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### Kelly criterion

In probability theory, the Kelly criterion, or Kelly strategy or Kelly formula, or Kelly bet, is a formula 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, in a 1956 issue of the Bell System Technical Journal. Edward O. Thorp demonstrated the practical use of the formula in a 1961 address to the American Mathematical Society 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 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 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 and Bill Gross use Kelly methods. William Poundstone 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, the Kelly bet is:

f^{*} = \frac{bp - q}{b} , \!

where:

* f* is the fraction of the current bankroll to wager;
* b is the net odds received on the wager (that is, odds are usually quoted as "b to 1")
* 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. For even-money bets (i.e. when b = 1), the formula can be simplified to: f^{*} = p - q . \! Since q = 1-p, this simplifies further to f^{*} = 2p - 1 . \! Proof For a rigorous and general proof, see Kelly's original paper[1] 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: 2^Np^K(1-p)^{N-K}W \! Suppose another bettor bets a different amount, (2p - 1 + Δ)W for some positive or negative Δ. He will have (2p + Δ)W after a win and [2(1 - p)- Δ]W after a loss. After the same wins and losses as the Kelly bettor, he will have: (2p+\Delta)^K[2(1-p)-\Delta]^{N-K}W \! Take the derivative of this with respect to Δ and get: 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 turning point of the original function occurs when this derivative equals zero, which occurs at: K[2(1-p)-\Delta]=(N-K)(2p+\Delta) \! which implies: \Delta=2(\frac{K}{N}-p) \! but: \lim_{N \to +\infty}\frac{K}{N}=p \! so in the long run, final wealth is maximized by setting Δ 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: \left(2\frac{K}{N}-1\right)W \! 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 (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 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. Bernoulli In a 1738 article, Daniel Bernoulli suggested that when one has a choice of bets or investments that one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different (Bernoulli wanted to resolve the St. Petersburg paradox). 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 pk, the total of bets placed on k-th horse is Bk (in dollars), and \beta_k=\frac{B_k}{\sum_i B_i}=\frac{1}{1+Q_k}, where Qk are the pay-off odds. D = 1 − tt, is the dividend rate where tt is the track take or tax, \frac{D}{\beta_k} 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 fk. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set So of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions f^o_k of bettor's wealth to be bet on the outcomes included in the optimal set So. 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: er_k=\frac{D}{\beta_k}p_k=D(1+Q_k)p_k. Step 2 Reorder the outcomes so that the new sequence erk is non-decreasing. Step 3 Set S = \varnothing (the empty set), k = 1, R(S) = 1. Step 4 Repeat: if er_k=\frac{D}{\beta_k}p_k > R(S) then insert k-th outcome into the set: S = S \cup \{k\}, recalculate R(S) according to the formula: R(S)=\frac{1-\sum_{i \in S}{p_i}}{1-\sum_{i \in S } \frac{\beta_i}{D}} and then set k = k + 1, else set So = S and then stop the repetition.

If the optimal set So is empty then do not bet at all. If the set So of optimal outcomes is not empty then the optimal fraction f^o_k to bet on k-th outcome may be calculated from this formula: f^o_k=\frac{er_k - R(S^o)}{\frac{D}{\beta_k}}=p_k-\frac{R(S^o)}{\frac{D}{\beta_k}}.

One may prove that R(S^o)=1-\sum_{i \in S^o}{f^o_i} is the reserve rate. Therefore the requirement er_k=\frac{D}{\beta_k}p_k > R(S) may be interpreted as follows: k-th outcome is included into the set So of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction f^o_k 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 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 G^o=\sum_{i \in S}{p_ilog_2{(er_i)}}+(1-\sum_{i \in S}{p_i})log_2{(R(S^o))} and the doubling time is T_d=\frac{1}{G^o}.

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 pk are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that
∑ pi < 1
i

and
∑ βi < 1
i