of the ratio of B/M and time remaining, however, is new.

9 Conceptually, this is a similar stochastic process to Bellman’s dynamic programming in reinforcement learning, where the optimal path through states is determined by maximizing the reward of actions, including a cost function of movement, when advancing backwards from the end goal [14, 15].

10 Note that the bankroll B is a “closed system,” that is, winnings flow into the account, losses flow out and there is no replenishment from the outside.

11 Unless, of course, the maximum payout is much greater than the starting capital, i.e., $M>{(2p)}^{N}B$.

Additional information

Notes on contributors

Nick Rubino

NICK RUBINO is a public equities healthcare and biotechnology investor, also completing his masters in mathematics from Villanova University. A love of math and making connections through interdisciplinary learning have always been his primary motivations for investing and applied interests in finance, economics, biology, and beyond. Nick is otherwise happiest surrounded by family/friends in the gym, playing jazz, or hiking.

Maya Rebholz

MAYA REBHOLZ is a student at the Massachusetts Institute of Technology. She grew up exploring math from the unseen equations which govern our world to its more tangible applications such as her research in gut microbiome data at the University of Pennsylvania and the Gini index with Professor Volpert. She is currently moving to take her love of analytically breaking down complex problems to the field of computer science in university. Beyond her studies, Maya finds joy spending time with her family and doing TaeKwonDo and skateboarding.

Klaus Volpert

KLAUS VOLPERT is an associate professor at Villanova University. He received the university’s Lindback Award for Distinguished Teaching in 2009. His research interests have changed over time from acoustics (during his high school days in Lindau, Germany) to spectral sequences (in his graduate work) to problems at the intersection of mathematics with economics and finance.