Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the Kelly criterion, or Kelly strategy or Kelly formula, or Kelly bet, is a formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run. It was described by J. L. Kelly, Jr
John Larry Kelly, Jr
John Larry Kelly, Jr. , was a scientist who worked at Bell Labs. He is best known for formulating the Kelly criterion, an algorithm for maximally investing money....
Edward Oakley Thorp is an American mathematics professor, author, hedge fund manager, and blackjack player. He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.He was the author of Beat the Dealer, the first...
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
and later in his books Beat the Dealer (for gambling) and Beat the Market (with Sheen Kassouf, for investing).
Although the Kelly strategy's promise of doing better than any other strategy seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints override the desire for optimal growth rate. The conventional alternative is utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
theory which says bets should be sized to maximize
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
In the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...
utility of the outcome (to an individual with logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
ic utility, the Kelly bet maximizes utility, so there is no conflict). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
In recent years, Kelly has become a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett
Warren Buffett
Warren Edward Buffett is an American business magnate, investor, and philanthropist. He is widely regarded as one of the most successful investors in the world. Often introduced as "legendary investor, Warren Buffett", he is the primary shareholder, chairman and CEO of Berkshire Hathaway. He is...
William Poundstone is an American author, columnist, and skeptic. He has written a number of books including the Big Secrets series and a biography of Carl Sagan...
wrote an extensive popular account of the history of Kelly betting in Fortune's Formula. But as Kelly's original paper demonstrates, the criterion is only valid when the investment or "game" is played many times over, with the same probability of winning or losing each time, and the same payout ratio.
Statement
For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds
Odds
The odds in favor of an event or a proposition are expressed as the ratio of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen...
, the Kelly bet is:
where:
f* is the fraction of the current bankroll to wager;
b is the net odds received on the wager ("b to 1"); that is, you could win $b (plus the $1 wagered) for a $1 bet
p is the probability of winning;
q is the probability of losing, which is 1 − p.
As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), but the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of the bankroll at each opportunity (f* = 0.20), in order to maximize the long-run growth rate of the bankroll.
If the gambler has zero edge, i.e. if b = q / p, then the criterion will usually recommend the gambler bets nothing (although in more complex scenarios such a bet may help ensure the best compounding rate of return: for instance a short-priced favourite in a horse race may be worth covering to provide downside protection even though the only advantageous bet is on another outsider). If the edge is negative (b < q / p) the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in standard American roulette, the bettor is offered an even money payoff (b = 1) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (p = 18/38). The Kelly bet is -1/19, meaning the gambler should bet one-nineteenth of the bankroll that red will not come up. Unfortunately, the casino doesn't allow betting against red, so a Kelly gambler could not bet.
The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win $b with probability p, or lose the $1 wagered, i.e. win $-1, with probability q. Hence:
For even-money bets (i.e. when b = 1), the first formula can be simplified to:
Since q = 1-p, this simplifies further to
Proof
Heuristic proofs of the Kelly criterion are straightforward.
For a symbolic verification with Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
SymPy is a Python library for symbolic computation. The stated goals of the library are to become a full-featured computer algebra system and to keep a simple code base to promote extensibility and comprehensibility. SymPy is written in Python....
one would set the derivative y'(x) of the expected value of the logarithmic bankroll y(x) to 0 and solve x:
>>> from sympy import *
>>> x,b,p = symbols('xbp')
>>> y = p*log(1+b*x) + (1-p)*log(1-x)
>>> solve(diff(y,x), x)
[-(1 - p - b*p)/b]
John Larry Kelly, Jr. , was a scientist who worked at Bell Labs. He is best known for formulating the Kelly criterion, an algorithm for maximally investing money....
original paper or some of the other references listed below. Some corrections can be found in: Optimal Gambling Systems For Favourable Games:- L. Breiman, University of California, Los Angeles.
We give the following non-rigorous argument for the case b = 1 (a 50:50 "even money" bet) to show the general idea and provide some insights.
When b = 1, the Kelly bettor bets 2p - 1 times initial wealth, W, as shown above. If he wins, he has 2pW. If he loses, he has 2(1 - p)W. Suppose he makes N bets like this, and wins K of them. The order of the wins and losses doesn't matter, he will have:
Suppose another bettor bets a different amount, (2p - 1 + )W for some positive or negative . He will have (2p + )W after a win and [2(1 - p)- ]W after a loss. After the same wins and losses as the Kelly bettor, he will have:
Take the derivative of this with respect to and get:
The turning point of the original function occurs when this derivative equals zero, which occurs at:
which implies:
but:
so in the long run, final wealth is maximized by setting to zero, which means following Kelly strategy.
This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win knowing that the rest of the bets will lose), one will end up with the most money if one bets:
each time. This is true whether N is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as N gets large, K will approach pN. Someone who bets more than Kelly can do better if K > pN for a stretch; someone who bets less than Kelly can do better if K < pN for a stretch, but in the long run, Kelly always wins.
Reasons to bet less than Kelly
A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount decreases the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since overbetting is worse than underbetting, it makes sense to err on the side of caution and bet less than the Kelly amount.
Kelly assumes sequential bets that are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
(later work generalizes to bets that have sufficient independence). That may be a good model for some gambling games, but generally does not apply in investing and other forms of risk-taking.
The Kelly property appears "in the long run" (that is, it is an asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
property). To a person, it matters whether the property emerges over a small number or a large number of bets. It makes sense to consider not just the long run, but where losing a bet might leave one in the short and medium term as well. A related point is that Kelly assumes the only important thing is long-term wealth. Most people also care about the path to get there. Kelly betting leads to highly volatile short-term outcomes which many people find unpleasant, even if they believe they will do well in the end.
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
suggested that when one has a choice of bets or investments that one should choose that with the highest geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different (Bernoulli wanted to resolve the St. Petersburg paradox
St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
). The Bernoulli article was not translated into English until 1956, but the work was well-known among mathematicians and economists.
Many horses
Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, like in horse races. Suppose there are several mutually exclusive outcomes. Probability of winning the race by k-th horse is , the total of bets placed on k-th horse is (in dollars), and , where are the pay-off odds. , is the dividend rate where is the track take or tax, is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor's funds to bet on k-th horse is . Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions of bettor's wealth to be bet on the outcomes included in the optimal set .
The algorithm for the optimal set of outcomes consists of four steps:
Step 1 calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:
Step 2 Reorder the outcomes so that the new sequence is non-decreasing.
Step 3 Set (the empty set), , .
Step 4 Repeat:
if then insert k-th outcome into the set: , recalculate according to the formula: and then set , else set and then stop the repetition.
If the optimal set is empty then do not bet at all. If the set of optimal outcomes is not empty then the optimal fraction to bet on k-th outcome may be calculated from this formula: .
One may prove that is the reserve rate. Therefore the requirement may be interpreted as follows: k-th outcome is included into the set of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction may be interpreted as the excess of the expected revenue rate of k-th horse over the reserve rate divided by the revenue after deduction of the track take when k-th horse wins or as the excess of the probability of k-th horse winning over the reserve rate divided by revenue after deduction of the track take when k-th horse wins. The binary growth exponent is
and the doubling time is .
Details of the proof may be found in "An explicit solution to the problem of optimizing the allocations of a bettor's wealth when wagering on horse races", Mathematical Scientist, volume 35 (2010), no. 1, pg. 10—17.
This method of selection of optimal bets may be applied also when probabilities are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that and .
)W for some positive or negative 
. He will have (2p + 
)W after a win and [2(1 - p)- 
]W after a loss. After the same wins and losses as the Kelly bettor, he will have:
and get:
to zero, which means following Kelly strategy.
, the total of bets placed on k-th horse is 
(in dollars), and 
, where 
are the pay-off odds. 
, is the dividend rate where 
is the track take or tax, 
is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor's funds to bet on k-th horse is 
. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set 
of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions 
of bettor's wealth to be bet on the outcomes included in the optimal set 
.
is non-decreasing.
(the empty set), 
, 
.
then insert k-th outcome into the set: 
, recalculate 
according to the formula:
and then set 
, else set 
and then stop the repetition.
is empty then do not bet at all. If the set 
of optimal outcomes is not empty then the optimal fraction 
to bet on k-th outcome may be calculated from this formula: 
.
is the reserve rate. Therefore the requirement 
may be interpreted as follows: k-th outcome is included into the set 
of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction 
may be interpreted as the excess of the expected revenue rate of k-th horse over the reserve rate divided by the revenue after deduction of the track take when k-th horse wins or as the excess of the probability of k-th horse winning over the reserve rate divided by revenue after deduction of the track take when k-th horse wins. The binary growth exponent is
.
are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that 
and 
.
