# Kelly criterion

In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula for sizing a bet. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. It assumes that the expected returns are known and is optimal for a bettor who values their wealth logarithmically. J. L. Kelly Jr, a researcher at Bell Labs, described the criterion in 1956.[1] Because the Kelly Criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity), it is a scientific gambling method.

Example of the optimal Kelly betting fraction, versus expected return of other fractional bets.

The practical use of the formula has been demonstrated for gambling[2][3] and the same idea was used to explain diversification in investment management.[4] In the 2000s, Kelly-style analysis became a part of mainstream investment theory[5] and the claim has been made that well-known successful investors including Warren Buffett[6] and Bill Gross[7] use Kelly methods. [8] Also see Intertemporal portfolio choice.

## Optimal betting example

In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at$250. But the behavior of the test subjects was far from optimal:

Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment.[9][10] Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of$250 and the finite duration of the test), the right approach would be to bet 20% of one's bankroll on each toss of the coin, which works out to a 2.034% average gain each round. This is a geometric mean, not the arithmetic rate of 4% (${\displaystyle r=(1+0.2\cdot 1.0)^{0.6}\cdot (1-0.2\cdot 1.0)^{0.4}}$ ). The theoretical expected wealth after 300 rounds works out to $10,505 (${\displaystyle =25\cdot (1.02034)^{300}}$ ) if it were not capped. In this particular game, because of the cap, a strategy of betting only 12% of the pot on each toss would have even better results (a 95% probability of reaching the cap and an average payout of$242.03).

## Gambling formula

Where losing the bet involves losing the entire wager, the Kelly bet is:

${\displaystyle f^{*}=p-{\frac {q}{b}}=p-{\frac {1-p}{b}}}$

where:

• ${\displaystyle f^{*}}$  is the fraction of the current bankroll to wager.
• ${\displaystyle p}$  is the probability of a win.
• ${\displaystyle q}$  is the probability of a loss (${\displaystyle q=1-p}$ ).
• ${\displaystyle b}$  is the proportion of the bet gained with a win. E.g. If betting $10 on a 2-to-1 odds bet, (upon win you are returned$30, winning you $20), then ${\displaystyle b=\20/\10=2.0}$ . The figure plots the amount gained with a win on the x-axis against the fraction of portfolio to bet on the y-axis. This figure assumes p=0.5 (that the probability of both a win and a loss is 50%). If the amount gained with a win is 1, then the Kelly betting amount is$0, which makes sense in a fair bet with no expected gain.

As an example, if a gamble has a 60% chance of winning (${\displaystyle p=0.6}$ , ${\displaystyle q=0.4}$ ), and the gambler receives 1-to-1 odds on a winning bet (${\displaystyle b=1}$ ), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (${\textstyle f^{*}=0.6-{\frac {0.4}{1}}=0.2}$ ).

The figure plots the amount gained with a win on the x-axis against the fraction of portfolio to bet on the y-axis. This figure assumes p=0.6 (that the probability of both a win is 60%).

3D figure representing the optimal Kelly bet size (vertical axis) as a function of win probability and amount gained with win.

If the gambler has zero edge, i.e. if ${\displaystyle b=q/p}$ , then the criterion recommends for the gambler to bet nothing.

If the edge is negative (${\displaystyle b ) the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in American roulette, the bettor is offered an even money payoff (${\displaystyle b=1}$ ) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (${\displaystyle p=18/38}$ ). The Kelly bet is ${\displaystyle -1/19}$ , meaning the gambler should bet one-nineteenth of their bankroll that red will not come up. There is no explicit anti-red bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.

## Investment formula

A more general form of the Kelly formula allows for partial losses, which is relevant for investments:

${\displaystyle f^{*}={\frac {p}{a}}-{\frac {q}{b}}}$

where:

• ${\displaystyle f^{*}}$  is the fraction of the assets to apply to the security.
• ${\displaystyle p}$  is the probability that the investment increases in value.
• ${\displaystyle q}$  is the probability that the investment decreases in value (${\displaystyle q=1-p}$ ).
• ${\displaystyle a}$  is the fraction that is lost in a negative outcome. If the security price falls 10%, then ${\displaystyle a=0.1}$
• ${\displaystyle b}$  is the fraction that is gained in a positive outcome. If the security price rises 10%, then ${\displaystyle b=0.1}$ .

Note that the Kelly Criterion is valid only for known outcome probabilities, which is not the case with investments. Risk averse investors should not invest the full Kelly fraction.

This formula can result in Kelly fractions higher than 1. In this case, it is theoretically advantageous to use leverage to purchase additional securities on margin.

## Proof

Heuristic proofs of the Kelly criterion are straightforward.[11] The Kelly criterion maximizes the expected value of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of the probability of each particular outcome multiplied by the value of the function in the event of that outcome). We start with 1 unit of wealth and bet a fraction ${\displaystyle f}$  of that wealth on an outcome that occurs with probability ${\displaystyle p}$  and offers odds of ${\displaystyle b}$ . The probability of winning is ${\displaystyle p}$ , and in that case the resulting wealth is equal to ${\displaystyle 1+fb}$ . The probability of losing is ${\displaystyle q=1-p}$ , and in that case the resulting wealth is equal to ${\displaystyle 1-fa}$ . Therefore, the expected geometric growth rate ${\displaystyle r}$  is:

${\displaystyle r=(1+fb)^{p}\cdot (1-fa)^{q}}$

We want to find the maximum r of this curve (as a function of f), which involves finding the derivative of the equation. This is more easily accomplished by taking the logarithm of each side first. The resulting equation is:

${\displaystyle E=\log(r)=p\log(1+fb)+q\log(1-fa)}$

with ${\displaystyle E}$  denoting logarithmic wealth growth. To find the value of ${\displaystyle f}$  for which the growth rate is maximized, denoted as ${\displaystyle f^{*}}$ , we differentiate the above expression and set this equal to zero. This gives:

${\displaystyle \left.{\frac {dE}{df}}\right|_{f=f^{*}}={\frac {pb}{1+f^{*}b}}+{\frac {-qa}{1-f^{*}a}}=0}$

Rearranging this equation to solve for the value of ${\displaystyle f^{*}}$  gives the Kelly criterion:

${\displaystyle f^{*}={\frac {p}{a}}-{\frac {q}{b}}}$

Notice that this expression reduces to the simple gambling formula when ${\displaystyle a=1=100\%}$ , when a loss results in full loss of the wager.

## Bernoulli

In a 1738 article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is different (Bernoulli wanted to resolve the St. Petersburg paradox).

An English-language translation of the Bernoulli article was not published until 1954,[12] but the work was well known among mathematicians and economists.

## Application to the stock market

In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal.

Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion (e.g., half).

## Criticism

Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate.[8] The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times[1]). 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.[13]

## Advanced mathematics

For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published.[14] We give the following non-rigorous argument for the case with ${\displaystyle b=1}$  (a 50:50 "even money" bet) to show the general idea and provide some insights.[1] When ${\displaystyle b=1}$ , a Kelly bettor bets ${\displaystyle 2p-1}$  times their initial wealth ${\displaystyle W}$ , as shown above. If they win, they have ${\displaystyle 2pW}$  after one bet. If they lose, they have ${\displaystyle 2(1-p)W}$ . Suppose they make ${\displaystyle N}$  bets like this, and win ${\displaystyle K}$  times out of this series of ${\displaystyle N}$  bets. The resulting wealth will be:

${\displaystyle 2^{N}p^{K}(1-p)^{N-K}W\!.}$

Note that the ordering of the wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount, ${\displaystyle (2p-1+\Delta )W}$  for some value of ${\displaystyle \Delta }$  (where ${\displaystyle \Delta }$  may be positive or negative). They will have ${\displaystyle (2p+\Delta )W}$  after a win and ${\displaystyle [2(1-p)-\Delta ]W}$  after a loss. After the same series of wins and losses as the Kelly bettor, they will have:

${\displaystyle (2p+\Delta )^{K}[2(1-p)-\Delta ]^{N-K}W}$

Take the derivative of this with respect to ${\displaystyle \Delta }$  and get:

${\displaystyle K(2p+\Delta )^{K-1}[2(1-p)-\Delta ]^{N-K}W-(N-K)(2p+\Delta )^{K}[2(1-p)-\Delta ]^{N-K-1}W}$

The function is maximized when this derivative is equal to zero, which occurs at:

${\displaystyle K[2(1-p)-\Delta ]=(N-K)(2p+\Delta )}$

which implies that

${\displaystyle \Delta =2\left({\frac {K}{N}}-p\right)}$

but the proportion of winning bets will eventually converge to:

${\displaystyle \lim _{N\to +\infty }{\frac {K}{N}}=p}$

according to the weak law of large numbers. So in the long run, final wealth is maximized by setting ${\displaystyle \Delta }$  to zero, which means following the 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:

${\displaystyle \left(2{\frac {K}{N}}-1\right)W}$

each time. This is true whether ${\displaystyle N}$  is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as ${\displaystyle N}$  gets large, ${\displaystyle K}$  will approach ${\displaystyle pN}$ . Someone who bets more than Kelly can do better if ${\displaystyle K>pN}$  for a stretch; someone who bets less than Kelly can do better if ${\displaystyle K  for a stretch, but in the long run, Kelly always wins. The heuristic proof for the general case proceeds as follows.[citation needed] In a single trial, if you invest the fraction ${\displaystyle f}$  of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor ${\displaystyle 1-f+f(1+b)=1+fb}$ , and, likewise, if the strategy fails, you end up having your capital decreased by the factor ${\displaystyle 1-fa}$ . Thus at the end of ${\displaystyle N}$  trials (with ${\displaystyle pN}$  successes and ${\displaystyle qN}$  failures), the starting capital of \$1 yields

${\displaystyle C_{N}=(1+fb)^{pN}(1-fa)^{qN}.}$

Maximizing ${\displaystyle \log(C_{N})/N}$ , and consequently ${\displaystyle C_{N}}$ , with respect to ${\displaystyle f}$  leads to the desired result

${\displaystyle f^{*}=p/a-q/b.}$

Edward O. Thorp provided a more detailed discussion of this formula for the general case.[15] There, it can be seen that the substitution of ${\displaystyle p}$  for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since ${\displaystyle p}$  is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting ${\displaystyle f^{*}}$  each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and ${\displaystyle p}$  and ${\displaystyle b}$  are the same for each trial. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. In the heuristic proof above, ${\displaystyle pN}$  successes and ${\displaystyle qN}$  failures are highly likely only for very large ${\displaystyle N}$ .

### Multiple outcomes

Kelly's criterion may be generalized[16] on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the ${\displaystyle k}$ -th horse wins the race is ${\displaystyle p_{k}}$ , the total amount of bets placed on ${\displaystyle k}$ -th horse is ${\displaystyle B_{k}}$ , and

${\displaystyle \beta _{k}={\frac {B_{k}}{\sum _{i}B_{i}}}={\frac {D}{1+Q_{k}}},}$

where ${\displaystyle Q_{k}}$  are the pay-off odds. ${\displaystyle D=1-tt}$ , is the dividend rate where ${\displaystyle tt}$  is the track take or tax, ${\displaystyle {\frac {D}{\beta _{k}}}}$  is the revenue rate after deduction of the track take when ${\displaystyle k}$ -th horse wins. The fraction of the bettor's funds to bet on ${\displaystyle k}$ -th horse is ${\displaystyle f_{k}}$ . Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set ${\displaystyle S^{o}}$  of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions ${\displaystyle f_{k}^{o}}$  of bettor's wealth to be bet on the outcomes included in the optimal set ${\displaystyle S^{o}}$ . The algorithm for the optimal set of outcomes consists of four steps:[16]

1. Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:
${\displaystyle er_{i}={\frac {Dp_{i}}{\beta _{i}}}=p_{i}(Q_{i}+1)}$

2. Reorder the outcomes so that the new sequence ${\displaystyle er_{k}}$  is non-increasing. Thus ${\displaystyle er_{1}}$  will be the best bet.
3. Set ${\displaystyle S=\varnothing }$  (the empty set), ${\displaystyle k=1}$ , ${\displaystyle R(S)=1}$ . Thus the best bet ${\displaystyle er_{k}=er_{1}}$  will be considered first.
4. Repeat:
If ${\displaystyle er_{k}={\frac {D}{\beta _{k}}}p_{k}>R(S)}$  then insert ${\displaystyle k}$ -th outcome into the set: ${\displaystyle S=S\cup \{k\}}$ , recalculate ${\displaystyle R(S)}$  according to the formula:
${\displaystyle R(S)={\frac {D\sum _{k\notin S}p_{k}}{D-\sum _{k\in S}\beta _{k}}}}$

and then set ${\displaystyle k=k+1}$ ,
Otherwise, set ${\displaystyle S^{o}=S}$  and stop the repetition.

If the optimal set ${\displaystyle S^{o}}$  is empty then do not bet at all. If the set ${\displaystyle S^{o}}$  of optimal outcomes is not empty, then the optimal fraction ${\displaystyle f_{k}^{o}}$  to bet on ${\displaystyle k}$ -th outcome may be calculated from this formula:

${\displaystyle f_{i}=p_{i}-\beta _{i}{\frac {\sum _{k\notin S}p_{k}}{\left(D-\sum _{k\in S}\beta _{k}\right)}}.}$

One may prove[16] that

${\displaystyle R(S^{o})=1-\sum _{i\in S^{o}}{f_{i}^{o}}}$

where the right hand-side is the reserve rate[clarification needed]. Therefore, the requirement ${\displaystyle er_{k}={\frac {D}{\beta _{k}}}p_{k}>R(S)}$  may be interpreted[16] as follows: ${\displaystyle k}$ -th outcome is included in the set ${\displaystyle S^{o}}$  of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction ${\displaystyle f_{k}^{o}}$  may be interpreted as the excess of the expected revenue rate of ${\displaystyle k}$ -th horse over the reserve rate divided by the revenue after deduction of the track take when ${\displaystyle k}$ -th horse wins or as the excess of the probability of ${\displaystyle k}$ -th horse winning over the reserve rate divided by revenue after deduction of the track take when ${\displaystyle k}$ -th horse wins. The binary growth exponent is

${\displaystyle G^{o}=\sum _{i\in S}{p_{i}\log _{2}(er_{i})}+\left(1-\sum _{i\in S}{p_{i}}\right)\log _{2}(R(S^{o})),}$

and the doubling time is

${\displaystyle T_{d}={\frac {1}{G^{o}}}.}$

This method of selection of optimal bets may be applied also when probabilities ${\displaystyle p_{k}}$  are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that

${\displaystyle \sum _{i}{p_{i}}<1,}$  and
${\displaystyle \sum _{i}{\beta _{i}}<1}$ .

### Stock investments

The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data – expected value and variance. This approximation leads to results that are robust and offer similar results as the original criterion.[17]

For single assets(stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through geometric Brownian motion. The stochastic differential equation governing the evolution of a lognormally distributed asset ${\displaystyle S}$  at time ${\displaystyle t}$  (${\displaystyle S_{t}}$ ) is

${\displaystyle dS_{t}/S_{t}=\mu dt+\sigma dW_{t}}$

whose solution is

${\displaystyle S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right)}$

where ${\displaystyle W_{t}}$  is a Wiener process, and ${\displaystyle \mu }$  (percentage drift) and ${\displaystyle \sigma }$  (the percentage volatility) are constants. Taking expectations of the logarithm:

${\displaystyle \mathbb {E} \log(S_{t})=\log(S_{0})+\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t.}$

Then the expected log return ${\displaystyle R_{s}}$  is

${\displaystyle R_{s}=\left(\mu -{\frac {\sigma ^{2}}{2}}\,\right)s.}$

Consider a portfolio made of an asset ${\displaystyle S}$  and a bond paying risk-free rate ${\displaystyle r}$ , with fraction ${\displaystyle f}$  invested in ${\displaystyle S}$  and ${\displaystyle (1-f)}$  in the bond. The aforementioned equation for ${\displaystyle dS_{t}}$  must be modified by this fraction, ie ${\displaystyle dS_{t}'=fdS_{t}}$ , with associated solution

${\displaystyle S'_{t}=S'_{0}\exp \left(\left(f\mu -{\frac {(f\sigma )^{2}}{2}}\right)t+f\sigma W_{t}\right)}$

the expected one-period return is given by

${\displaystyle \mathbb {E} {\left(\left[{\frac {S'_{1}}{S'_{0}}}-1\right]+(1-f)r\right)}=\mathbb {E} {\left(\left[\exp \left(\left(f\mu -{\frac {(f\sigma )^{2}}{2}}\right)+f\sigma W_{1}\right)-1\right]\right)}+(1-f)r}$

For small ${\displaystyle \mu }$ , ${\displaystyle \sigma }$ , and ${\displaystyle W_{t}}$ , the solution can be expanded to first order to yield an approximate increase in wealth

${\displaystyle G(f)=f\mu -{\frac {(f\sigma )^{2}}{2}}+((1-f)\ r).}$

Solving ${\displaystyle \max(G(f))}$  we obtain

${\displaystyle f^{*}={\frac {\mu -r}{\sigma ^{2}}}.}$

${\displaystyle f^{*}}$  is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp[15] arrived at the same result but through a different derivation. Remember that ${\displaystyle \mu }$  is different from the asset log return ${\displaystyle R_{s}}$ . Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion.

For multiple assets, consider a market with ${\displaystyle n}$  correlated stocks ${\displaystyle S_{k}}$  with stochastic returns ${\displaystyle r_{k}}$ , ${\displaystyle k=1,\dots ,n,}$  and a riskless bond with return ${\displaystyle r}$ . An investor puts a fraction ${\displaystyle u_{k}}$  of their capital in ${\displaystyle S_{k}}$  and the rest is invested in the bond. Without loss of generality, assume that investor's starting capital is equal to 1. According to the Kelly criterion one should maximize

${\displaystyle \mathbb {E} \left[\ln \left((1+r)+\sum \limits _{k=1}^{n}u_{k}(r_{k}-r)\right)\right].}$

Expanding this with a Taylor series around ${\displaystyle {\vec {u_{0}}}=(0,\ldots ,0)}$  we obtain

${\displaystyle \mathbb {E} \left[\ln(1+r)+\sum \limits _{k=1}^{n}{\frac {u_{k}(r_{k}-r)}{1+r}}-{\frac {1}{2}}\sum \limits _{k=1}^{n}\sum \limits _{j=1}^{n}u_{k}u_{j}{\frac {(r_{k}-r)(r_{j}-r)}{(1+r)^{2}}}\right].}$

Thus we reduce the optimization problem to quadratic programming and the unconstrained solution is

${\displaystyle {\vec {u^{\star }}}=(1+r)({\widehat {\Sigma }})^{-1}({\widehat {\vec {r}}}-r)}$

where ${\displaystyle {\widehat {\vec {r}}}}$  and ${\displaystyle {\widehat {\Sigma }}}$  are the vector of means and the matrix of second mixed noncentral moments of the excess returns. There is also a numerical algorithm for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.[18]

## References

1. ^ a b c d 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.
2. ^ Thorp, E. O. (January 1961), "Fortune's Formula: The Game of Blackjack", American Mathematical Society
3. ^ Thorp, E. O. (1962), Beat the dealer: a winning strategy for the game of twenty-one. A scientific analysis of the world-wide game known variously as blackjack, twenty-one, vingt-et-un, pontoon or Van John, Blaisdell Pub. Co
4. ^ Thorp, Edward O.; Kassouf, Sheen T. (1967), Beat the Market: A Scientific Stock Market System (PDF), Random House, ISBN 0-394-42439-5, archived from the original (PDF) on 2009-10-07, page 184f.
5. ^ Zenios, S. A.; Ziemba, W. T. (2006), Handbook of Asset and Liability Management, North Holland, ISBN 978-0-444-50875-1
6. ^ Pabrai, Mohnish (2007), The Dhandho Investor: The Low-Risk Value Method to High Returns, Wiley, ISBN 978-0-470-04389-9
7. ^ Thorp, E. O. (September 2008), "The Kelly Criterion: Part II", Wilmott Magazine
8. ^ a b Poundstone, William (2005), Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, New York: Hill and Wang, ISBN 0-8090-4637-7
9. ^ Haghani, Victor; Dewey, Richard (19 October 2016). "Rational Decision-Making under Uncertainty: Observed Betting Patterns on a Biased Coin". SSRN 2856963. arXiv:1701.01427
10. ^ "Buttonwood", "Irrational tossers", The Economist Newspaper Limited 2016, Nov 1st 2016.
11. ^ Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 14.7 (Example 2.)", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
12. ^ Bernoulli, Daniel (1954) [1738]. "Exposition of a New Theory on the Measurement of Risk". Econometrica. The Econometric Society. 22 (1): 22–36. doi:10.2307/1909829. JSTOR 1909829.
13. ^ Thorp, E. O. (May 2008), "The Kelly Criterion: Part I", Wilmott Magazine
14. ^ Thorp, E. O. (1969). "Optimal Gambling Systems for Favorable Games". Revue de l'Institut International de Statistique / Review of the International Statistical Institute. International Statistical Institute (ISI). 37 (3): 273–293. doi:10.2307/1402118. JSTOR 1402118. MR 0135630.
15. ^ a b Thorp, Edward O. (June 1997). "The Kelly criterion in blackjack, sports betting, and the stock market" (PDF). 10th International Conference on Gambling and Risk Taking. Montreal. Archived from the original (PDF) on 2009-03-20. Retrieved 2009-03-20.
16. ^ a b c d Smoczynski, Peter; Tomkins, Dave (2010) "An explicit solution to the problem of optimizing the allocations of a bettor’s wealth when wagering on horse races", Mathematical Scientist", 35 (1), 10-17
17. ^ Marek, Patrice; Ťoupal, Tomáš; Vávra, František (2016). "Efficient Distribution of Investment Capital". 34th International Conference Mathematical Methods in Economics, MME2016, Conference Proceedings: 540–545. Retrieved 24 January 2018.
18. ^ Nekrasov, Vasily (2013). "Kelly Criterion for Multivariate Portfolios: A Model-Free Approach". SSRN 2259133.