Statistical inference might be thought of as gambling theory applied to the world around us. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information.^[1] In that sense, information theory might be considered a formal expression of the theory of gambling. It is no surprise, therefore, that information theory has applications to games of chance.^[2]

The Kelly calculator is one of our most popular bet calculators, because it makes a complex mathematical equation easy to understand. This sports betting calculator works out your ideal stake for. Betting less than the Kelly amount corrects for this. The other is that the Kelly formula leads to extreme volatility, and you should underbet to limit the chance of being badly down for unacceptably long stretches. It is true that gamblers often overestimate their. Because the Kelly Criterion seeks to calculate the optimum stake for any value bet so as to maximise that value as well as maximise the growth of your betting bankroll.

Kelly Betting[edit]

Kelly betting or proportional betting is an application of information theory to investing and gambling. Its discoverer was John Larry Kelly, Jr.

Part of Kelly's insight was to have the gambler maximize the expectation of the logarithm of his capital, rather than the expected profit from each bet. This is important, since in the latter case, one would be led to gamble all he had when presented with a favorable bet, and if he lost, would have no capital with which to place subsequent bets. Kelly realized that it was the logarithm of the gambler's capital which is additive in sequential bets, and 'to which the law of large numbers applies.'

Side information[edit]

A bit is the amount of entropy in a bettable event with two possible outcomes and even odds. Obviously we could double our money if we knew beforehand for certain what the outcome of that event would be. Kelly's insight was that no matter how complicated the betting scenario is, we can use an optimum betting strategy, called the Kelly criterion, to make our money grow exponentially with whatever side information we are able to obtain. The value of this 'illicit' side information is measured as mutual information relative to the outcome of the betable event:

where Y is the side information, X is the outcome of the betable event, and I is the state of the bookmaker's knowledge. This is the average Kullback–Leibler divergence, or information gain, of the a posteriori probability distribution of X given the value of Y relative to the a priori distribution, or stated odds, on X. Notice that the expectation is taken over Y rather than X: we need to evaluate how accurate, in the long term, our side information Y is before we start betting real money on X. This is a straightforward application of Bayesian inference. Note that the side information Y might affect not just our knowledge of the event X but also the event itself. For example, Y might be a horse that had too many oats or not enough water. The same mathematics applies in this case, because from the bookmaker's point of view, the occasional race fixing is already taken into account when he makes his odds.

The nature of side information is extremely finicky. We have already seen that it can affect the actual event as well as our knowledge of the outcome. Suppose we have an informer, who tells us that a certain horse is going to win. We certainly do not want to bet all our money on that horse just upon a rumor: that informer may be betting on another horse, and may be spreading rumors just so he can get better odds himself. Instead, as we have indicated, we need to evaluate our side information in the long term to see how it correlates with the outcomes of the races. This way we can determine exactly how reliable our informer is, and place our bets precisely to maximize the expected logarithm of our capital according to the Kelly criterion. Even if our informer is lying to us, we can still profit from his lies if we can find some reverse correlation between his tips and the actual race results.

Doubling rate[edit]

Doubling rate in gambling on a horse race is ^[3]

${\displaystyle W(b,p)=\mathbb{E}[{\log }_{2}S(X)]=\sum _{i=1}^m{p}_i{\log }_2{b}_i{o}_i}$

where there are ${\displaystyle m}$ horses, the probability of the ${\displaystyle i}$th horse winning being ${\displaystyle p_i}$, the proportion of wealth bet on the horse being ${\displaystyle b_i}$, and the odds (payoff) being ${\displaystyle o_i}$ (e.g., ${\displaystyle o_i=2}$ if the ${\displaystyle i}$th horse winning pays double the amount bet). This quantity is maximized by proportional (Kelly) gambling:

${\displaystyle b=p}$

for which

${\displaystyle \underset{b}{max}W(b,p)=\sum _i{p}_i{\log }_2{o}_i-H(p)}$

where ${\displaystyle H(p)}$ is information entropy.

Expected gains[edit]

An important but simple relation exists between the amount of side information a gambler obtains and the expected exponential growth of his capital (Kelly):

${\displaystyle \mathbb{E}\log K_t=\log K_0+\sum _{i=1}^t{H}_i}$

for an optimal betting strategy, where ${\displaystyle K_0}$ is the initial capital, ${\displaystyle K_t}$ is the capital after the tth bet, and ${\displaystyle H_i}$ is the amount of side information obtained concerning the ith bet (in particular, the mutual information relative to the outcome of each betable event). This equation applies in the absence of any transaction costs or minimum bets. When these constraints apply (as they invariably do in real life), another important gambling concept comes into play: the gambler (or unscrupulous investor) must face a certain probability of ultimate ruin, which is known as the gambler's ruin scenario. Note that even food, clothing, and shelter can be considered fixed transaction costs and thus contribute to the gambler's probability of ultimate ruin.

This equation was the first application of Shannon's theory of information outside its prevailing paradigm of data communications (Pierce).

Applications for self-information[edit]

The logarithmic probability measure self-information or surprisal,^[4] whose average is information entropy/uncertainty and whose average difference is KL-divergence, has applications to odds-analysis all by itself. Its two primary strengths are that surprisals: (i) reduce minuscule probabilities to numbers of manageable size, and (ii) add whenever probabilities multiply.

For example, one might say that 'the number of states equals two to the number of bits' i.e. #states = 2^#bits. Here the quantity that's measured in bits is the logarithmic information measure mentioned above. Hence there are N bits of surprisal in landing all heads on one's first toss of N coins.

The additive nature of surprisals, and one's ability to get a feel for their meaning with a handful of coins, can help one put improbable events (like winning the lottery, or having an accident) into context. For example if one out of 17 million tickets is a winner, then the surprisal of winning from a single random selection is about 24 bits. Tossing 24 coins a few times might give you a feel for the surprisal of getting all heads on the first try.

The additive nature of this measure also comes in handy when weighing alternatives. For example, imagine that the surprisal of harm from a vaccination is 20 bits. If the surprisal of catching a disease without it is 16 bits, but the surprisal of harm from the disease if you catch it is 2 bits, then the surprisal of harm from NOT getting the vaccination is only 16+2=18 bits. Whether or not you decide to get the vaccination (e.g. the monetary cost of paying for it is not included in this discussion), you can in that way at least take responsibility for a decision informed to the fact that not getting the vaccination involves more than one bit of additional risk.

More generally, one can relate probability p to bits of surprisal sbits as probability = 1/2^sbits. As suggested above, this is mainly useful with small probabilities. However, Jaynes pointed out that with true-false assertions one can also define bits of evidence ebits as the surprisal against minus the surprisal for. This evidence in bits relates simply to the odds ratio = p/(1-p) = 2^ebits, and has advantages similar to those of self-information itself.

Applications in games of chance[edit]

Information theory can be thought of as a way of quantifying information so as to make the best decision in the face of imperfect information. That is, how to make the best decision using only the information you have available. The point of betting is to rationally assess all relevant variables of an uncertain game/race/match, then compare them to the bookmaker's assessments, which usually comes in the form of odds or spreads and place the proper bet if the assessments differ sufficiently.^[5] The area of gambling where this has the most use is sports betting. Sports handicapping lends itself to information theory extremely well because of the availability of statistics. For many years noted economists have tested different mathematical theories using sports as their laboratory, with vastly differing results.

One theory regarding sports betting is that it is a random walk. Random walk is a scenario where new information, prices and returns will fluctuate by chance, this is part of the efficient market hypothesis. The underlying belief of the efficient market hypothesis is that the market will always make adjustments for any new information. Therefore no one can beat the market because they are trading on the same information from which the market adjusted. However, according to Fama,^[6] to have an efficient market three qualities need to be met:

• There are no transaction costs in trading securities
• All available information is costlessly available to all market participants
• All agree on the implications of the current information for the current price and distributions of future prices of each security

Statisticians have shown that it's the third condition which allows for information theory to be useful in sports handicapping. When everyone doesn't agree on how information will affect the outcome of the event, we get differing opinions.

See also[edit]

References[edit]

1. ^Jaynes, E.T. (1998/2003) Probability Theory: The Logic of Science (Cambridge U. Press, New York).
2. ^Kelly, J. L. (1956). 'A New Interpretation of Information Rate'(PDF). Bell System Technical Journal. 35 (4): 917–926. doi:10.1002/j.1538-7305.1956.tb03809.x.
3. ^Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN0-471-06259-6, Chapter 6.
4. ^Tribus, Myron (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand Company Inc., 24 West 40 Street, New York 18, New York, U.S.A) ASIN: B000ARSH5S.
5. ^Hansen, Kristen Brinch. (2006) Sports Betting from a Behavioral Finance Point of View (Arhus School of Business).
6. ^Fama, E.F. (1970) 'Efficient Capital Markets: A Review of Theory and Independent Work', Journal of Financial Economics Volume 25, 383-417

External links[edit]

The Kelly Calculator (or Kelly Criterion Calculator) can help a sports bettor decide how much of their bankroll to risk on a wager. The amount recommended is based on the odds offered by the sportsbook as well as an understanding of your predicted winning percentage.

Originally applied to the stock market, the Kelly Calculator quickly moved to horse betting and found its most successful use in poker. But this aggressive betting strategy can be used with any form of wagering to maximize profit based on the information at hand.

Kelly Calculator

Try out the Kelly Criterion Calculator below, but pay careful attention to sure things (like -200 odds or above) because that is where Kelly can get you in trouble.

What Is A Kelly Criterion Calculator?

A Kelly Criterion Calculator helps you decide what percentage of your bankroll you should wager on a sports bet. So you first need to decide your bankroll size and the length of time you’ll be using the Kelly method.

The easiest is to say you’ll be using the Kelly Criterion for one year or the length of a sports season. A timeframe is important because the goal of the Kelly Calculator is to profit over a given period. Once that time has elapsed, you can see your profit percentage, then adjust your Kelly Criterion approach accordingly.

You’ll also want a good idea of your win percentage. But if you’re exclusively a -110 bettor, the minimum win percentage is 53% for the Kelly Calculator to recommend betting any amount.

If your win percentage is lower than 53% on -110 wagers but you still want to use the Kelly Criterion, you’ll need to look for bets with longer odds (or bets that you think would have a higher win percentage).

How To Use A Kelly Criterion Calculator To Place A Sports Bet

To use the Kelly Calculator for sports betting, you need a few pieces of information. The odds, of course, but then you also need your winning percentage. Unfortunately, this is where some go wrong.

Kelly Betting System For Blackjack

You need the winning percentage of the specific odds you are betting on. If you put in your overall winning percentage, you are in trouble.

Imagine that you are a 55% winning sports bettor at -110 odds. Good for you! But if you put that 55% in the Kelly calculator on a +150 dog, Kelly will advise you a ridiculous 25% of your bankroll because it is looking to maximize your profit. If you put that much into a single bet, you risk losing an enormous amount of your bankroll.

Instead, you want to be as conservative as possible. If your win percentage is already 45% or lower, then just use that. But if it’s higher than 50%, you want to be realistic when you’re betting on odds longer than -110.

Kelly Multiplier

You also need to decide the Kelly Multiplier you’re going to use. Basically, this is how much of the Kelly Calculator recommended amount you want to wager. While the calculator is automatically set at 1, we recommend adjusting it to no more than 0.5 for long-term betting.

Most bettors apply a factor to the Kelly calculator (the Kelly multiplier) to take advantage of the theory’s betting advice while limiting risk. This means a much less aggressive potential growth while keeping the volatility down by a significantly lower margin.

You Can Never Guarantee A Profit

There is a huge drawback that you must understand and be aware of before using Kelly Criterion in your betting. The catch is always the win percentage. In sports betting, as with investing, your personal win percentage at different odds is virtually impossible to get accurate. And if it’s not accurate, the volatility in your betting will evaporate your bankroll.

So approach this knowing that you can never assure that you’ll make a profit.

But if you are a strict -110 bettor, then, over time, Kelly Criterion can help give you the ideal betting outcome.

Looking for other calculators to use when sports betting? Check out:

How The Kelly Criterion Calculator Math Works

While you can simply enter the information into the Gaming Today online Kelly Calculator, it can be helpful to know how the math works. Here’s a step-by-step guide.

Step 1: Convert Odds To Decimal

The easiest way to convert American odds to Decimal would be to use the Odds Calculator. But it can also be done manually.

To convert positive odds, the equation is:

(Odds divided by 100) + 1

Kelly Betting Calculator

To convert negative odds, the equation is:

(100 divided by odds) + 1

Step 2: Use The Kelly Criterion Formula

This long but easy formula is how the Kelly Calculator creates its results:

((Decimal Odds – 1) * Decimal Winning Percentage – (1 – Winning Percentage)) / (Decimal Odds – 1) * Kelly Multiplier

Kelly Criterion Example

Let’s take the basic case of -110 odds and a winning percentage of 55% with 0.5 Kelly multiplier, which is also known as a half Kelly. While it’s not the simplest situation, it’s one of the most likely scenarios when utilizing this betting strategy.

So let’s add a bit of simplicity and say that your bankroll is $1,000. That way, we can do the math and see exactly how much you would wager in this scenario.

Step 1: Converting -110 American Odds To Decimal Odds

Because it’s a negative number, you’ll use the equation (100 divided by odds) + 1 = decimal odds. 100 divided by 110 is 0.9091. Plus one, and you get 1.9091 for the decimal odds. Here it is another way:

(100/110) + 1 = 1.9091

Step 2: Plugging Decimal Odds Into The Kelly Criterion Formula

With 1.9091 decimal odds, a 55% winning percentage as a decimal (0.55), and a half Kelly (0.5), the equation would look like this:

((1.9091 – 1) * 0.55 – (1 – 0.55)) / (1.9091 – 1) * 0.5 = 0.0275 (2.75%)

If we do the math in the parentheticals first, it would be:

(0.9091 * 0.55 – 0.45) / 0.9091 * 0.5 = 0.0275 (2.75%)

Broken down again:

0.050005/0.9091 = 0.0550049499505 * 0.5 = 0.0275 (2.75%)

Using the Kelly Criterion, you should use 2.75% of your $1,000 bankroll, or $27.50. Good luck!