The Kelly Criterion

In this horse-race betting system, placing a wager is an attempt to maximize a utility function. All successful wagering schemes are Markov processes based on bankroll, the amount of money in your pocket; probabilities, the likelihood of each horse winning the race; and the odds, the payoffs determined by the wagering and take-out. (Technically, what makes them Markov Processes is that the next step is independent of the previous step, i.e. how much you bet has nothing to do with your previous bets. All rational wagering systems are Markov Processes.)

The Kelly Criterion provides us with the utility function. The Kelly Criterion is "wager to maximize the expected growth of the bankroll". It is not the only way to bet, but I like it. The math has been worked out by others. It provides an objective way to grade a system, at least a system that produces estimates of probabilities. Many horse players have different preferences; maximizing return on investment and maximizing the amount of action is probably the most common. The Kelly Criterion has problems. The two best know problems are (1) the assumption that going broke is infinitely bad and (2) that doubling your wealth has the same utility whether your have two dollars, two hundred dollars, or two thousand dollars.

There are problems with the Kelly Criterion. In his original paper, Kelly went through some artificial mechanisms to simplify. To avoid dealing with the effects of wagers on the payoffs, all the wagers were placed in a "wire room", that is through a bookie. To avoid having to have a mechanism to predict the outcomes of races, before they happened, he assumed that the better had access to a transmitted signal giving the bettor the results before the bookie. Kelly's calculations are based on the probability of the bettors signal being garbled, not on the probability of a horse winning the race. The article may indeed been intended as illustrating how to financially evaluate the need for improvements in telecommunications equipment, though horse-players doubt this.

A definition of terms

p(h) 

the probability of horse h winning a race.

p(4)=.27. I'll take the probability estimates as a given. Accurately deriving such figures is hard. Many systems simply and sanely try to find horses with a better than 1 in 3 chance of winning going off at odds over 3-1. Correctly estimate the chances of a horse winning is the key to horseplaying; the Kelly Criterion can only be used after that step.

o(h) 

the odds-for-a-dollar on horse h winning.

o(4)=9.04. To get the more conventional odds-to-a-dollar subtract one. To get the odds as displayed on the totalizator, subtract one and round down. "odds-for-a-dollar" is not used at U.S. tracks. They give odds-to-a-dollar or odds-for-two-dollars. The latter are called "will-pays" by horse-players and most commonly used for exotic wagers. At the track the odds on the totalizator for Falconese would most likely be displayed as 8-1.

s(h) 

the probability suggested by the odds.

s(4)=0.111. Odds, probabilities and payoffs all express the same notion of the probable outcome of an event. The probable outcome of heads, in a coin toss, is 0.5, the odds are even or 1-1, the payoff would be $2 for a dollar bet. All of these suggest the same outcome. At the track the odds are determined by the wagers and the track take-out; the take-out prevents the probabilities suggested by the tote from totaling to 1. I tend to use probabilities for my assessment of the chances of the runners, and the odds to refer to crowds assessment.

expected gain 

amount returned times the probability of winning.

The amount returned for a dollar times the probability of winning the wager is how much you should expect to win or lose on average if you repeatedly make the same bet. This classical notion of expectation is the amount won times the probability of winning less the amount lost times the probability of losing. Since horse races are not Bernouli trials, we need to consider the expected gain separately.

The steps

1. Order the entries: Place all the entries in descending order of expected gain. The table below has them already in order.

2. Find b: b is the minimum positive value of 1 - (the sum of the ) p(h) divided by 1- (the sum of the ) s(h). [ I really could use a sigma here]. Look at the columns labeled sigma p(h) and sigma s(h). In this case the we can use the first three horses. The sum of p(4) + P(7) + p(10) = 0.27 + 0.24 + 0.23 = 0.74 and the sum of s(4) + s(7) + s(10) = 0.555. b = (1 - 0.740) / ( 1 - 0.555) = 0.584

3. Find the bets:For each horse in the race the bet is p(h) - b/o(h) which is listed in the last column. For Island of Silver the numbers are .240 - 0.584 / 0.510

Some things worth noting: Kelly wagering is a Dutching System, that is a betting system where it is common to bet more than one entry. There is a nasty potential for division by zero, whenever sigma s(h) is one, if this algorithm is used unmodified. I thank Dave Jones for pointing this out to me; the problem exists with the original algorithm.

Thanks to the many readers of the html authoring newsgroups who told me all the things wrong with my table under various browsers.

Copyright ©1997 Josh Kuperman, All rights reserved.