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Taking Advantage of an Advantage - Part 1: Introduction, References

6 February 2011

By Donald Catlin

In 1956 J. L. Kelly published a paper [3] in which he suggested an "optimal" betting system for positive games. This paper was a very technical piece of work and was generally unknown to the gambling community until the publication of Beat the Dealer [5] by Dr. Edward Thorp. This means of betting caught on with savvy gamblers, primarily blackjack players, and became known as Kelly betting. I qualified the word "optimal" in the first line because not all practitioners believe that Kelly betting is the best way to approach a positive game [1]. Nevertheless, it is an approach that is used by many and in these three articles I would like to try to explain to my readers the idea behind Kelly betting. A version of this article was first published in Frank's magazine "The New Chance and Circumstance" back in 2001. I decided to rewrite that article for my readers here because of a question about craps that was posed by one of my readers named John. I'll get to that in a future article.

The original article by Kelly and subsequent articles, such as those in my brief bibliography, use phrases like "logarithmic utility function," "exponential growth rate," or "Shannon entropy." Because of such terms I think that the average gambler avoids the subject and has no idea what this simple betting procedure is all about.

Although the subject cannot be understood without digging into some mathematics, I am going to explain it to you without the above technical phrases. At one point I will need some calculus, but the conclusion of the calculation can be understood without understanding calculus. All you will need to read this article is a bit of high school algebra. The result will be a somewhat naïve approach that some experts may find a bit too glib, but will at least be available to those without training in advanced mathematics.

Some gambles in this world have a positive expectation for the gambler. If one counts cards, some blackjack games have a positive expectation. Some video poker machines have a positive expectation. In reference [6] Thorp indicates that one can get an edge in sports betting as well. Both Griffin and Thorp [2] document a promotion by the now defunct Klondike Casino in Las Vegas that gave the players an edge in their blackjack game. There are other instances as well. The question we address here is as follows: When facing a positive game, how can one make optimal use of their bankroll? As we will see, answering this question is a bit tricky.

To begin with, let us postulate a simple game consisting of a series of independent win/lose trials that has a positive edge of 2%. If p is the probability of winning and q is the probability of losing, then the edge e in decimal form is just e = p - q and since p + q = 1 we have e = 2p - 1 = 0.02. It follows that p = 0.51 and q = 0.49. It would seem reasonable that as the player's stake grows (or falls) that the player's bet should raise (or lower) accordingly. For example, if one has a stake of $50 and is willing to risk $5, then if the stake reaches $1000 the player should be willing to risk $100. On the other hand if the stake falls to $10 then the player should reduce the wager to $1. Such betting is called proportional betting and is the basis of the Kelly betting scheme.

If f represents the fraction of the player's stake that he or she is willing to risk, then f is obviously a number between 0 and 1. Our problem is to select f so that the player's actual return after a finite number of steps is both profitable and likely. Does picking f to maximize the player's expected return sound reasonable? It does, but in my next article I will show you that, surprisingly, this is how not to solve the problem. This discussion in turn will show us a reasonable path towards a solution.

Here is my bibliography for these three articles. See you next month.

References

References [1], [2], [4], and [6] are all found in the book Finding the Edge published by The Institute for the Study of Gambling and Commercial Gaming at the University of Nevada, Reno. Editors are Olaf Vancura, Judy A. Cornelius, and William R. Eadington.

[1] Brown, Sid (2000) Can You Do Better Than Kelly in the Short Run?, pp 215-231

[2] Griffin, Peter ND Thorp, E.O. (2000), Blackjack: Betting the Klondike's Free Ride, pp 215-272

[3] Kelly, J.L. (1956), A New Interpretation of Information Rate, Bell Systems Technical Journal, July 1956, pp 917-926

[4] Leib, John (2000) Limitations on Kelly or The Ubiquitous "n approaches infinity," pp 233-253

[5] Thorp, E.O., (1962), Beat the Dealer: A Winning Strategy for the Game of Twenty One, Blaisdell Publishing Company, New York, page 89

[6] Thorp, E.O. (2000), The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market, pp 163-213


Don Catlin can be reached at 711cat@comcast.net

Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers