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Kelly Criterion

About 6 months before I started Focus Ratings I also started a small blog which detailed my learning curve with all things Horse Racing.

Once Focus Ratings was under way though, I found that I didn't have time to do both and so I stopped updating the blog.

I am now shutting down the blog and have decided to move some of the content over here - purely for your amusement only.

The Kelly Criterion is a formula used to determine the optimal size of a series of bets.

Generally, in gambling scenarios (and some investing scenarios), the Kelly strategy will do better than any other strategy in the long run.

However, personal attitude to risk can conflict with Kelly and even Kelly supporters usually choose to bet a fixed fraction of the amount recommended by Kelly in order to reduce volatility, or to protect against calculation errors.

Kelly Criterion in detail...

Where a bet has two outcomes (either losing the stake or winning the stake multiplied by the odds offered by the bookmaker or casino), the Kelly bet is as shown in the formula to the left....

where f* is the fraction of the bank to stake;
b is the odds for the bet ("b to 1");
p is the probability of winning;
q is the probability of losing, which is 1 − p.

The Kelly Criterion only works where we have value in the bet.

it won't work in a casino as the house always has the edge.

In order for f* to be positive (in other words, for you to actually have a bet) the potential payout has to be higher than one divided by the actual odds of winning.

Example 1: Black Beauty can either win a race or lose it.

Black Beauty has a price of 5/1.

You believe (after extensive form study etc...) that Black Beauty has a probability of winning of 0.2 and thus, a probability of losing of 0.8

f* = (0.2 x (5 + 1) - 1) / 5 = 0.04

In other words, in this scenario we should (according to Kelly) use a stake of 4% of our bank.

Example 2: Black Beauty can either win a race or lose it.

Black Beauty has a price of 5/1.

You believe (after extensive form study etc...) that Black Beauty has a probability of winning of 0.5 and thus, a probability of losing of 0.5

f* = (0.5 x (5 + 1) - 1) / 5 = 0.40

In other words, in this scenario we should (according to Kelly) use a stake of 40% of our bank.

Example 2: Black Beauty can either win a race or lose it.

Black Beauty has a price of 5/1.

You believe (after extensive form study etc...) that Black Beauty has a probability of winning of 0.1 and thus, a probability of losing of 0.9

f* = (0.1 x (5 + 1) - 1) / 5 = -0.08

In other words, in this scenario we should (according to Kelly) use a stake of negative 8% of our bank.

In other words, we shouldn't bet on Black Beauty at those odds.

But wait....

Kelly himself said that his calculations only work where you have a high number of similar "games."

Now, no two races are identical, although system followers/users assume that they are.

Also, we can never be absolutely exact in our estimation of p (the probability of winning.)

We can study form all day long but we can never absolutely accurately predict how Black Beauty is going to perform.

She may have just had an argument with Mr Black Beauty?

She may be worried about Baby Black Beauty's school grades?

A car might backfire and scare her?

It is because we don't have an infinite number of identical events and that we can never absolutely predict the possibility of winning (as we would be able to if we were rolling a dice or flipping a coin) that we bet to Fractional Kelly.

Typically we would be to half of the computed Kelly Stake but...

It can be seen that the probability of halving the bank, using full Kelly Stakes, is 50% (you have a 1/n chance of reducing your bank to 1/n of its original value.)

However, for the Half Kelly Stake, the probability of reducing the bank to half its original value is square root of what it was previously - in other words... 1/9

The downside to this sensible way of handling risk (the inevitable losing runs which will happen)is that the bank will increase more slowly, but the risks of losing it all are diminished.

Also, by using a Half Kelly Stake the volatility experienced is greatly reduced although....

So is the eventual return over any set period of time.

Betting a Half Kelly Stake, for example, reduces bank volatility by 50%, but growth by only 25%.


The Kelly Criterion has many critics; primarily because we cannot exactly measure the probability of winning (in horse racing, at least) and is only works where we have a large number of identical events (no two horse races are identical.)


It is worth having a think about the maths behind the Kelly Criterion purely because....

It displays, quite simply, the need to have value in each and every bet.

It's not enough just to have value (i.e. don't back Dennis the Donkey just because you can get him at Ladbrokes at 200/1 when his real odds should be 50/1) - you can't eat value!

You need to back the winning horse (or lay the losing one) but...

The Kelly Criterion indicates that you should stake a percentage of your bank that varies according to the confidence you have in winning the bet.

That is why, at - I use advised stakes as well as fixed stakes; the advised stakes refelct the confidence in the selection and...

Thus the higher percentage of the bank to stake.

keith-eckstein  the horse racing scholar



Do you agree?

Or do you think that this is a recipe for disaster?

Feel free to leave a comment if you have something to say.

Your opinion is valued!


Links and further study...

1). Taking Chances: Winning with Probability - What are the odds against winning the Lottery, making money in a casino, or backing the right horse? Every day, people make judgements on these matters and face other decisions that rest on their understanding of probability: buying insurance, following medical advice, carrying an umbrella. Yet many of us have a frightening ignorance of how probability works. Taking Chances presents an entertaining and fascinating exploration of probability, revealing traps and fallacies in the field. It describes and analyses a remarkable variety of situations where chance plays a role, including football pools, the Lottery, TV games, sport, cards, roulette, coins, and dice. The book guides the reader round common pitfalls, demonstrates how to make better informed decisions, and shows where the odds can be unexpectedly in your favour. This new edition has been fully updated, and includes information on "Who Wants to be a Millionaire?" and "The Weakest Link", plus a new chapter on Probability for Lawyers.

Get the book here... Taking Chances: Winning with Probability.

2). Understanding a Wager - Have you ever wondered how to calculate your odds and expected return on various gambles, or how to manage your bankroll and avoid gambler's ruin? This book will use simple mathematics and evidence-based research to answer these questions and many others you have not yet considered. So if you want to understand wagers and how to profit from them, or reduce your gambling losses, or learn about bankroll management, then this book is essential reading. "Understanding a Wager sets out the basic mathematical guidelines that will enable players to determine their long-term chances of winning, and to reduce, as much as possible, their losses ... In doing so, it examines issues such as probability theory, randomness and bankroll management, as well as differentiating 'good' bets from 'bad' bets on the basis of fairly simple mathematical formulae.

Get the book here... Understanding a Wager

3). Probability Guide to Gambling - Over the past few decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory that can model the hazards of gambling, even in idealized conditions, numerical probabilities are viewed not only as purely theoretical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability (definitions, properties, theorems and calculus formulas), combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the "Guide to Numerical Results" to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled "The Mathematics of Games of Chance" presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what experiments, events and probability fields mean in games of chance and why and how probability formulas can be applied there. The main portion of this work is a collection of probability results for each type of game. Each game's section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Five Card Draw Poker, Texas Hold'em Poker, Lottery and Sport Bets. No formal background in mathematics is necessary to read these sections, although familiarity with some probability and set theory notions is helpful. In most cases, readers will benefit from having some college math to follow the formulas here, but this is not a requirement because the numerical results are given as summaries and tables at the end of each section. Most of casino games are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one (like many other titles published on this subject), but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal. The author is a recognized authority on casino mathematics.

Get the book here... Probability Guide to Gambling

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