9059:
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 colloquial terms, the Kelly criterion requires accurate probability values, which isn't always possible for real-world event outcomes. When a gambler overestimates their true probability of winning, the criterion value calculated will diverge from the optimal, increasing the risk of ruin.
499:
355:
507:
20:
1559:). This happens somewhat counterintuitively, because the Kelly fraction formula compensates for a small losing size with a larger bet. However, in most real situations, there is high uncertainty about all parameters entering the Kelly formula. In the case of a Kelly fraction higher than 1, it is theoretically advantageous to use
3802:
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 K win knowing that
358:
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
9058:
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). Even Kelly supporters usually argue for fractional Kelly (betting a fixed
2597:
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
9041:
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. The conventional
1213:
1652:
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
1589:
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
7560:
953:
6468:
1047:
5959:
9009:
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.
2766:
and a co-author estimates Kelly fraction to be 117% for the
American stock market SP500 index. Significant downside tail-risk for equity markets is another reason to reduce Kelly fraction from naive estimate (for instance, to reduce to half-Kelly).
2285:
106:
In a system where the return on an investment or a bet is binary, so an interested party either wins or loses a fixed percentage of their bet, the expected growth rate coefficient yields a very specific solution for an optimal betting percentage.
2613:
16%, then yearly Sharpe ratio and Kelly ratio are calculated to be 25% and 150%. Daily Sharpe ratio and Kelly ratio are 1.7% and 150%. Sharpe ratio implies daily win probability of p=(50% + 1.7%/4), where we assumed that probability bandwidth is
6941:
1584:
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
9062:
Kelly formula can be thought as 'time diversification', which is taking equal risk during different sequential time periods (as opposed to taking equal risk in different assets for asset diversification). There is clearly a difference between
6667:
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 –
7129:
7804:
8939:
5649:
4806:
3561:
1579:
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:
1069:
7229:
5182:
7973:
1639:
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).
4539:
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,
6128:
5461:
189:
9071:. There is also a difference between ensemble-averaging (utility calculation) and time-averaging (Kelly multi-period betting over a single time path in real life). The debate was renewed by envoking
6050:
3711:
3773:
8075:
2488:
2371:
750:
6827:
2067:
4614:
Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the
3853:
4266:
886:
8535:
6295:
1940:
980:
493:
6652:
9188:
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
6524:
3077:
1634:
5834:
4928:
3343:
2606:
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.
6605:
502:
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 a win is 60%).
8978:
9007:
7426:
1266:
8106:
is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp arrived at the same result but through a different derivation. Remember that
7434:
3653:
8816:
8468:
2721:
1307:
350:
8345:
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
4079:
3124:
2145:
5273:
4424:
8269:
5519:
5380:
4316:
3202:
6838:
2644:
1476:
2564:
8014:
2757:
1400:
6207:
5806:
5553:
5054:
2409:
1432:
583:
7847:
7023:
7015:
5714:
3997:
3968:
3800:
3366:
3164:
3144:
2942:
2759:. Both of these estimates of Kelly fraction appear quite reasonable, yet a prudent approach suggest a further multiplication of Kelly ratio by 50% (i.e. half-Kelly).
647:
7571:
7374:
5334:
5244:
5214:
4871:
1505:
8343:
8316:
8228:
8201:
8151:
8104:
7874:
7344:
7159:
6971:
6748:
6554:
6175:
5774:
5747:
5681:
5081:
5022:
4995:
4836:
4706:
4659:
4497:
4343:
3999:
for a stretch, but in the long run, Kelly always wins. The heuristic proof for the general case proceeds as follows. In a single trial, if one invests the fraction
2779:
original paper or in some of the other references listed below. Some corrections have been published. We give the following non-rigorous argument for the case with
2671:
2517:
2315:
2137:
1792:
859:
783:
678:
546:
415:
389:
278:
222:
8831:
8124:
7827:
6995:
5548:
4108:
2858:
2537:
1841:
1760:
5299:
2904:
2829:
2803:
1557:
1531:
1367:
1341:
613:
441:
4894:
4584:
4561:
4174:
4151:
3939:
8289:
8174:
7312:
7292:
7272:
7252:
6721:
6701:
6287:
6267:
6247:
6227:
6148:
5826:
5481:
4968:
4948:
4679:
4632:
4604:
4537:
4517:
4470:
4450:
4363:
4128:
4017:
3916:
3896:
3876:
3252:
3002:
2982:
2962:
2878:
2588:
2110:
2090:
1861:
1812:
1731:
1711:
1691:
1671:
975:
881:
827:
805:
300:
244:
9220:
78:. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known successful investors including
2427:
In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is
5090:
2419:
If the return rates on an investment or a bet are continuous in nature the optimal growth rate coefficient must take all possible events into account.
4714:
9033:
An
English translation of the Bernoulli article was not published until 1954, but the work was well known among mathematicians and economists.
3374:
1208:{\displaystyle f^{*}={\frac {p}{l}}\left(1-{\frac {1-p}{p}}{\frac {l}{g}}\right)={\frac {p}{l}}\left(1-{\frac {1}{WLP}}{\frac {1}{WLR}}\right)}
9390:
Haghani, Victor; Dewey, Richard (19 October 2016). "Rational
Decision-Making under Uncertainty: Observed Betting Patterns on a Biased Coin".
9026:
of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is different (Bernoulli wanted to resolve the
7167:
1594:, not the arithmetic rate of 4% (r = 0.2 x (0.6 - 0.4) = 0.04). The theoretical expected wealth after 300 rounds works out to $ 10,505 (
7882:
6676:. This approximation may offer similar results as the original criterion, but in some cases the solution obtained may be infeasible.
6556:
are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that
6177:
of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction
9464:
6058:
5391:
6679:
For single assets (stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through
4452:
for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since
121:
5970:
510:
3D figure representing the optimal Kelly bet size (vertical axis) as a function of win probability and amount gained with win.
443:), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (
9448:
9292:
9266:
3664:
3726:
585:), the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in
9362:
8022:
4997:. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set
3082:
The ordering of the wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount,
4432:
provided a more detailed discussion of this formula for the general case. There, it can be seen that the substitution of
2438:
2323:
702:
6756:
1974:
948:{\displaystyle g={\frac {{\text{final value}}-{\text{original value}}}{\text{original value}}}={\frac {1.1-1}{1}}=0.1}
9669:
9593:"An explicit solution to the problem of optimizing the allocations of a bettor's wealth when wagering on horse races"
9337:
9233:
9195:
6463:{\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})),}
4499:
each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and
3809:
1042:{\displaystyle l={\frac {{\text{original value}}-{\text{final value}}}{\text{original value}}}={\frac {1-.9}{1}}=0.1}
9239:
9635:. 53rd Annual Allerton Conference on Communication, Control, and Computing. Monticello, IL: IEEE. pp. 165–172.
4182:
8480:
1869:
9097:
6612:
5954:{\displaystyle f_{i}=p_{i}-\beta _{i}{\frac {\sum _{k\notin S}p_{k}}{\left(D-\sum _{k\in S}\beta _{k}\right)}}.}
446:
6479:
3010:
87:
1597:
9854:
9145:
9107:
5024:
of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions
4899:
3878:
is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as
3260:
9811:
Psychology is
Fundamental: The Limitations of Growth-Optimal Approaches to Decision Making under Uncertainty
7555:{\displaystyle S'_{t}=S'_{0}\exp \left(\left(f\mu -{\frac {(f\sigma )^{2}}{2}}\right)t+f\sigma W_{t}\right)}
6562:
8947:
5087:
Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:
2591:
9329:
Fortune's
Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street
8983:
1224:
9874:
9869:
9859:
3572:
8543:
8351:
4019:
of their capital, if the strategy succeeds, the capital at the end of the trial increases by the factor
2676:
2280:{\displaystyle \left.{\frac {dE}{df}}\right|_{f=f^{*}}={\frac {pb}{1+f^{*}b}}+{\frac {-qa}{1-f^{*}a}}=0}
1273:
9849:
9047:
696:
A more general form of the Kelly formula allows for partial losses, which is relevant for investments:
309:
9022:
suggested that, when one has a choice of bets or investments, one should choose that with the highest
8153:. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion.
7379:
6936:{\displaystyle S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right)}
6680:
4022:
3779:
3085:
71:
9543:
Revue de l'Institut
International de Statistique / Review of the International Statistical Institute
5252:
4371:
8233:
5486:
5339:
4274:
3169:
6683:. The stochastic differential equation governing the evolution of a lognormally distributed asset
6269:-th horse winning over the reserve rate divided by revenue after deduction of the track take when
4472:
is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting
2617:
1437:
9686:
9102:
2542:
2519:
is the fraction of available capital invested that maximizes the expected geometric growth rate,
688:
bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.
7981:
7124:{\displaystyle \mathbb {E} \log(S_{t})=\log(S_{0})+\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t.}
2726:
1372:
9646:
Nekrasov, Vasily (2013). "Kelly
Criterion for Multivariate Portfolios: A Model-Free Approach".
9416:
9027:
7799:{\displaystyle \mathbb {E} {\left(\left+(1-f)r\right)}=\mathbb {E} {\left(\left\right)}+(1-f)r}
6180:
5779:
5027:
2379:
1648:
Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the
23:
Example of the optimal Kelly betting fraction, versus expected return of other fractional bets.
9651:
9395:
6229:-th horse over the reserve rate divided by the revenue after deduction of the track take when
1405:
554:
9620:. 34th International Conference Mathematical Methods in Economics. pp. 540–545. MME2016.
9080:
8934:{\displaystyle {\vec {u^{\star }}}=(1+r)({\widehat {\Sigma }})^{-1}({\widehat {\vec {r}}}-r)}
8822:
7832:
7000:
5686:
3973:
3944:
3785:
3351:
3149:
3129:
2909:
883:
is the fraction that is gained in a positive outcome. If the security price rises 10%, then
618:
75:
9137:
7349:
5304:
5219:
5189:
4841:
3254:
after a loss. After the same series of wins and losses as the Kelly bettor, they will have:
1484:
9570:
9133:
8321:
8294:
8206:
8179:
8129:
8082:
7852:
7317:
7137:
6949:
6726:
6532:
6153:
5752:
5725:
5654:
5059:
5000:
4973:
4814:
4684:
4637:
4475:
4321:
2776:
2649:
2610:
2495:
2293:
2115:
1765:
977:
is the fraction that is lost in a negative outcome. If the security price falls 10%, then
832:
761:
652:
517:
394:
368:
251:
200:
56:
8109:
7812:
6980:
5524:
4084:
2834:
2522:
1817:
1736:
1481:
The Kelly formula can easily result in a fraction higher than 1, such as with losing size
8:
7876:, the solution can be expanded to first order to yield an approximate increase in wealth
5278:
2883:
2808:
2782:
1536:
1510:
1346:
1320:
592:
420:
4876:
4566:
4543:
4156:
4133:
3921:
9822:
9792:
9764:
9727:
9706:
9698:
9558:
9506:
9400:
9309:
9282:
9158:
8274:
8159:
7297:
7277:
7257:
7237:
6706:
6686:
6272:
6252:
6232:
6212:
6133:
5811:
5466:
4953:
4933:
4664:
4617:
4589:
4522:
4502:
4455:
4435:
4348:
4113:
4002:
3901:
3881:
3861:
3207:
2987:
2967:
2947:
2863:
2573:
2095:
2075:
1965:
1846:
1797:
1716:
1696:
1676:
1656:
1560:
960:
866:
812:
790:
285:
229:
28:
1268:
is the win-loss probability (WLP) ratio, which is the ratio of winning to losing bets.
9879:
9864:
9826:
9784:
9647:
9615:
9592:
9444:
9391:
9333:
9327:
9288:
9262:
9229:
9201:
9191:
9068:
1946:
51:
of the logarithm of wealth, which is equivalent to maximizing the long-term expected
2805:(a 50:50 "even money" bet) to show the general idea and provide some insights. When
2433:
The Kelly
Criterion shows that for a given volatile security this is satisfied when
9814:
9796:
9774:
9710:
9690:
9665:
9550:
9538:
9516:
9483:
9479:
9436:
9154:
9043:
9019:
1968:, it does not change the locations of function extrema. The resulting equation is:
1564:
302:
is the proportion of the bet gained with a win. E.g., if betting $ 10 on a 2-to-1
83:
5644:{\displaystyle R(S)={\frac {D\sum _{k\notin S}p_{k}}{D-\sum _{k\in S}\beta _{k}}}}
3803:
the rest of the bets will lose), one will end up with the most money if one bets:
3566:
The function is maximized when this derivative is equal to zero, which occurs at:
9566:
9534:
7017:(the percentage volatility) are constants. Taking expectations of the logarithm:
4429:
2763:
91:
9741:
Samuelson, Paul. A. (1963), "Risk and uncertainty: a fallacy of large numbers",
6529:
This method of selection of optimal bets may be applied also when probabilities
4801:{\displaystyle \beta _{k}={\frac {B_{k}}{\sum _{i}B_{i}}}={\frac {D}{1+Q_{k}}},}
2139:, we differentiate the above expression and set this equal to zero. This gives:
680:, meaning the gambler should bet one-nineteenth of their bankroll that red will
9051:
9023:
6974:
6669:
4081:, and, likewise, if the strategy fails, the capital is decreased by the factor
1649:
1591:
79:
52:
48:
9843:
9421:
8474:
6664:
1057:
615:) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (
9755:
Peters, Ole; Gell-Mann, Murray (2015), "Evaluating gambles using dynamics",
2602:
performance of a growth-optimal portfolio may differ fantastically from the
1056:
outcome probabilities, which is not the case with investments. In addition,
9788:
9677:
9520:
9435:
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007),
9092:
9205:
6055:
where the right hand-side is the reserve rate. Therefore, the requirement
1402:). It is even possible that the win-loss probability ratio is unfavorable
498:
354:
9818:
9250:
9172:
Thorp, E. O. (January 1961), "Fortune's
Formula: The Game of Blackjack",
5056:
of bettor's wealth to be bet on the outcomes included in the optimal set
3556:{\displaystyle K(2p+\Delta )^{K-1}^{N-K}W-(N-K)(2p+\Delta )^{K}^{N-K-1}W}
115:
Where losing the bet involves losing the entire wager, the Kelly bet is:
9363:"The Kelly criterion in blackjack, sports betting, and the stock market"
5083:. The algorithm for the optimal set of outcomes consists of four steps:
2376:
Notice that this expression reduces to the simple gambling formula when
9702:
9562:
9076:
9072:
3717:
1957:
1576:
9779:
94:
in anytime-valid statistical tests and confidence intervals, based on
9055:
1961:
506:
60:
9694:
9554:
9511:
9404:
7224:{\displaystyle R_{s}=\left(\mu -{\frac {\sigma ^{2}}{2}}\,\right).}
6673:
2567:
586:
95:
67:
9769:
5177:{\displaystyle er_{i}={\frac {Dp_{i}}{\beta _{i}}}=p_{i}(Q_{i}+1)}
2609:
Rough estimates are still useful. If we take excess return 4% and
6209:
may be interpreted as the excess of the expected revenue rate of
3970:
for a stretch; someone who bets less than Kelly can do better if
44:
9434:
7968:{\displaystyle G(f)=f\mu -{\frac {(f\sigma )^{2}}{2}}+(1-f)\ r.}
1960:
of the equation. This is more easily accomplished by taking the
9307:
Thorp, E. O. (September 2008), "The Kelly
Criterion: Part II",
9284:
The Dhandho Investor: The Low-Risk Value Method to High Returns
1575:
In a study, each participant was given $ 25 and asked to place
2414:
4950:-th horse wins. The fraction of the bettor's funds to bet on
306:
bet (upon win you are returned $ 30, winning you $ 20), then
5776:
of optimal outcomes is not empty, then the optimal fraction
19:
6123:{\displaystyle er_{k}={\frac {D}{\beta _{k}}}p_{k}>R(S)}
5456:{\displaystyle er_{k}={\frac {D}{\beta _{k}}}p_{k}>R(S)}
4930:
is the revenue rate after deduction of the track take when
3782:. So in the long run, final wealth is maximized by setting
2151:
1570:
829:
is the probability that the investment decreases in value (
303:
66:
The practical use of the formula has been demonstrated for
2598:
weights are largely a function of estimation errors, then
2092:
denoting logarithmic wealth growth. To find the value of
1673:
of that wealth on an outcome that occurs with probability
1309:
is the win-loss ratio (WLR) of bet outcomes, which is the
807:
is the probability that the investment increases in value.
548:), then the criterion recommends the gambler bet nothing.
417:), and the gambler receives 1-to-1 odds on a winning bet (
101:
47:
for sizing a sequence of bets by maximizing the long-term
9549:(3). International Statistical Institute (ISI): 273–293.
9370:
10th International Conference on Gambling and Risk Taking
9321:
9319:
184:{\displaystyle f^{*}=p-{\frac {q}{b}}=p-{\frac {1-p}{b}}}
9757:
Chaos: An Interdisciplinary Journal of Nonlinear Science
9725:
Thorp, E. O. (May 2008), "The Kelly Criterion: Part I",
9614:
Marek, Patrice; Ťoupal, Tomáš; Vávra, František (2016).
9128:
9126:
9124:
9122:
6045:{\displaystyle R(S^{o})=1-\sum _{i\in S^{o}}{f_{i}^{o}}}
365:
As an example, if a gamble has a 60% chance of winning (
9670:"Exposition of a New Theory on the Measurement of Risk"
2723:, and we get another rough estimate for Kelly fraction
2590:
is the risk-free rate of return. Note that a symmetric
785:
is the fraction of the assets to apply to the security.
9443:(3rd ed.), New York: Cambridge University Press,
9316:
6249:-th horse wins or as the excess of the probability of
3706:{\displaystyle \Delta =2\left({\frac {K}{N}}-p\right)}
449:
9658:
9497:
Turlakov, Mihail (2017). "Leverage and Uncertainty".
9119:
8986:
8950:
8834:
8546:
8483:
8354:
8324:
8297:
8277:
8236:
8209:
8182:
8162:
8132:
8112:
8085:
8025:
7984:
7885:
7855:
7835:
7815:
7574:
7437:
7382:
7352:
7320:
7300:
7280:
7260:
7240:
7170:
7140:
7026:
7003:
6983:
6952:
6841:
6759:
6729:
6709:
6689:
6615:
6565:
6535:
6482:
6298:
6275:
6255:
6235:
6215:
6183:
6156:
6136:
6061:
5973:
5837:
5814:
5782:
5755:
5728:
5689:
5657:
5556:
5527:
5489:
5469:
5394:
5342:
5307:
5281:
5255:
5222:
5192:
5093:
5062:
5030:
5003:
4976:
4956:
4936:
4902:
4879:
4844:
4817:
4717:
4687:
4667:
4640:
4620:
4592:
4569:
4546:
4525:
4505:
4478:
4458:
4438:
4374:
4351:
4324:
4277:
4185:
4159:
4136:
4116:
4087:
4025:
4005:
3976:
3947:
3924:
3904:
3884:
3864:
3812:
3788:
3768:{\displaystyle \lim _{N\to +\infty }{\frac {K}{N}}=p}
3729:
3667:
3575:
3377:
3354:
3263:
3210:
3172:
3152:
3132:
3088:
3013:
2990:
2970:
2950:
2912:
2886:
2866:
2837:
2811:
2785:
2729:
2679:
2652:
2620:
2576:
2545:
2525:
2498:
2441:
2382:
2326:
2296:
2148:
2118:
2098:
2078:
1977:
1872:
1849:
1820:
1800:
1768:
1739:
1719:
1699:
1679:
1659:
1600:
1539:
1513:
1487:
1440:
1408:
1375:
1349:
1323:
1276:
1227:
1072:
1060:
investors should not invest the full Kelly fraction.
983:
963:
889:
869:
835:
815:
793:
764:
705:
655:
621:
595:
557:
520:
423:
397:
371:
312:
288:
254:
232:
203:
124:
9743:
Scientia (6th Series, 57th year, April-May), 153–158
8070:{\displaystyle f^{*}={\frac {\mu -r}{\sigma ^{2}}}.}
3941:. Someone who bets more than Kelly can do better if
2290:
Rearranging this equation to solve for the value of
1733:, and in that case the resulting wealth is equal to
9274:
2483:{\displaystyle f^{*}={\frac {\mu -r}{\sigma ^{2}}}}
2366:{\displaystyle f^{*}={\frac {p}{a}}-{\frac {q}{b}}}
2112:for which the growth rate is maximized, denoted as
745:{\displaystyle f^{*}={\frac {p}{l}}-{\frac {q}{g}}}
9441:Numerical Recipes: The Art of Scientific Computing
9356:
9354:
9352:
9350:
9348:
9001:
8972:
8933:
8810:
8529:
8462:
8337:
8310:
8283:
8263:
8222:
8195:
8168:
8145:
8118:
8098:
8069:
8008:
7967:
7868:
7841:
7821:
7798:
7554:
7420:
7368:
7338:
7306:
7286:
7266:
7246:
7223:
7153:
7123:
7009:
6989:
6965:
6935:
6821:
6742:
6715:
6695:
6646:
6599:
6548:
6518:
6462:
6281:
6261:
6241:
6221:
6201:
6169:
6142:
6122:
6044:
5953:
5820:
5800:
5768:
5741:
5708:
5675:
5643:
5542:
5513:
5475:
5455:
5374:
5328:
5293:
5267:
5238:
5208:
5176:
5075:
5048:
5016:
4989:
4962:
4942:
4922:
4888:
4865:
4830:
4800:
4700:
4673:
4653:
4626:
4598:
4578:
4555:
4531:
4511:
4491:
4464:
4444:
4418:
4357:
4337:
4310:
4260:
4168:
4145:
4122:
4102:
4073:
4011:
3991:
3962:
3933:
3910:
3890:
3870:
3847:
3794:
3767:
3705:
3647:
3555:
3360:
3337:
3246:
3196:
3158:
3138:
3118:
3071:
2996:
2976:
2956:
2936:
2898:
2872:
2852:
2823:
2797:
2751:
2715:
2665:
2638:
2582:
2558:
2531:
2511:
2482:
2422:
2403:
2365:
2309:
2279:
2131:
2104:
2084:
2061:
1934:
1855:
1835:
1806:
1786:
1754:
1725:
1705:
1685:
1665:
1628:
1551:
1525:
1499:
1470:
1426:
1394:
1361:
1335:
1301:
1260:
1207:
1041:
969:
947:
875:
853:
821:
799:
777:
744:
672:
641:
607:
577:
540:
487:
435:
409:
383:
344:
294:
272:
238:
216:
183:
9590:
9222:Beat the Market: A Scientific Stock Market System
6822:{\displaystyle dS_{t}/S_{t}=\mu dt+\sigma dW_{t}}
5828:-th outcome may be calculated from this formula:
3065:
2411:, when a loss results in full loss of the wager.
2062:{\displaystyle E=\log(r)=p\log(1+fb)+q\log(1-fa)}
1369:needs to be larger than 1 for having an edge (so
224:is the fraction of the current bankroll to wager.
9841:
9613:
7985:
3731:
1843:. Therefore, the expected geometric growth rate
1814:. In that case the resulting wealth is equal to
1052:Note that the Kelly criterion is valid only for
9754:
9345:
9067:and asset diversification, which was raised by
4586:failures are highly likely only for very large
3848:{\displaystyle \left(2{\frac {K}{N}}-1\right)W}
1317:It is clear that, at least, one of the factors
9586:
9584:
9582:
9580:
9539:"Optimal Gambling Systems for Favorable Games"
9212:
9179:
9054:utility of the outcome (to an individual with
6289:-th horse wins. The binary growth exponent is
5186:Reorder the outcomes so that the new sequence
4176:failures), the starting capital of $ 1 yields
2646:. Now we can apply discrete Kelly formula for
1063:The general form can be rewritten as follows
589:, the bettor is offered an even money payoff (
9734:
7346:in the bond. The aforementioned equation for
4261:{\displaystyle C_{N}=(1+fb)^{pN}(1-fa)^{qN}.}
3166:may be positive or negative). They will have
2775:A rigorous and general proof can be found in
1964:of each side first; because the logarithm is
9630:
9617:Efficient Distribution of Investment Capital
9490:
9462:
9389:
9256:
9219:Thorp, Edward O.; Kassouf, Sheen T. (1967),
9218:
8530:{\displaystyle {\vec {u_{0}}}=(0,\ldots ,0)}
8156:For multiple assets, consider a market with
5749:is empty then do not bet at all. If the set
5508:
5502:
3348:Take the derivative of this with respect to
1935:{\displaystyle r=(1+fb)^{p}\cdot (1-fa)^{q}}
9631:Hsieh, Chung-Han; Barmish, B. Ross (2015).
9577:
8821:Thus we reduce the optimization problem to
7565:the expected one-period return is given by
2415:Kelly criterion for non-binary return rates
488:{\textstyle f^{*}=0.6-{\frac {0.4}{1}}=0.2}
9499:Journal of Investment Strategies, Pp 81-97
9465:"The Kelly criterion and the Stock Market"
9325:
9259:Handbook of Asset and Liability Management
9138:"A New Interpretation of Information Rate"
9046:theory which says bets should be sized to
6647:{\displaystyle \sum _{i}{\beta _{i}}<1}
1507:(see the above expression with factors of
9778:
9768:
9740:
9718:
9664:
9591:Smoczynski, Peter; Tomkins, Dave (2010).
9510:
9456:
9300:
9165:
8548:
8356:
7663:
7576:
7212:
7028:
6519:{\displaystyle T_{d}={\frac {1}{G^{o}}}.}
3072:{\displaystyle 2^{N}p^{K}(1-p)^{N-K}W\!.}
2880:, as shown above. If they win, they have
2539:is the expected growth rate coefficient,
90:. It is also the standard replacement of
9802:
9748:
9645:
9496:
9463:Thorp, E. O.; Rotando, Louis M. (1992).
3716:but the proportion of winning bets will
1629:{\displaystyle =25\cdot (1.02034)^{300}}
1571:Betting example – behavioural experiment
505:
497:
353:
70:, and the same idea was used to explain
18:
9808:
8126:is different from the asset log return
4923:{\displaystyle {\frac {D}{\beta _{k}}}}
3338:{\displaystyle (2p+\Delta )^{K}^{N-K}W}
2906:after one bet. If they lose, they have
514:If the gambler has zero edge (i.e., if
102:Kelly criterion for binary return rates
16:Bet sizing formula for long-term growth
9842:
9280:
7376:must be modified by this fraction, ie
7234:Consider a portfolio made of an asset
6600:{\displaystyle \sum _{i}{p_{i}}<1,}
1794:and the odds of a negative outcome is
9724:
9533:
9360:
9306:
9257:Zenios, S. A.; Ziemba, W. T. (2006),
9185:
9171:
9132:
9075:breaking. Yet the difference between
8973:{\displaystyle {\widehat {\vec {r}}}}
4661:, the total amount of bets placed on
1563:to purchase additional securities on
691:
9002:{\displaystyle {\widehat {\Sigma }}}
6658:
4609:
3004:bets. The resulting wealth will be:
1261:{\displaystyle WLP={\frac {p}{1-p}}}
63:, described the criterion in 1956.
8686:
8665:
8582:
8395:
6150:-th outcome is included in the set
3648:{\displaystyle K=(N-K)(2p+\Delta )}
2570:of the growth rate coefficient and
110:
13:
9715:English translation of 1738 paper.
9633:On Kelly betting: some limitations
9159:10.1002/j.1538-7305.1956.tb03809.x
8990:
8878:
8825:and the unconstrained solution is
8811:{\displaystyle \mathbb {E} \left.}
8463:{\displaystyle \mathbb {E} \left.}
3789:
3744:
3668:
3639:
3603:
3525:
3488:
3436:
3393:
3355:
3313:
3276:
3235:
3185:
3153:
3133:
3107:
2746:
2716:{\displaystyle p=50.425\%,a=b=1\%}
2710:
2689:
2633:
2398:
1302:{\displaystyle WLR={\frac {g}{l}}}
330:
319:
14:
9891:
9835:
9809:Ford, Matthew; Kay, John (2022),
9472:The American Mathematical Monthly
7254:and a bond paying risk-free rate
5262:
1434:, but one has an edge as long as
345:{\displaystyle b=\$ 20/\$ 10=2.0}
8271:and a riskless bond with return
3858:each time. This is true whether
1952:of this curve (as a function of
1713:. The probability of winning is
9639:
9624:
9607:
9527:
9428:
9409:
9383:
9098:Gambling and information theory
7421:{\displaystyle dS_{t}'=fdS_{t}}
6130:may be interpreted as follows:
4074:{\displaystyle 1-f+f(1+b)=1+fb}
3119:{\displaystyle (2p-1+\Delta )W}
2423:Application to the stock market
1762:. The probability of losing is
9484:10.1080/00029890.1992.11995955
9361:Thorp, Edward O. (June 1997).
8959:
8928:
8911:
8900:
8888:
8872:
8869:
8857:
8848:
8788:
8775:
8770:
8751:
8748:
8729:
8634:
8615:
8575:
8563:
8524:
8506:
8497:
8444:
8425:
8388:
8376:
8291:. An investor puts a fraction
8003:
8000:
7994:
7988:
7953:
7941:
7923:
7913:
7895:
7889:
7790:
7778:
7715:
7705:
7647:
7635:
7505:
7495:
7333:
7321:
7076:
7063:
7051:
7038:
6454:
6451:
6438:
6432:
6368:
6352:
6117:
6111:
5990:
5977:
5566:
5560:
5537:
5531:
5450:
5444:
5317:
5311:
5268:{\displaystyle S=\varnothing }
5171:
5152:
4419:{\displaystyle f^{*}=p/a-q/b.}
4297:
4284:
4243:
4227:
4215:
4199:
4053:
4041:
3738:
3642:
3627:
3624:
3612:
3606:
3597:
3585:
3579:
3529:
3519:
3507:
3501:
3492:
3476:
3473:
3461:
3440:
3430:
3418:
3412:
3397:
3381:
3317:
3307:
3295:
3289:
3280:
3264:
3238:
3229:
3217:
3211:
3188:
3173:
3110:
3089:
3047:
3034:
2928:
2916:
2056:
2041:
2026:
2011:
1996:
1990:
1956:), which involves finding the
1923:
1907:
1895:
1879:
1617:
1610:
684:come up. There is no explicit
88:intertemporal portfolio choice
1:
9174:American Mathematical Society
9146:Bell System Technical Journal
9113:
8264:{\displaystyle k=1,\dots ,n,}
7134:Then the expected log return
5514:{\displaystyle S=S\cup \{k\}}
5375:{\displaystyle er_{k}=er_{1}}
4873:, is the dividend rate where
4311:{\displaystyle \log(C_{N})/N}
3197:{\displaystyle (2p+\Delta )W}
280:is the probability of a loss.
9326:Poundstone, William (2005),
9036:
9013:
4365:leads to the desired result
2984:times out of this series of
2639:{\displaystyle 4\sigma =4\%}
2592:probability density function
1471:{\displaystyle WLP*WLR>1}
246:is the probability of a win.
86:use Kelly methods. Also see
7:
9437:"Section 14.7 (Example 2.)"
9332:, New York: Hill and Wang,
9086:
7428:, with associated solution
4634:-th horse wins the race is
2860:times their initial wealth
2559:{\displaystyle \sigma ^{2}}
2317:gives the Kelly criterion:
10:
9896:
9190:. New York: Random House.
9108:Merton's portfolio problem
8009:{\displaystyle \max(G(f))}
5550:according to the formula:
5483:-th outcome into the set:
4896:is the track take or tax,
2770:
2752:{\displaystyle f^{*}=85\%}
1636:) if it were not capped.
1395:{\displaystyle f^{*}>0}
359:bet with no expected gain.
9186:Thorp, Edward O. (1966).
6681:geometric Brownian motion
6473:and the doubling time is
6202:{\displaystyle f_{k}^{o}}
5801:{\displaystyle f_{k}^{o}}
5382:will be considered first.
5216:is non-increasing. Thus
5049:{\displaystyle f_{k}^{o}}
3780:weak law of large numbers
2404:{\displaystyle a=1=100\%}
551:If the edge is negative (
9281:Pabrai, Mohnish (2007),
8203:with stochastic returns
5716:and stop the repetition.
2964:bets like this, and win
1643:
1427:{\displaystyle WLP<1}
578:{\displaystyle b<q/p}
96:e-values and e-processes
9687:The Econometric Society
7842:{\displaystyle \sigma }
7010:{\displaystyle \sigma }
6997:(percentage drift) and
5709:{\displaystyle S^{o}=S}
3992:{\displaystyle K<pN}
3963:{\displaystyle K>pN}
3795:{\displaystyle \Delta }
3361:{\displaystyle \Delta }
3159:{\displaystyle \Delta }
3139:{\displaystyle \Delta }
2937:{\displaystyle 2(1-p)W}
642:{\displaystyle p=18/38}
9597:Mathematical Scientist
9521:10.21314/JOIS.2017.087
9083:should be recognized.
9028:St. Petersburg paradox
9003:
8974:
8935:
8812:
8705:
8684:
8601:
8531:
8473:Expanding this with a
8464:
8414:
8339:
8312:
8285:
8265:
8224:
8197:
8170:
8147:
8120:
8100:
8071:
8010:
7969:
7870:
7843:
7823:
7800:
7556:
7422:
7370:
7369:{\displaystyle dS_{t}}
7340:
7308:
7288:
7268:
7248:
7225:
7155:
7125:
7011:
6991:
6967:
6937:
6823:
6744:
6717:
6697:
6648:
6601:
6550:
6520:
6464:
6283:
6263:
6243:
6223:
6203:
6171:
6144:
6124:
6046:
5955:
5822:
5802:
5770:
5743:
5710:
5677:
5645:
5544:
5515:
5477:
5457:
5376:
5330:
5329:{\displaystyle R(S)=1}
5295:
5269:
5240:
5239:{\displaystyle er_{1}}
5210:
5209:{\displaystyle er_{k}}
5178:
5077:
5050:
5018:
4991:
4964:
4944:
4924:
4890:
4867:
4866:{\displaystyle D=1-tt}
4838:are the pay-off odds.
4832:
4802:
4702:
4675:
4655:
4628:
4600:
4580:
4557:
4533:
4513:
4493:
4466:
4446:
4420:
4359:
4339:
4312:
4262:
4170:
4147:
4124:
4104:
4075:
4013:
3993:
3964:
3935:
3912:
3892:
3872:
3849:
3796:
3769:
3707:
3649:
3557:
3362:
3339:
3248:
3198:
3160:
3140:
3120:
3073:
2998:
2978:
2958:
2938:
2900:
2874:
2854:
2831:, a Kelly bettor bets
2825:
2799:
2753:
2717:
2667:
2640:
2584:
2560:
2533:
2513:
2484:
2405:
2367:
2311:
2281:
2133:
2106:
2086:
2063:
1936:
1857:
1837:
1808:
1788:
1756:
1727:
1707:
1687:
1667:
1630:
1587:
1553:
1527:
1501:
1500:{\displaystyle l\ll 1}
1472:
1428:
1396:
1363:
1337:
1303:
1262:
1209:
1043:
971:
949:
877:
855:
823:
801:
779:
746:
674:
643:
609:
579:
542:
511:
503:
489:
437:
411:
385:
360:
346:
296:
274:
240:
218:
185:
24:
9081:Knightian uncertainty
9004:
8975:
8936:
8823:quadratic programming
8813:
8685:
8664:
8581:
8532:
8465:
8394:
8340:
8338:{\displaystyle S_{k}}
8313:
8311:{\displaystyle u_{k}}
8286:
8266:
8225:
8223:{\displaystyle r_{k}}
8198:
8196:{\displaystyle S_{k}}
8171:
8148:
8146:{\displaystyle R_{s}}
8121:
8101:
8099:{\displaystyle f^{*}}
8072:
8011:
7970:
7871:
7869:{\displaystyle W_{t}}
7844:
7824:
7801:
7557:
7423:
7371:
7341:
7339:{\displaystyle (1-f)}
7309:
7289:
7269:
7249:
7226:
7156:
7154:{\displaystyle R_{s}}
7126:
7012:
6992:
6968:
6966:{\displaystyle W_{t}}
6938:
6824:
6745:
6743:{\displaystyle S_{t}}
6718:
6698:
6649:
6602:
6551:
6549:{\displaystyle p_{k}}
6521:
6465:
6284:
6264:
6244:
6224:
6204:
6172:
6170:{\displaystyle S^{o}}
6145:
6125:
6047:
5956:
5823:
5803:
5771:
5769:{\displaystyle S^{o}}
5744:
5742:{\displaystyle S^{o}}
5711:
5678:
5676:{\displaystyle k=k+1}
5646:
5545:
5516:
5478:
5458:
5377:
5336:. Thus the best bet
5331:
5296:
5270:
5246:will be the best bet.
5241:
5211:
5179:
5078:
5076:{\displaystyle S^{o}}
5051:
5019:
5017:{\displaystyle S^{o}}
4992:
4990:{\displaystyle f_{k}}
4965:
4945:
4925:
4891:
4868:
4833:
4831:{\displaystyle Q_{k}}
4803:
4703:
4701:{\displaystyle B_{k}}
4676:
4656:
4654:{\displaystyle p_{k}}
4629:
4601:
4581:
4558:
4534:
4514:
4494:
4492:{\displaystyle f^{*}}
4467:
4447:
4421:
4360:
4340:
4338:{\displaystyle C_{N}}
4313:
4263:
4171:
4148:
4125:
4110:. Thus at the end of
4105:
4076:
4014:
3994:
3965:
3936:
3913:
3893:
3873:
3850:
3797:
3770:
3708:
3650:
3558:
3363:
3340:
3249:
3199:
3161:
3141:
3121:
3074:
2999:
2979:
2959:
2939:
2901:
2875:
2855:
2826:
2800:
2754:
2718:
2668:
2666:{\displaystyle f^{*}}
2641:
2585:
2561:
2534:
2514:
2512:{\displaystyle f^{*}}
2485:
2406:
2368:
2312:
2310:{\displaystyle f^{*}}
2282:
2134:
2132:{\displaystyle f^{*}}
2107:
2087:
2064:
1937:
1858:
1838:
1809:
1789:
1787:{\displaystyle q=1-p}
1757:
1728:
1708:
1688:
1668:
1631:
1582:
1554:
1528:
1502:
1473:
1429:
1397:
1364:
1338:
1304:
1263:
1210:
1044:
972:
950:
878:
856:
854:{\displaystyle q=1-p}
824:
802:
780:
778:{\displaystyle f^{*}}
747:
675:
673:{\displaystyle -1/19}
644:
610:
580:
543:
541:{\displaystyle b=q/p}
509:
501:
490:
438:
412:
410:{\displaystyle q=0.4}
386:
384:{\displaystyle p=0.6}
357:
347:
297:
275:
273:{\displaystyle q=1-p}
241:
219:
217:{\displaystyle f^{*}}
186:
76:investment management
53:geometric growth rate
22:
9855:Gambling mathematics
9819:10.2139/ssrn.4140625
9417:"Irrational tossers"
9103:Proebsting's paradox
9065:time diversification
8984:
8948:
8832:
8544:
8481:
8352:
8322:
8318:of their capital in
8295:
8275:
8234:
8207:
8180:
8160:
8130:
8119:{\displaystyle \mu }
8110:
8083:
8023:
7982:
7883:
7853:
7833:
7822:{\displaystyle \mu }
7813:
7572:
7435:
7380:
7350:
7318:
7298:
7278:
7258:
7238:
7168:
7138:
7024:
7001:
6990:{\displaystyle \mu }
6981:
6950:
6839:
6757:
6727:
6707:
6687:
6613:
6563:
6533:
6480:
6296:
6273:
6253:
6233:
6213:
6181:
6154:
6134:
6059:
5971:
5835:
5812:
5780:
5753:
5726:
5687:
5655:
5554:
5543:{\displaystyle R(S)}
5525:
5487:
5467:
5392:
5340:
5305:
5279:
5253:
5220:
5190:
5091:
5060:
5028:
5001:
4974:
4954:
4934:
4900:
4877:
4842:
4815:
4715:
4685:
4665:
4638:
4618:
4590:
4567:
4544:
4523:
4503:
4476:
4456:
4436:
4372:
4349:
4322:
4275:
4183:
4157:
4134:
4114:
4103:{\displaystyle 1-fa}
4085:
4023:
4003:
3974:
3945:
3922:
3902:
3882:
3862:
3810:
3786:
3727:
3665:
3573:
3375:
3352:
3261:
3208:
3170:
3150:
3130:
3086:
3011:
2988:
2968:
2948:
2944:. Suppose they make
2910:
2884:
2864:
2853:{\displaystyle 2p-1}
2835:
2809:
2783:
2762:A detailed paper by
2727:
2677:
2650:
2618:
2574:
2543:
2532:{\displaystyle \mu }
2523:
2496:
2439:
2380:
2324:
2294:
2146:
2116:
2096:
2076:
1975:
1945:We want to find the
1870:
1847:
1836:{\displaystyle 1-fa}
1818:
1798:
1766:
1755:{\displaystyle 1+fb}
1737:
1717:
1697:
1677:
1657:
1598:
1537:
1511:
1485:
1438:
1406:
1373:
1347:
1321:
1274:
1225:
1070:
981:
961:
887:
867:
833:
813:
791:
762:
703:
653:
649:). The Kelly bet is
619:
593:
555:
518:
447:
421:
395:
369:
310:
286:
252:
230:
201:
122:
57:John Larry Kelly Jr.
9018:In a 1738 article,
7618:
7605:
7466:
7450:
7398:
6198:
6040:
5964:One may prove that
5797:
5722:If the optimal set
5294:{\displaystyle k=1}
5045:
4318:, and consequently
3718:eventually converge
3658:which implies that
2899:{\displaystyle 2pW}
2824:{\displaystyle b=1}
2798:{\displaystyle b=1}
2594:was assumed here.
1693:and offers odds of
1552:{\displaystyle WLP}
1526:{\displaystyle WLR}
1362:{\displaystyle WLR}
1336:{\displaystyle WLP}
608:{\displaystyle b=1}
436:{\displaystyle b=1}
9875:Portfolio theories
9870:1956 introductions
9860:Information theory
9424:, 1 November 2016.
8999:
8970:
8931:
8808:
8527:
8460:
8335:
8308:
8281:
8261:
8220:
8193:
8176:correlated stocks
8166:
8143:
8116:
8096:
8067:
8006:
7965:
7866:
7839:
7819:
7796:
7606:
7593:
7552:
7454:
7438:
7418:
7386:
7366:
7336:
7304:
7284:
7264:
7244:
7221:
7151:
7121:
7007:
6987:
6963:
6933:
6832:whose solution is
6819:
6740:
6713:
6693:
6644:
6625:
6597:
6575:
6546:
6516:
6460:
6401:
6327:
6279:
6259:
6239:
6219:
6199:
6184:
6167:
6140:
6120:
6042:
6026:
6024:
5951:
5930:
5892:
5818:
5798:
5783:
5766:
5739:
5706:
5683:, Otherwise, set
5673:
5641:
5627:
5593:
5540:
5511:
5473:
5453:
5372:
5326:
5291:
5265:
5236:
5206:
5174:
5073:
5046:
5031:
5014:
4987:
4960:
4940:
4920:
4889:{\displaystyle tt}
4886:
4863:
4828:
4798:
4753:
4698:
4671:
4651:
4624:
4596:
4579:{\displaystyle qN}
4576:
4556:{\displaystyle pN}
4553:
4529:
4509:
4489:
4462:
4442:
4416:
4355:
4345:, with respect to
4335:
4308:
4258:
4169:{\displaystyle qN}
4166:
4146:{\displaystyle pN}
4143:
4120:
4100:
4071:
4009:
3989:
3960:
3934:{\displaystyle pN}
3931:
3908:
3888:
3868:
3845:
3792:
3765:
3748:
3703:
3645:
3553:
3358:
3335:
3244:
3194:
3156:
3136:
3126:for some value of
3116:
3069:
2994:
2974:
2954:
2934:
2896:
2870:
2850:
2821:
2795:
2749:
2713:
2663:
2636:
2580:
2556:
2529:
2509:
2480:
2401:
2363:
2307:
2277:
2129:
2102:
2082:
2059:
1932:
1853:
1833:
1804:
1784:
1752:
1723:
1703:
1683:
1663:
1626:
1549:
1523:
1497:
1468:
1424:
1392:
1359:
1333:
1299:
1258:
1205:
1039:
967:
945:
873:
851:
819:
797:
775:
742:
692:Investment formula
670:
639:
605:
575:
538:
512:
504:
485:
433:
407:
381:
361:
342:
292:
270:
236:
214:
181:
59:, a researcher at
29:probability theory
25:
9850:Optimal decisions
9780:10.1063/1.4940236
9666:Bernoulli, Daniel
9450:978-0-521-88068-8
9294:978-0-470-04389-9
9268:978-0-444-50875-1
9261:, North Holland,
9069:Paul A. Samuelson
8996:
8967:
8962:
8919:
8914:
8884:
8851:
8798:
8662:
8649:
8500:
8284:{\displaystyle r}
8169:{\displaystyle n}
8062:
7958:
7936:
7728:
7619:
7518:
7307:{\displaystyle S}
7287:{\displaystyle f}
7267:{\displaystyle r}
7247:{\displaystyle S}
7210:
7108:
6902:
6716:{\displaystyle t}
6696:{\displaystyle S}
6665:Taylor polynomial
6663:The second-order
6659:Stock investments
6616:
6566:
6511:
6386:
6312:
6282:{\displaystyle k}
6262:{\displaystyle k}
6242:{\displaystyle k}
6222:{\displaystyle k}
6143:{\displaystyle k}
6093:
6002:
5946:
5915:
5877:
5821:{\displaystyle k}
5639:
5612:
5578:
5476:{\displaystyle k}
5426:
5275:(the empty set),
5137:
4963:{\displaystyle k}
4943:{\displaystyle k}
4918:
4793:
4765:
4744:
4674:{\displaystyle k}
4627:{\displaystyle k}
4610:Multiple outcomes
4599:{\displaystyle N}
4532:{\displaystyle b}
4512:{\displaystyle p}
4465:{\displaystyle p}
4445:{\displaystyle p}
4358:{\displaystyle f}
4123:{\displaystyle N}
4012:{\displaystyle f}
3911:{\displaystyle K}
3891:{\displaystyle N}
3871:{\displaystyle N}
3829:
3778:according to the
3757:
3730:
3690:
3247:{\displaystyle W}
2997:{\displaystyle N}
2977:{\displaystyle K}
2957:{\displaystyle N}
2873:{\displaystyle W}
2583:{\displaystyle r}
2478:
2361:
2348:
2269:
2230:
2171:
2105:{\displaystyle f}
2085:{\displaystyle E}
1856:{\displaystyle r}
1807:{\displaystyle a}
1726:{\displaystyle p}
1706:{\displaystyle b}
1686:{\displaystyle p}
1666:{\displaystyle f}
1297:
1256:
1198:
1180:
1151:
1133:
1123:
1094:
1031:
1010:
1009:
1004:
996:
970:{\displaystyle l}
937:
916:
915:
910:
902:
876:{\displaystyle g}
822:{\displaystyle q}
800:{\displaystyle p}
740:
727:
587:American roulette
477:
295:{\displaystyle b}
239:{\displaystyle p}
179:
152:
92:statistical power
9887:
9830:
9829:
9806:
9800:
9799:
9782:
9772:
9752:
9746:
9745:
9738:
9732:
9731:
9728:Wilmott Magazine
9722:
9716:
9714:
9674:
9662:
9656:
9655:
9643:
9637:
9636:
9628:
9622:
9621:
9611:
9605:
9604:
9588:
9575:
9574:
9531:
9525:
9524:
9514:
9494:
9488:
9487:
9469:
9460:
9454:
9453:
9432:
9426:
9413:
9407:
9399:
9387:
9381:
9380:
9378:
9377:
9367:
9358:
9343:
9342:
9323:
9314:
9313:
9310:Wilmott Magazine
9304:
9298:
9297:
9278:
9272:
9271:
9254:
9248:
9246:
9244:
9238:, archived from
9228:, Random House,
9227:
9216:
9210:
9209:
9183:
9177:
9176:
9169:
9163:
9162:
9142:
9130:
9044:expected utility
9020:Daniel Bernoulli
9008:
9006:
9005:
9000:
8998:
8997:
8989:
8979:
8977:
8976:
8971:
8969:
8968:
8963:
8955:
8953:
8940:
8938:
8937:
8932:
8921:
8920:
8915:
8907:
8905:
8899:
8898:
8886:
8885:
8877:
8853:
8852:
8847:
8846:
8837:
8817:
8815:
8814:
8809:
8804:
8800:
8799:
8797:
8796:
8795:
8773:
8763:
8762:
8741:
8740:
8727:
8725:
8724:
8715:
8714:
8704:
8699:
8683:
8678:
8663:
8655:
8650:
8648:
8637:
8627:
8626:
8614:
8613:
8603:
8600:
8595:
8551:
8536:
8534:
8533:
8528:
8502:
8501:
8496:
8495:
8486:
8469:
8467:
8466:
8461:
8456:
8452:
8451:
8447:
8437:
8436:
8424:
8423:
8413:
8408:
8359:
8344:
8342:
8341:
8336:
8334:
8333:
8317:
8315:
8314:
8309:
8307:
8306:
8290:
8288:
8287:
8282:
8270:
8268:
8267:
8262:
8229:
8227:
8226:
8221:
8219:
8218:
8202:
8200:
8199:
8194:
8192:
8191:
8175:
8173:
8172:
8167:
8152:
8150:
8149:
8144:
8142:
8141:
8125:
8123:
8122:
8117:
8105:
8103:
8102:
8097:
8095:
8094:
8076:
8074:
8073:
8068:
8063:
8061:
8060:
8051:
8040:
8035:
8034:
8015:
8013:
8012:
8007:
7974:
7972:
7971:
7966:
7956:
7937:
7932:
7931:
7930:
7911:
7875:
7873:
7872:
7867:
7865:
7864:
7848:
7846:
7845:
7840:
7828:
7826:
7825:
7820:
7805:
7803:
7802:
7797:
7774:
7773:
7769:
7765:
7758:
7754:
7753:
7752:
7734:
7730:
7729:
7724:
7723:
7722:
7703:
7666:
7658:
7657:
7653:
7631:
7627:
7620:
7614:
7601:
7592:
7579:
7561:
7559:
7558:
7553:
7551:
7547:
7546:
7545:
7524:
7520:
7519:
7514:
7513:
7512:
7493:
7462:
7446:
7427:
7425:
7424:
7419:
7417:
7416:
7394:
7375:
7373:
7372:
7367:
7365:
7364:
7345:
7343:
7342:
7337:
7313:
7311:
7310:
7305:
7293:
7291:
7290:
7285:
7274:, with fraction
7273:
7271:
7270:
7265:
7253:
7251:
7250:
7245:
7230:
7228:
7227:
7222:
7217:
7213:
7211:
7206:
7205:
7196:
7180:
7179:
7160:
7158:
7157:
7152:
7150:
7149:
7130:
7128:
7127:
7122:
7114:
7110:
7109:
7104:
7103:
7094:
7075:
7074:
7050:
7049:
7031:
7016:
7014:
7013:
7008:
6996:
6994:
6993:
6988:
6972:
6970:
6969:
6964:
6962:
6961:
6942:
6940:
6939:
6934:
6932:
6928:
6927:
6926:
6908:
6904:
6903:
6898:
6897:
6888:
6864:
6863:
6851:
6850:
6828:
6826:
6825:
6820:
6818:
6817:
6787:
6786:
6777:
6772:
6771:
6749:
6747:
6746:
6741:
6739:
6738:
6722:
6720:
6719:
6714:
6702:
6700:
6699:
6694:
6653:
6651:
6650:
6645:
6637:
6636:
6635:
6624:
6606:
6604:
6603:
6598:
6587:
6586:
6585:
6574:
6555:
6553:
6552:
6547:
6545:
6544:
6525:
6523:
6522:
6517:
6512:
6510:
6509:
6497:
6492:
6491:
6469:
6467:
6466:
6461:
6450:
6449:
6428:
6427:
6418:
6414:
6413:
6412:
6411:
6400:
6371:
6367:
6366:
6348:
6347:
6338:
6337:
6326:
6308:
6307:
6288:
6286:
6285:
6280:
6268:
6266:
6265:
6260:
6248:
6246:
6245:
6240:
6228:
6226:
6225:
6220:
6208:
6206:
6205:
6200:
6197:
6192:
6176:
6174:
6173:
6168:
6166:
6165:
6149:
6147:
6146:
6141:
6129:
6127:
6126:
6121:
6104:
6103:
6094:
6092:
6091:
6079:
6074:
6073:
6051:
6049:
6048:
6043:
6041:
6039:
6034:
6023:
6022:
6021:
5989:
5988:
5960:
5958:
5957:
5952:
5947:
5945:
5941:
5940:
5939:
5929:
5903:
5902:
5901:
5891:
5875:
5873:
5872:
5860:
5859:
5847:
5846:
5827:
5825:
5824:
5819:
5807:
5805:
5804:
5799:
5796:
5791:
5775:
5773:
5772:
5767:
5765:
5764:
5748:
5746:
5745:
5740:
5738:
5737:
5715:
5713:
5712:
5707:
5699:
5698:
5682:
5680:
5679:
5674:
5650:
5648:
5647:
5642:
5640:
5638:
5637:
5636:
5626:
5604:
5603:
5602:
5592:
5573:
5549:
5547:
5546:
5541:
5520:
5518:
5517:
5512:
5482:
5480:
5479:
5474:
5462:
5460:
5459:
5454:
5437:
5436:
5427:
5425:
5424:
5412:
5407:
5406:
5381:
5379:
5378:
5373:
5371:
5370:
5355:
5354:
5335:
5333:
5332:
5327:
5300:
5298:
5297:
5292:
5274:
5272:
5271:
5266:
5245:
5243:
5242:
5237:
5235:
5234:
5215:
5213:
5212:
5207:
5205:
5204:
5183:
5181:
5180:
5175:
5164:
5163:
5151:
5150:
5138:
5136:
5135:
5126:
5125:
5124:
5111:
5106:
5105:
5082:
5080:
5079:
5074:
5072:
5071:
5055:
5053:
5052:
5047:
5044:
5039:
5023:
5021:
5020:
5015:
5013:
5012:
4996:
4994:
4993:
4988:
4986:
4985:
4969:
4967:
4966:
4961:
4949:
4947:
4946:
4941:
4929:
4927:
4926:
4921:
4919:
4917:
4916:
4904:
4895:
4893:
4892:
4887:
4872:
4870:
4869:
4864:
4837:
4835:
4834:
4829:
4827:
4826:
4807:
4805:
4804:
4799:
4794:
4792:
4791:
4790:
4771:
4766:
4764:
4763:
4762:
4752:
4742:
4741:
4732:
4727:
4726:
4707:
4705:
4704:
4699:
4697:
4696:
4680:
4678:
4677:
4672:
4660:
4658:
4657:
4652:
4650:
4649:
4633:
4631:
4630:
4625:
4605:
4603:
4602:
4597:
4585:
4583:
4582:
4577:
4562:
4560:
4559:
4554:
4538:
4536:
4535:
4530:
4518:
4516:
4515:
4510:
4498:
4496:
4495:
4490:
4488:
4487:
4471:
4469:
4468:
4463:
4451:
4449:
4448:
4443:
4425:
4423:
4422:
4417:
4409:
4395:
4384:
4383:
4364:
4362:
4361:
4356:
4344:
4342:
4341:
4336:
4334:
4333:
4317:
4315:
4314:
4309:
4304:
4296:
4295:
4267:
4265:
4264:
4259:
4254:
4253:
4226:
4225:
4195:
4194:
4175:
4173:
4172:
4167:
4152:
4150:
4149:
4144:
4129:
4127:
4126:
4121:
4109:
4107:
4106:
4101:
4080:
4078:
4077:
4072:
4018:
4016:
4015:
4010:
3998:
3996:
3995:
3990:
3969:
3967:
3966:
3961:
3940:
3938:
3937:
3932:
3917:
3915:
3914:
3909:
3897:
3895:
3894:
3889:
3877:
3875:
3874:
3869:
3854:
3852:
3851:
3846:
3841:
3837:
3830:
3822:
3801:
3799:
3798:
3793:
3774:
3772:
3771:
3766:
3758:
3750:
3747:
3712:
3710:
3709:
3704:
3702:
3698:
3691:
3683:
3654:
3652:
3651:
3646:
3562:
3560:
3559:
3554:
3549:
3548:
3500:
3499:
3454:
3453:
3411:
3410:
3367:
3365:
3364:
3359:
3344:
3342:
3341:
3336:
3331:
3330:
3288:
3287:
3253:
3251:
3250:
3245:
3204:after a win and
3203:
3201:
3200:
3195:
3165:
3163:
3162:
3157:
3145:
3143:
3142:
3137:
3125:
3123:
3122:
3117:
3078:
3076:
3075:
3070:
3061:
3060:
3033:
3032:
3023:
3022:
3003:
3001:
3000:
2995:
2983:
2981:
2980:
2975:
2963:
2961:
2960:
2955:
2943:
2941:
2940:
2935:
2905:
2903:
2902:
2897:
2879:
2877:
2876:
2871:
2859:
2857:
2856:
2851:
2830:
2828:
2827:
2822:
2804:
2802:
2801:
2796:
2758:
2756:
2755:
2750:
2739:
2738:
2722:
2720:
2719:
2714:
2672:
2670:
2669:
2664:
2662:
2661:
2645:
2643:
2642:
2637:
2589:
2587:
2586:
2581:
2565:
2563:
2562:
2557:
2555:
2554:
2538:
2536:
2535:
2530:
2518:
2516:
2515:
2510:
2508:
2507:
2489:
2487:
2486:
2481:
2479:
2477:
2476:
2467:
2456:
2451:
2450:
2410:
2408:
2407:
2402:
2372:
2370:
2369:
2364:
2362:
2354:
2349:
2341:
2336:
2335:
2316:
2314:
2313:
2308:
2306:
2305:
2286:
2284:
2283:
2278:
2270:
2268:
2264:
2263:
2247:
2236:
2231:
2229:
2225:
2224:
2208:
2200:
2195:
2194:
2193:
2192:
2176:
2172:
2170:
2162:
2154:
2138:
2136:
2135:
2130:
2128:
2127:
2111:
2109:
2108:
2103:
2091:
2089:
2088:
2083:
2068:
2066:
2065:
2060:
1941:
1939:
1938:
1933:
1931:
1930:
1903:
1902:
1862:
1860:
1859:
1854:
1842:
1840:
1839:
1834:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1761:
1759:
1758:
1753:
1732:
1730:
1729:
1724:
1712:
1710:
1709:
1704:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1635:
1633:
1632:
1627:
1625:
1624:
1558:
1556:
1555:
1550:
1532:
1530:
1529:
1524:
1506:
1504:
1503:
1498:
1477:
1475:
1474:
1469:
1433:
1431:
1430:
1425:
1401:
1399:
1398:
1393:
1385:
1384:
1368:
1366:
1365:
1360:
1342:
1340:
1339:
1334:
1308:
1306:
1305:
1300:
1298:
1290:
1267:
1265:
1264:
1259:
1257:
1255:
1241:
1214:
1212:
1211:
1206:
1204:
1200:
1199:
1197:
1183:
1181:
1179:
1165:
1152:
1144:
1139:
1135:
1134:
1126:
1124:
1119:
1108:
1095:
1087:
1082:
1081:
1048:
1046:
1045:
1040:
1032:
1027:
1016:
1011:
1007:
1006:
1005:
1002:
997:
994:
991:
976:
974:
973:
968:
954:
952:
951:
946:
938:
933:
922:
917:
913:
912:
911:
908:
903:
900:
897:
882:
880:
879:
874:
860:
858:
857:
852:
828:
826:
825:
820:
806:
804:
803:
798:
784:
782:
781:
776:
774:
773:
751:
749:
748:
743:
741:
733:
728:
720:
715:
714:
679:
677:
676:
671:
666:
648:
646:
645:
640:
635:
614:
612:
611:
606:
584:
582:
581:
576:
571:
547:
545:
544:
539:
534:
494:
492:
491:
486:
478:
470:
459:
458:
442:
440:
439:
434:
416:
414:
413:
408:
390:
388:
387:
382:
351:
349:
348:
343:
329:
301:
299:
298:
293:
279:
277:
276:
271:
245:
243:
242:
237:
223:
221:
220:
215:
213:
212:
190:
188:
187:
182:
180:
175:
164:
153:
145:
134:
133:
111:Gambling formula
9895:
9894:
9890:
9889:
9888:
9886:
9885:
9884:
9840:
9839:
9838:
9833:
9807:
9803:
9753:
9749:
9739:
9735:
9723:
9719:
9695:10.2307/1909829
9672:
9663:
9659:
9644:
9640:
9629:
9625:
9612:
9608:
9589:
9578:
9555:10.2307/1402118
9532:
9528:
9495:
9491:
9478:(10): 922–931.
9467:
9461:
9457:
9451:
9433:
9429:
9414:
9410:
9388:
9384:
9375:
9373:
9365:
9359:
9346:
9340:
9324:
9317:
9305:
9301:
9295:
9279:
9275:
9269:
9255:
9251:
9242:
9236:
9225:
9217:
9213:
9198:
9184:
9180:
9170:
9166:
9140:
9131:
9120:
9116:
9089:
9042:alternative is
9039:
9016:
8988:
8987:
8985:
8982:
8981:
8954:
8952:
8951:
8949:
8946:
8945:
8906:
8904:
8903:
8891:
8887:
8876:
8875:
8842:
8838:
8836:
8835:
8833:
8830:
8829:
8791:
8787:
8774:
8758:
8754:
8736:
8732:
8728:
8726:
8720:
8716:
8710:
8706:
8700:
8689:
8679:
8668:
8654:
8638:
8622:
8618:
8609:
8605:
8604:
8602:
8596:
8585:
8556:
8552:
8547:
8545:
8542:
8541:
8491:
8487:
8485:
8484:
8482:
8479:
8478:
8432:
8428:
8419:
8415:
8409:
8398:
8375:
8371:
8364:
8360:
8355:
8353:
8350:
8349:
8329:
8325:
8323:
8320:
8319:
8302:
8298:
8296:
8293:
8292:
8276:
8273:
8272:
8235:
8232:
8231:
8214:
8210:
8208:
8205:
8204:
8187:
8183:
8181:
8178:
8177:
8161:
8158:
8157:
8137:
8133:
8131:
8128:
8127:
8111:
8108:
8107:
8090:
8086:
8084:
8081:
8080:
8056:
8052:
8041:
8039:
8030:
8026:
8024:
8021:
8020:
7983:
7980:
7979:
7926:
7922:
7912:
7910:
7884:
7881:
7880:
7860:
7856:
7854:
7851:
7850:
7834:
7831:
7830:
7814:
7811:
7810:
7748:
7744:
7718:
7714:
7704:
7702:
7692:
7688:
7687:
7683:
7676:
7672:
7668:
7667:
7662:
7610:
7597:
7591:
7590:
7586:
7585:
7581:
7580:
7575:
7573:
7570:
7569:
7541:
7537:
7508:
7504:
7494:
7492:
7482:
7478:
7477:
7473:
7458:
7442:
7436:
7433:
7432:
7412:
7408:
7390:
7381:
7378:
7377:
7360:
7356:
7351:
7348:
7347:
7319:
7316:
7315:
7299:
7296:
7295:
7279:
7276:
7275:
7259:
7256:
7255:
7239:
7236:
7235:
7201:
7197:
7195:
7188:
7184:
7175:
7171:
7169:
7166:
7165:
7145:
7141:
7139:
7136:
7135:
7099:
7095:
7093:
7086:
7082:
7070:
7066:
7045:
7041:
7027:
7025:
7022:
7021:
7002:
6999:
6998:
6982:
6979:
6978:
6957:
6953:
6951:
6948:
6947:
6922:
6918:
6893:
6889:
6887:
6880:
6876:
6875:
6871:
6859:
6855:
6846:
6842:
6840:
6837:
6836:
6813:
6809:
6782:
6778:
6773:
6767:
6763:
6758:
6755:
6754:
6734:
6730:
6728:
6725:
6724:
6708:
6705:
6704:
6688:
6685:
6684:
6661:
6631:
6627:
6626:
6620:
6614:
6611:
6610:
6581:
6577:
6576:
6570:
6564:
6561:
6560:
6540:
6536:
6534:
6531:
6530:
6505:
6501:
6496:
6487:
6483:
6481:
6478:
6477:
6445:
6441:
6423:
6419:
6407:
6403:
6402:
6390:
6379:
6375:
6362:
6358:
6343:
6339:
6333:
6329:
6328:
6316:
6303:
6299:
6297:
6294:
6293:
6274:
6271:
6270:
6254:
6251:
6250:
6234:
6231:
6230:
6214:
6211:
6210:
6193:
6188:
6182:
6179:
6178:
6161:
6157:
6155:
6152:
6151:
6135:
6132:
6131:
6099:
6095:
6087:
6083:
6078:
6069:
6065:
6060:
6057:
6056:
6035:
6030:
6025:
6017:
6013:
6006:
5984:
5980:
5972:
5969:
5968:
5935:
5931:
5919:
5908:
5904:
5897:
5893:
5881:
5876:
5874:
5868:
5864:
5855:
5851:
5842:
5838:
5836:
5833:
5832:
5813:
5810:
5809:
5792:
5787:
5781:
5778:
5777:
5760:
5756:
5754:
5751:
5750:
5733:
5729:
5727:
5724:
5723:
5694:
5690:
5688:
5685:
5684:
5656:
5653:
5652:
5632:
5628:
5616:
5605:
5598:
5594:
5582:
5574:
5572:
5555:
5552:
5551:
5526:
5523:
5522:
5488:
5485:
5484:
5468:
5465:
5464:
5432:
5428:
5420:
5416:
5411:
5402:
5398:
5393:
5390:
5389:
5366:
5362:
5350:
5346:
5341:
5338:
5337:
5306:
5303:
5302:
5280:
5277:
5276:
5254:
5251:
5250:
5230:
5226:
5221:
5218:
5217:
5200:
5196:
5191:
5188:
5187:
5159:
5155:
5146:
5142:
5131:
5127:
5120:
5116:
5112:
5110:
5101:
5097:
5092:
5089:
5088:
5067:
5063:
5061:
5058:
5057:
5040:
5035:
5029:
5026:
5025:
5008:
5004:
5002:
4999:
4998:
4981:
4977:
4975:
4972:
4971:
4955:
4952:
4951:
4935:
4932:
4931:
4912:
4908:
4903:
4901:
4898:
4897:
4878:
4875:
4874:
4843:
4840:
4839:
4822:
4818:
4816:
4813:
4812:
4786:
4782:
4775:
4770:
4758:
4754:
4748:
4743:
4737:
4733:
4731:
4722:
4718:
4716:
4713:
4712:
4692:
4688:
4686:
4683:
4682:
4666:
4663:
4662:
4645:
4641:
4639:
4636:
4635:
4619:
4616:
4615:
4612:
4591:
4588:
4587:
4568:
4565:
4564:
4545:
4542:
4541:
4524:
4521:
4520:
4504:
4501:
4500:
4483:
4479:
4477:
4474:
4473:
4457:
4454:
4453:
4437:
4434:
4433:
4430:Edward O. Thorp
4405:
4391:
4379:
4375:
4373:
4370:
4369:
4350:
4347:
4346:
4329:
4325:
4323:
4320:
4319:
4300:
4291:
4287:
4276:
4273:
4272:
4246:
4242:
4218:
4214:
4190:
4186:
4184:
4181:
4180:
4158:
4155:
4154:
4135:
4132:
4131:
4115:
4112:
4111:
4086:
4083:
4082:
4024:
4021:
4020:
4004:
4001:
4000:
3975:
3972:
3971:
3946:
3943:
3942:
3923:
3920:
3919:
3903:
3900:
3899:
3883:
3880:
3879:
3863:
3860:
3859:
3821:
3817:
3813:
3811:
3808:
3807:
3787:
3784:
3783:
3749:
3734:
3728:
3725:
3724:
3682:
3681:
3677:
3666:
3663:
3662:
3574:
3571:
3570:
3532:
3528:
3495:
3491:
3443:
3439:
3400:
3396:
3376:
3373:
3372:
3353:
3350:
3349:
3320:
3316:
3283:
3279:
3262:
3259:
3258:
3209:
3206:
3205:
3171:
3168:
3167:
3151:
3148:
3147:
3131:
3128:
3127:
3087:
3084:
3083:
3050:
3046:
3028:
3024:
3018:
3014:
3012:
3009:
3008:
2989:
2986:
2985:
2969:
2966:
2965:
2949:
2946:
2945:
2911:
2908:
2907:
2885:
2882:
2881:
2865:
2862:
2861:
2836:
2833:
2832:
2810:
2807:
2806:
2784:
2781:
2780:
2773:
2764:Edward O. Thorp
2734:
2730:
2728:
2725:
2724:
2678:
2675:
2674:
2657:
2653:
2651:
2648:
2647:
2619:
2616:
2615:
2575:
2572:
2571:
2550:
2546:
2544:
2541:
2540:
2524:
2521:
2520:
2503:
2499:
2497:
2494:
2493:
2472:
2468:
2457:
2455:
2446:
2442:
2440:
2437:
2436:
2429:growth optimal.
2425:
2417:
2381:
2378:
2377:
2353:
2340:
2331:
2327:
2325:
2322:
2321:
2301:
2297:
2295:
2292:
2291:
2259:
2255:
2248:
2237:
2235:
2220:
2216:
2209:
2201:
2199:
2188:
2184:
2177:
2163:
2155:
2153:
2150:
2149:
2147:
2144:
2143:
2123:
2119:
2117:
2114:
2113:
2097:
2094:
2093:
2077:
2074:
2073:
1976:
1973:
1972:
1926:
1922:
1898:
1894:
1871:
1868:
1867:
1848:
1845:
1844:
1819:
1816:
1815:
1799:
1796:
1795:
1767:
1764:
1763:
1738:
1735:
1734:
1718:
1715:
1714:
1698:
1695:
1694:
1678:
1675:
1674:
1658:
1655:
1654:
1646:
1620:
1616:
1599:
1596:
1595:
1577:even-money bets
1573:
1538:
1535:
1534:
1512:
1509:
1508:
1486:
1483:
1482:
1439:
1436:
1435:
1407:
1404:
1403:
1380:
1376:
1374:
1371:
1370:
1348:
1345:
1344:
1322:
1319:
1318:
1289:
1275:
1272:
1271:
1245:
1240:
1226:
1223:
1222:
1187:
1182:
1169:
1164:
1157:
1153:
1143:
1125:
1109:
1107:
1100:
1096:
1086:
1077:
1073:
1071:
1068:
1067:
1017:
1015:
1001:
993:
992:
990:
982:
979:
978:
962:
959:
958:
923:
921:
907:
899:
898:
896:
888:
885:
884:
868:
865:
864:
834:
831:
830:
814:
811:
810:
792:
789:
788:
769:
765:
763:
760:
759:
732:
719:
710:
706:
704:
701:
700:
694:
662:
654:
651:
650:
631:
620:
617:
616:
594:
591:
590:
567:
556:
553:
552:
530:
519:
516:
515:
469:
454:
450:
448:
445:
444:
422:
419:
418:
396:
393:
392:
370:
367:
366:
325:
311:
308:
307:
287:
284:
283:
253:
250:
249:
231:
228:
227:
208:
204:
202:
199:
198:
165:
163:
144:
129:
125:
123:
120:
119:
113:
104:
72:diversification
33:Kelly criterion
17:
12:
11:
5:
9893:
9883:
9882:
9877:
9872:
9867:
9862:
9857:
9852:
9837:
9836:External links
9834:
9832:
9831:
9801:
9747:
9733:
9717:
9657:
9638:
9623:
9606:
9576:
9526:
9489:
9455:
9449:
9427:
9415:"Buttonwood",
9408:
9382:
9344:
9338:
9315:
9299:
9293:
9273:
9267:
9249:
9234:
9211:
9196:
9178:
9164:
9153:(4): 917–926.
9117:
9115:
9112:
9111:
9110:
9105:
9100:
9095:
9088:
9085:
9038:
9035:
9024:geometric mean
9015:
9012:
8995:
8992:
8966:
8961:
8958:
8942:
8941:
8930:
8927:
8924:
8918:
8913:
8910:
8902:
8897:
8894:
8890:
8883:
8880:
8874:
8871:
8868:
8865:
8862:
8859:
8856:
8850:
8845:
8841:
8819:
8818:
8807:
8803:
8794:
8790:
8786:
8783:
8780:
8777:
8772:
8769:
8766:
8761:
8757:
8753:
8750:
8747:
8744:
8739:
8735:
8731:
8723:
8719:
8713:
8709:
8703:
8698:
8695:
8692:
8688:
8682:
8677:
8674:
8671:
8667:
8661:
8658:
8653:
8647:
8644:
8641:
8636:
8633:
8630:
8625:
8621:
8617:
8612:
8608:
8599:
8594:
8591:
8588:
8584:
8580:
8577:
8574:
8571:
8568:
8565:
8562:
8559:
8555:
8550:
8526:
8523:
8520:
8517:
8514:
8511:
8508:
8505:
8499:
8494:
8490:
8471:
8470:
8459:
8455:
8450:
8446:
8443:
8440:
8435:
8431:
8427:
8422:
8418:
8412:
8407:
8404:
8401:
8397:
8393:
8390:
8387:
8384:
8381:
8378:
8374:
8370:
8367:
8363:
8358:
8332:
8328:
8305:
8301:
8280:
8260:
8257:
8254:
8251:
8248:
8245:
8242:
8239:
8217:
8213:
8190:
8186:
8165:
8140:
8136:
8115:
8093:
8089:
8078:
8077:
8066:
8059:
8055:
8050:
8047:
8044:
8038:
8033:
8029:
8005:
8002:
7999:
7996:
7993:
7990:
7987:
7976:
7975:
7964:
7961:
7955:
7952:
7949:
7946:
7943:
7940:
7935:
7929:
7925:
7921:
7918:
7915:
7909:
7906:
7903:
7900:
7897:
7894:
7891:
7888:
7863:
7859:
7838:
7818:
7807:
7806:
7795:
7792:
7789:
7786:
7783:
7780:
7777:
7772:
7768:
7764:
7761:
7757:
7751:
7747:
7743:
7740:
7737:
7733:
7727:
7721:
7717:
7713:
7710:
7707:
7701:
7698:
7695:
7691:
7686:
7682:
7679:
7675:
7671:
7665:
7661:
7656:
7652:
7649:
7646:
7643:
7640:
7637:
7634:
7630:
7626:
7623:
7617:
7613:
7609:
7604:
7600:
7596:
7589:
7584:
7578:
7563:
7562:
7550:
7544:
7540:
7536:
7533:
7530:
7527:
7523:
7517:
7511:
7507:
7503:
7500:
7497:
7491:
7488:
7485:
7481:
7476:
7472:
7469:
7465:
7461:
7457:
7453:
7449:
7445:
7441:
7415:
7411:
7407:
7404:
7401:
7397:
7393:
7389:
7385:
7363:
7359:
7355:
7335:
7332:
7329:
7326:
7323:
7303:
7283:
7263:
7243:
7232:
7231:
7220:
7216:
7209:
7204:
7200:
7194:
7191:
7187:
7183:
7178:
7174:
7148:
7144:
7132:
7131:
7120:
7117:
7113:
7107:
7102:
7098:
7092:
7089:
7085:
7081:
7078:
7073:
7069:
7065:
7062:
7059:
7056:
7053:
7048:
7044:
7040:
7037:
7034:
7030:
7006:
6986:
6975:Wiener process
6960:
6956:
6944:
6943:
6931:
6925:
6921:
6917:
6914:
6911:
6907:
6901:
6896:
6892:
6886:
6883:
6879:
6874:
6870:
6867:
6862:
6858:
6854:
6849:
6845:
6830:
6829:
6816:
6812:
6808:
6805:
6802:
6799:
6796:
6793:
6790:
6785:
6781:
6776:
6770:
6766:
6762:
6737:
6733:
6712:
6692:
6670:expected value
6660:
6657:
6656:
6655:
6643:
6640:
6634:
6630:
6623:
6619:
6608:
6596:
6593:
6590:
6584:
6580:
6573:
6569:
6543:
6539:
6527:
6526:
6515:
6508:
6504:
6500:
6495:
6490:
6486:
6471:
6470:
6459:
6456:
6453:
6448:
6444:
6440:
6437:
6434:
6431:
6426:
6422:
6417:
6410:
6406:
6399:
6396:
6393:
6389:
6385:
6382:
6378:
6374:
6370:
6365:
6361:
6357:
6354:
6351:
6346:
6342:
6336:
6332:
6325:
6322:
6319:
6315:
6311:
6306:
6302:
6278:
6258:
6238:
6218:
6196:
6191:
6187:
6164:
6160:
6139:
6119:
6116:
6113:
6110:
6107:
6102:
6098:
6090:
6086:
6082:
6077:
6072:
6068:
6064:
6053:
6052:
6038:
6033:
6029:
6020:
6016:
6012:
6009:
6005:
6001:
5998:
5995:
5992:
5987:
5983:
5979:
5976:
5962:
5961:
5950:
5944:
5938:
5934:
5928:
5925:
5922:
5918:
5914:
5911:
5907:
5900:
5896:
5890:
5887:
5884:
5880:
5871:
5867:
5863:
5858:
5854:
5850:
5845:
5841:
5817:
5795:
5790:
5786:
5763:
5759:
5736:
5732:
5720:
5719:
5718:
5717:
5705:
5702:
5697:
5693:
5672:
5669:
5666:
5663:
5660:
5635:
5631:
5625:
5622:
5619:
5615:
5611:
5608:
5601:
5597:
5591:
5588:
5585:
5581:
5577:
5571:
5568:
5565:
5562:
5559:
5539:
5536:
5533:
5530:
5521:, recalculate
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5472:
5452:
5449:
5446:
5443:
5440:
5435:
5431:
5423:
5419:
5415:
5410:
5405:
5401:
5397:
5383:
5369:
5365:
5361:
5358:
5353:
5349:
5345:
5325:
5322:
5319:
5316:
5313:
5310:
5290:
5287:
5284:
5264:
5261:
5258:
5247:
5233:
5229:
5225:
5203:
5199:
5195:
5184:
5173:
5170:
5167:
5162:
5158:
5154:
5149:
5145:
5141:
5134:
5130:
5123:
5119:
5115:
5109:
5104:
5100:
5096:
5070:
5066:
5043:
5038:
5034:
5011:
5007:
4984:
4980:
4959:
4939:
4915:
4911:
4907:
4885:
4882:
4862:
4859:
4856:
4853:
4850:
4847:
4825:
4821:
4809:
4808:
4797:
4789:
4785:
4781:
4778:
4774:
4769:
4761:
4757:
4751:
4747:
4740:
4736:
4730:
4725:
4721:
4695:
4691:
4670:
4648:
4644:
4623:
4611:
4608:
4595:
4575:
4572:
4563:successes and
4552:
4549:
4528:
4508:
4486:
4482:
4461:
4441:
4427:
4426:
4415:
4412:
4408:
4404:
4401:
4398:
4394:
4390:
4387:
4382:
4378:
4354:
4332:
4328:
4307:
4303:
4299:
4294:
4290:
4286:
4283:
4280:
4269:
4268:
4257:
4252:
4249:
4245:
4241:
4238:
4235:
4232:
4229:
4224:
4221:
4217:
4213:
4210:
4207:
4204:
4201:
4198:
4193:
4189:
4165:
4162:
4153:successes and
4142:
4139:
4119:
4099:
4096:
4093:
4090:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4008:
3988:
3985:
3982:
3979:
3959:
3956:
3953:
3950:
3930:
3927:
3918:will approach
3907:
3887:
3867:
3856:
3855:
3844:
3840:
3836:
3833:
3828:
3825:
3820:
3816:
3791:
3776:
3775:
3764:
3761:
3756:
3753:
3746:
3743:
3740:
3737:
3733:
3714:
3713:
3701:
3697:
3694:
3689:
3686:
3680:
3676:
3673:
3670:
3656:
3655:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3564:
3563:
3552:
3547:
3544:
3541:
3538:
3535:
3531:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3498:
3494:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3452:
3449:
3446:
3442:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3409:
3406:
3403:
3399:
3395:
3392:
3389:
3386:
3383:
3380:
3357:
3346:
3345:
3334:
3329:
3326:
3323:
3319:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3286:
3282:
3278:
3275:
3272:
3269:
3266:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3155:
3135:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3080:
3079:
3068:
3064:
3059:
3056:
3053:
3049:
3045:
3042:
3039:
3036:
3031:
3027:
3021:
3017:
2993:
2973:
2953:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2895:
2892:
2889:
2869:
2849:
2846:
2843:
2840:
2820:
2817:
2814:
2794:
2791:
2788:
2772:
2769:
2748:
2745:
2742:
2737:
2733:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2660:
2656:
2635:
2632:
2629:
2626:
2623:
2579:
2553:
2549:
2528:
2506:
2502:
2475:
2471:
2466:
2463:
2460:
2454:
2449:
2445:
2424:
2421:
2416:
2413:
2400:
2397:
2394:
2391:
2388:
2385:
2374:
2373:
2360:
2357:
2352:
2347:
2344:
2339:
2334:
2330:
2304:
2300:
2288:
2287:
2276:
2273:
2267:
2262:
2258:
2254:
2251:
2246:
2243:
2240:
2234:
2228:
2223:
2219:
2215:
2212:
2207:
2204:
2198:
2191:
2187:
2183:
2180:
2175:
2169:
2166:
2161:
2158:
2152:
2126:
2122:
2101:
2081:
2070:
2069:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1943:
1942:
1929:
1925:
1921:
1918:
1915:
1912:
1909:
1906:
1901:
1897:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1852:
1832:
1829:
1826:
1823:
1803:
1783:
1780:
1777:
1774:
1771:
1751:
1748:
1745:
1742:
1722:
1702:
1682:
1662:
1650:expected value
1645:
1642:
1623:
1619:
1615:
1612:
1609:
1606:
1603:
1592:geometric mean
1572:
1569:
1548:
1545:
1542:
1522:
1519:
1516:
1496:
1493:
1490:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1423:
1420:
1417:
1414:
1411:
1391:
1388:
1383:
1379:
1358:
1355:
1352:
1332:
1329:
1326:
1315:
1314:
1296:
1293:
1288:
1285:
1282:
1279:
1269:
1254:
1251:
1248:
1244:
1239:
1236:
1233:
1230:
1216:
1215:
1203:
1196:
1193:
1190:
1186:
1178:
1175:
1172:
1168:
1163:
1160:
1156:
1150:
1147:
1142:
1138:
1132:
1129:
1122:
1118:
1115:
1112:
1106:
1103:
1099:
1093:
1090:
1085:
1080:
1076:
1050:
1049:
1038:
1035:
1030:
1026:
1023:
1020:
1014:
1008:original value
1000:
995:original value
989:
986:
966:
956:
944:
941:
936:
932:
929:
926:
920:
914:original value
909:original value
906:
895:
892:
872:
862:
850:
847:
844:
841:
838:
818:
808:
796:
786:
772:
768:
753:
752:
739:
736:
731:
726:
723:
718:
713:
709:
693:
690:
669:
665:
661:
658:
638:
634:
630:
627:
624:
604:
601:
598:
574:
570:
566:
563:
560:
537:
533:
529:
526:
523:
484:
481:
476:
473:
468:
465:
462:
457:
453:
432:
429:
426:
406:
403:
400:
380:
377:
374:
363:
362:
341:
338:
335:
332:
328:
324:
321:
318:
315:
291:
281:
269:
266:
263:
260:
257:
247:
235:
225:
211:
207:
192:
191:
178:
174:
171:
168:
162:
159:
156:
151:
148:
143:
140:
137:
132:
128:
112:
109:
103:
100:
80:Warren Buffett
49:expected value
37:Kelly strategy
15:
9:
6:
4:
3:
2:
9892:
9881:
9878:
9876:
9873:
9871:
9868:
9866:
9863:
9861:
9858:
9856:
9853:
9851:
9848:
9847:
9845:
9828:
9824:
9820:
9816:
9812:
9805:
9798:
9794:
9790:
9786:
9781:
9776:
9771:
9766:
9763:(2): 023103,
9762:
9758:
9751:
9744:
9737:
9730:
9729:
9721:
9712:
9708:
9704:
9700:
9696:
9692:
9688:
9684:
9680:
9679:
9671:
9667:
9661:
9653:
9649:
9642:
9634:
9627:
9619:
9618:
9610:
9602:
9598:
9594:
9587:
9585:
9583:
9581:
9572:
9568:
9564:
9560:
9556:
9552:
9548:
9544:
9540:
9536:
9530:
9522:
9518:
9513:
9508:
9504:
9500:
9493:
9485:
9481:
9477:
9473:
9466:
9459:
9452:
9446:
9442:
9438:
9431:
9425:
9423:
9422:The Economist
9418:
9412:
9406:
9402:
9397:
9393:
9386:
9371:
9364:
9357:
9355:
9353:
9351:
9349:
9341:
9339:0-8090-4637-7
9335:
9331:
9330:
9322:
9320:
9312:
9311:
9303:
9296:
9290:
9286:
9285:
9277:
9270:
9264:
9260:
9253:
9245:on 2009-10-07
9241:
9237:
9235:0-394-42439-5
9231:
9224:
9223:
9215:
9207:
9203:
9199:
9197:0-394-70310-3
9193:
9189:
9182:
9175:
9168:
9160:
9156:
9152:
9148:
9147:
9139:
9135:
9129:
9127:
9125:
9123:
9118:
9109:
9106:
9104:
9101:
9099:
9096:
9094:
9091:
9090:
9084:
9082:
9079:breaking and
9078:
9074:
9070:
9066:
9060:
9057:
9053:
9049:
9045:
9034:
9031:
9029:
9025:
9021:
9011:
8993:
8964:
8956:
8925:
8922:
8916:
8908:
8895:
8892:
8881:
8866:
8863:
8860:
8854:
8843:
8839:
8828:
8827:
8826:
8824:
8805:
8801:
8792:
8784:
8781:
8778:
8767:
8764:
8759:
8755:
8745:
8742:
8737:
8733:
8721:
8717:
8711:
8707:
8701:
8696:
8693:
8690:
8680:
8675:
8672:
8669:
8659:
8656:
8651:
8645:
8642:
8639:
8631:
8628:
8623:
8619:
8610:
8606:
8597:
8592:
8589:
8586:
8578:
8572:
8569:
8566:
8560:
8557:
8553:
8540:
8539:
8538:
8521:
8518:
8515:
8512:
8509:
8503:
8492:
8488:
8476:
8475:Taylor series
8457:
8453:
8448:
8441:
8438:
8433:
8429:
8420:
8416:
8410:
8405:
8402:
8399:
8391:
8385:
8382:
8379:
8372:
8368:
8365:
8361:
8348:
8347:
8346:
8330:
8326:
8303:
8299:
8278:
8258:
8255:
8252:
8249:
8246:
8243:
8240:
8237:
8215:
8211:
8188:
8184:
8163:
8154:
8138:
8134:
8113:
8091:
8087:
8064:
8057:
8053:
8048:
8045:
8042:
8036:
8031:
8027:
8019:
8018:
8017:
7997:
7991:
7962:
7959:
7950:
7947:
7944:
7938:
7933:
7927:
7919:
7916:
7907:
7904:
7901:
7898:
7892:
7886:
7879:
7878:
7877:
7861:
7857:
7836:
7816:
7793:
7787:
7784:
7781:
7775:
7770:
7766:
7762:
7759:
7755:
7749:
7745:
7741:
7738:
7735:
7731:
7725:
7719:
7711:
7708:
7699:
7696:
7693:
7689:
7684:
7680:
7677:
7673:
7669:
7659:
7654:
7650:
7644:
7641:
7638:
7632:
7628:
7624:
7621:
7615:
7611:
7607:
7602:
7598:
7594:
7587:
7582:
7568:
7567:
7566:
7548:
7542:
7538:
7534:
7531:
7528:
7525:
7521:
7515:
7509:
7501:
7498:
7489:
7486:
7483:
7479:
7474:
7470:
7467:
7463:
7459:
7455:
7451:
7447:
7443:
7439:
7431:
7430:
7429:
7413:
7409:
7405:
7402:
7399:
7395:
7391:
7387:
7383:
7361:
7357:
7353:
7330:
7327:
7324:
7301:
7281:
7261:
7241:
7218:
7214:
7207:
7202:
7198:
7192:
7189:
7185:
7181:
7176:
7172:
7164:
7163:
7162:
7146:
7142:
7118:
7115:
7111:
7105:
7100:
7096:
7090:
7087:
7083:
7079:
7071:
7067:
7060:
7057:
7054:
7046:
7042:
7035:
7032:
7020:
7019:
7018:
7004:
6984:
6976:
6958:
6954:
6929:
6923:
6919:
6915:
6912:
6909:
6905:
6899:
6894:
6890:
6884:
6881:
6877:
6872:
6868:
6865:
6860:
6856:
6852:
6847:
6843:
6835:
6834:
6833:
6814:
6810:
6806:
6803:
6800:
6797:
6794:
6791:
6788:
6783:
6779:
6774:
6768:
6764:
6760:
6753:
6752:
6751:
6735:
6731:
6710:
6690:
6682:
6677:
6675:
6671:
6666:
6641:
6638:
6632:
6628:
6621:
6617:
6609:
6594:
6591:
6588:
6582:
6578:
6571:
6567:
6559:
6558:
6557:
6541:
6537:
6513:
6506:
6502:
6498:
6493:
6488:
6484:
6476:
6475:
6474:
6457:
6446:
6442:
6435:
6429:
6424:
6420:
6415:
6408:
6404:
6397:
6394:
6391:
6387:
6383:
6380:
6376:
6372:
6363:
6359:
6355:
6349:
6344:
6340:
6334:
6330:
6323:
6320:
6317:
6313:
6309:
6304:
6300:
6292:
6291:
6290:
6276:
6256:
6236:
6216:
6194:
6189:
6185:
6162:
6158:
6137:
6114:
6108:
6105:
6100:
6096:
6088:
6084:
6080:
6075:
6070:
6066:
6062:
6036:
6031:
6027:
6018:
6014:
6010:
6007:
6003:
5999:
5996:
5993:
5985:
5981:
5974:
5967:
5966:
5965:
5948:
5942:
5936:
5932:
5926:
5923:
5920:
5916:
5912:
5909:
5905:
5898:
5894:
5888:
5885:
5882:
5878:
5869:
5865:
5861:
5856:
5852:
5848:
5843:
5839:
5831:
5830:
5829:
5815:
5793:
5788:
5784:
5761:
5757:
5734:
5730:
5703:
5700:
5695:
5691:
5670:
5667:
5664:
5661:
5658:
5651:and then set
5633:
5629:
5623:
5620:
5617:
5613:
5609:
5606:
5599:
5595:
5589:
5586:
5583:
5579:
5575:
5569:
5563:
5557:
5534:
5528:
5505:
5499:
5496:
5493:
5490:
5470:
5447:
5441:
5438:
5433:
5429:
5421:
5417:
5413:
5408:
5403:
5399:
5395:
5387:
5386:
5384:
5367:
5363:
5359:
5356:
5351:
5347:
5343:
5323:
5320:
5314:
5308:
5288:
5285:
5282:
5259:
5256:
5248:
5231:
5227:
5223:
5201:
5197:
5193:
5185:
5168:
5165:
5160:
5156:
5147:
5143:
5139:
5132:
5128:
5121:
5117:
5113:
5107:
5102:
5098:
5094:
5086:
5085:
5084:
5068:
5064:
5041:
5036:
5032:
5009:
5005:
4982:
4978:
4970:-th horse is
4957:
4937:
4913:
4909:
4905:
4883:
4880:
4860:
4857:
4854:
4851:
4848:
4845:
4823:
4819:
4795:
4787:
4783:
4779:
4776:
4772:
4767:
4759:
4755:
4749:
4745:
4738:
4734:
4728:
4723:
4719:
4711:
4710:
4709:
4693:
4689:
4681:-th horse is
4668:
4646:
4642:
4621:
4607:
4593:
4573:
4570:
4550:
4547:
4526:
4506:
4484:
4480:
4459:
4439:
4431:
4413:
4410:
4406:
4402:
4399:
4396:
4392:
4388:
4385:
4380:
4376:
4368:
4367:
4366:
4352:
4330:
4326:
4305:
4301:
4292:
4288:
4281:
4278:
4255:
4250:
4247:
4239:
4236:
4233:
4230:
4222:
4219:
4211:
4208:
4205:
4202:
4196:
4191:
4187:
4179:
4178:
4177:
4163:
4160:
4140:
4137:
4130:trials (with
4117:
4097:
4094:
4091:
4088:
4068:
4065:
4062:
4059:
4056:
4050:
4047:
4044:
4038:
4035:
4032:
4029:
4026:
4006:
3986:
3983:
3980:
3977:
3957:
3954:
3951:
3948:
3928:
3925:
3905:
3885:
3865:
3842:
3838:
3834:
3831:
3826:
3823:
3818:
3814:
3806:
3805:
3804:
3781:
3762:
3759:
3754:
3751:
3741:
3735:
3723:
3722:
3721:
3719:
3699:
3695:
3692:
3687:
3684:
3678:
3674:
3671:
3661:
3660:
3659:
3636:
3633:
3630:
3621:
3618:
3615:
3609:
3600:
3594:
3591:
3588:
3582:
3576:
3569:
3568:
3567:
3550:
3545:
3542:
3539:
3536:
3533:
3522:
3516:
3513:
3510:
3504:
3496:
3485:
3482:
3479:
3470:
3467:
3464:
3458:
3455:
3450:
3447:
3444:
3433:
3427:
3424:
3421:
3415:
3407:
3404:
3401:
3390:
3387:
3384:
3378:
3371:
3370:
3369:
3332:
3327:
3324:
3321:
3310:
3304:
3301:
3298:
3292:
3284:
3273:
3270:
3267:
3257:
3256:
3255:
3241:
3232:
3226:
3223:
3220:
3214:
3191:
3182:
3179:
3176:
3113:
3104:
3101:
3098:
3095:
3092:
3066:
3062:
3057:
3054:
3051:
3043:
3040:
3037:
3029:
3025:
3019:
3015:
3007:
3006:
3005:
2991:
2971:
2951:
2931:
2925:
2922:
2919:
2913:
2893:
2890:
2887:
2867:
2847:
2844:
2841:
2838:
2818:
2815:
2812:
2792:
2789:
2786:
2778:
2768:
2765:
2760:
2743:
2740:
2735:
2731:
2707:
2704:
2701:
2698:
2695:
2692:
2686:
2683:
2680:
2658:
2654:
2630:
2627:
2624:
2621:
2612:
2607:
2605:
2601:
2595:
2593:
2577:
2569:
2551:
2547:
2526:
2504:
2500:
2490:
2473:
2469:
2464:
2461:
2458:
2452:
2447:
2443:
2434:
2431:
2430:
2420:
2412:
2395:
2392:
2389:
2386:
2383:
2358:
2355:
2350:
2345:
2342:
2337:
2332:
2328:
2320:
2319:
2318:
2302:
2298:
2274:
2271:
2265:
2260:
2256:
2252:
2249:
2244:
2241:
2238:
2232:
2226:
2221:
2217:
2213:
2210:
2205:
2202:
2196:
2189:
2185:
2181:
2178:
2173:
2167:
2164:
2159:
2156:
2142:
2141:
2140:
2124:
2120:
2099:
2079:
2053:
2050:
2047:
2044:
2038:
2035:
2032:
2029:
2023:
2020:
2017:
2014:
2008:
2005:
2002:
1999:
1993:
1987:
1984:
1981:
1978:
1971:
1970:
1969:
1967:
1963:
1959:
1955:
1951:
1948:
1927:
1919:
1916:
1913:
1910:
1904:
1899:
1891:
1888:
1885:
1882:
1876:
1873:
1866:
1865:
1864:
1850:
1830:
1827:
1824:
1821:
1801:
1781:
1778:
1775:
1772:
1769:
1749:
1746:
1743:
1740:
1720:
1700:
1680:
1660:
1651:
1641:
1637:
1621:
1613:
1607:
1604:
1601:
1593:
1586:
1581:
1578:
1568:
1566:
1562:
1546:
1543:
1540:
1520:
1517:
1514:
1494:
1491:
1488:
1479:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1421:
1418:
1415:
1412:
1409:
1389:
1386:
1381:
1377:
1356:
1353:
1350:
1330:
1327:
1324:
1312:
1294:
1291:
1286:
1283:
1280:
1277:
1270:
1252:
1249:
1246:
1242:
1237:
1234:
1231:
1228:
1221:
1220:
1219:
1201:
1194:
1191:
1188:
1184:
1176:
1173:
1170:
1166:
1161:
1158:
1154:
1148:
1145:
1140:
1136:
1130:
1127:
1120:
1116:
1113:
1110:
1104:
1101:
1097:
1091:
1088:
1083:
1078:
1074:
1066:
1065:
1064:
1061:
1059:
1055:
1036:
1033:
1028:
1024:
1021:
1018:
1012:
998:
987:
984:
964:
957:
942:
939:
934:
930:
927:
924:
918:
904:
893:
890:
870:
863:
848:
845:
842:
839:
836:
816:
809:
794:
787:
770:
766:
758:
757:
756:
737:
734:
729:
724:
721:
716:
711:
707:
699:
698:
697:
689:
687:
683:
667:
663:
659:
656:
636:
632:
628:
625:
622:
602:
599:
596:
588:
572:
568:
564:
561:
558:
549:
535:
531:
527:
524:
521:
508:
500:
496:
482:
479:
474:
471:
466:
463:
460:
455:
451:
430:
427:
424:
404:
401:
398:
378:
375:
372:
356:
339:
336:
333:
326:
322:
316:
313:
305:
289:
282:
267:
264:
261:
258:
255:
248:
233:
226:
209:
205:
197:
196:
195:
176:
172:
169:
166:
160:
157:
154:
149:
146:
141:
138:
135:
130:
126:
118:
117:
116:
108:
99:
97:
93:
89:
85:
81:
77:
73:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
21:
9810:
9804:
9760:
9756:
9750:
9742:
9736:
9726:
9720:
9682:
9678:Econometrica
9676:
9660:
9641:
9632:
9626:
9616:
9609:
9600:
9596:
9546:
9542:
9535:Thorp, E. O.
9529:
9502:
9498:
9492:
9475:
9471:
9458:
9440:
9430:
9420:
9411:
9385:
9374:. Retrieved
9369:
9328:
9308:
9302:
9283:
9276:
9258:
9252:
9247:, page 184f.
9240:the original
9221:
9214:
9187:
9181:
9173:
9167:
9150:
9144:
9134:Kelly, J. L.
9093:Risk of ruin
9064:
9061:
9040:
9032:
9017:
8943:
8820:
8472:
8155:
8079:
7977:
7808:
7564:
7294:invested in
7233:
7133:
6945:
6831:
6678:
6662:
6528:
6472:
6054:
5963:
5721:
5463:then insert
4810:
4613:
4428:
4270:
3898:gets large,
3857:
3777:
3715:
3657:
3565:
3347:
3081:
2774:
2761:
2608:
2603:
2599:
2596:
2491:
2435:
2432:
2428:
2426:
2418:
2375:
2289:
2071:
1953:
1949:
1944:
1647:
1638:
1588:
1583:
1574:
1480:
1316:
1311:winning skew
1310:
1217:
1062:
1053:
1051:
754:
695:
685:
681:
550:
513:
364:
193:
114:
105:
65:
40:
36:
32:
26:
9603:(1): 10–17.
9056:logarithmic
4271:Maximizing
2673:above with
1585:experiment.
1058:risk averse
1003:final value
901:final value
9844:Categories
9512:1612.07194
9405:1701.01427
9376:2024-03-20
9372:. Montreal
9114:References
9077:ergodicity
9073:ergodicity
8537:we obtain
8016:we obtain
7809:For small
5808:to bet on
2611:volatility
1958:derivative
84:Bill Gross
9827:250132554
9770:1405.0585
9689:: 22–36.
9287:, Wiley,
9037:Criticism
9014:Bernoulli
8994:^
8991:Σ
8965:^
8960:→
8923:−
8917:^
8912:→
8893:−
8882:^
8879:Σ
8849:→
8844:⋆
8765:−
8743:−
8687:∑
8666:∑
8652:−
8629:−
8583:∑
8561:
8516:…
8498:→
8439:−
8396:∑
8369:
8250:…
8114:μ
8092:∗
8054:σ
8046:−
8043:μ
8032:∗
7948:−
7920:σ
7908:−
7905:μ
7837:σ
7817:μ
7785:−
7760:−
7742:σ
7712:σ
7700:−
7697:μ
7681:
7642:−
7622:−
7535:σ
7502:σ
7490:−
7487:μ
7471:
7328:−
7199:σ
7193:−
7190:μ
7097:σ
7091:−
7088:μ
7061:
7036:
7005:σ
6985:μ
6916:σ
6891:σ
6885:−
6882:μ
6869:
6804:σ
6792:μ
6629:β
6618:∑
6568:∑
6430:
6395:∈
6388:∑
6384:−
6350:
6321:∈
6314:∑
6085:β
6011:∈
6004:∑
6000:−
5933:β
5924:∈
5917:∑
5913:−
5886:∉
5879:∑
5866:β
5862:−
5630:β
5621:∈
5614:∑
5610:−
5587:∉
5580:∑
5500:∪
5418:β
5263:∅
5129:β
4910:β
4855:−
4746:∑
4720:β
4485:∗
4400:−
4381:∗
4282:
4234:−
4092:−
4030:−
3832:−
3790:Δ
3745:∞
3739:→
3693:−
3669:Δ
3640:Δ
3619:−
3604:Δ
3601:−
3592:−
3543:−
3537:−
3526:Δ
3523:−
3514:−
3489:Δ
3468:−
3459:−
3448:−
3437:Δ
3434:−
3425:−
3405:−
3394:Δ
3368:and get:
3356:Δ
3325:−
3314:Δ
3311:−
3302:−
3277:Δ
3236:Δ
3233:−
3224:−
3186:Δ
3154:Δ
3134:Δ
3108:Δ
3099:−
3055:−
3041:−
2923:−
2845:−
2747:%
2736:∗
2711:%
2690:%
2659:∗
2634:%
2625:σ
2548:σ
2527:μ
2505:∗
2470:σ
2462:−
2459:μ
2448:∗
2399:%
2351:−
2333:∗
2303:∗
2261:∗
2253:−
2239:−
2222:∗
2190:∗
2125:∗
2048:−
2039:
2009:
1988:
1966:monotonic
1962:logarithm
1914:−
1905:⋅
1825:−
1779:−
1608:⋅
1492:≪
1451:∗
1382:∗
1250:−
1162:−
1114:−
1105:−
1079:∗
1022:−
999:−
928:−
905:−
846:−
771:∗
730:−
712:∗
657:−
467:−
456:∗
331:$
320:$
265:−
210:∗
170:−
161:−
142:−
131:∗
61:Bell Labs
41:Kelly bet
9880:Formulas
9865:Wagering
9789:26931584
9668:(1954).
9537:(1969).
9136:(1956).
9087:See also
9052:expected
9048:maximize
7978:Solving
7616:′
7603:′
7464:′
7448:′
7396:′
6703:at time
6674:variance
5385:Repeat:
2568:variance
1561:leverage
686:anti-red
68:gambling
9797:9726238
9711:9165746
9703:1909829
9652:2259133
9571:0135630
9563:1402118
9396:2856963
8477:around
3146:(where
2777:Kelly's
2604:ex-ante
2600:Ex-post
2566:is the
1947:maximum
1614:1.02034
1218:where:
755:where:
194:where:
45:formula
43:) is a
9825:
9795:
9787:
9709:
9701:
9650:
9569:
9561:
9447:
9394:
9336:
9291:
9265:
9232:
9206:655875
9204:
9194:
8944:where
7957:
7849:, and
6977:, and
6946:where
4811:where
4708:, and
2687:50.425
2492:where
1565:margin
31:, the
9823:S2CID
9793:S2CID
9765:arXiv
9707:S2CID
9699:JSTOR
9685:(1).
9673:(PDF)
9559:JSTOR
9507:arXiv
9468:(PDF)
9401:arXiv
9366:(PDF)
9243:(PDF)
9226:(PDF)
9141:(PDF)
6973:is a
6750:) is
2771:Proof
2072:with
1644:Proof
1054:known
9785:PMID
9648:SSRN
9445:ISBN
9392:SSRN
9334:ISBN
9289:ISBN
9263:ISBN
9230:ISBN
9202:OCLC
9192:ISBN
9050:the
8980:and
7314:and
6672:and
6639:<
6589:<
6106:>
5439:>
5249:Set
4519:and
3981:<
3952:>
3720:to:
1863:is:
1533:and
1463:>
1419:<
1387:>
562:<
304:odds
82:and
35:(or
9815:doi
9775:doi
9691:doi
9551:doi
9517:doi
9480:doi
9155:doi
9030:).
7986:max
7678:exp
7468:exp
7161:is
7058:log
7033:log
6866:exp
6607:and
6421:log
6341:log
5388:If
4279:log
3732:lim
2396:100
2036:log
2006:log
1985:log
1622:300
1478:.
1343:or
1037:0.1
943:0.1
925:1.1
682:not
495:).
483:0.2
472:0.4
464:0.6
405:0.4
379:0.6
340:2.0
74:in
55:.
39:or
27:In
9846::
9821:,
9813:,
9791:,
9783:,
9773:,
9761:26
9759:,
9705:.
9697:.
9683:22
9681:.
9675:.
9601:35
9599:.
9595:.
9579:^
9567:MR
9565:.
9557:.
9547:37
9545:.
9541:.
9515:.
9505:.
9501:.
9476:99
9474:.
9470:.
9439:,
9419:,
9368:.
9347:^
9318:^
9200:.
9151:35
9149:.
9143:.
9121:^
8558:ln
8366:ln
8230:,
7829:,
5301:,
4606:.
2744:85
1605:25
1567:.
1025:.9
861:).
668:19
637:38
629:18
391:,
334:10
323:20
98:.
9817::
9777::
9767::
9713:.
9693::
9654:.
9573:.
9553::
9523:.
9519::
9509::
9503:6
9486:.
9482::
9403::
9398:.
9379:.
9208:.
9161:.
9157::
8957:r
8929:)
8926:r
8909:r
8901:(
8896:1
8889:)
8873:(
8870:)
8867:r
8864:+
8861:1
8858:(
8855:=
8840:u
8806:.
8802:]
8793:2
8789:)
8785:r
8782:+
8779:1
8776:(
8771:)
8768:r
8760:j
8756:r
8752:(
8749:)
8746:r
8738:k
8734:r
8730:(
8722:j
8718:u
8712:k
8708:u
8702:n
8697:1
8694:=
8691:j
8681:n
8676:1
8673:=
8670:k
8660:2
8657:1
8646:r
8643:+
8640:1
8635:)
8632:r
8624:k
8620:r
8616:(
8611:k
8607:u
8598:n
8593:1
8590:=
8587:k
8579:+
8576:)
8573:r
8570:+
8567:1
8564:(
8554:[
8549:E
8525:)
8522:0
8519:,
8513:,
8510:0
8507:(
8504:=
8493:0
8489:u
8458:.
8454:]
8449:)
8445:)
8442:r
8434:k
8430:r
8426:(
8421:k
8417:u
8411:n
8406:1
8403:=
8400:k
8392:+
8389:)
8386:r
8383:+
8380:1
8377:(
8373:(
8362:[
8357:E
8331:k
8327:S
8304:k
8300:u
8279:r
8259:,
8256:n
8253:,
8247:,
8244:1
8241:=
8238:k
8216:k
8212:r
8189:k
8185:S
8164:n
8139:s
8135:R
8088:f
8065:.
8058:2
8049:r
8037:=
8028:f
8004:)
8001:)
7998:f
7995:(
7992:G
7989:(
7963:.
7960:r
7954:)
7951:f
7945:1
7942:(
7939:+
7934:2
7928:2
7924:)
7917:f
7914:(
7902:f
7899:=
7896:)
7893:f
7890:(
7887:G
7862:t
7858:W
7794:r
7791:)
7788:f
7782:1
7779:(
7776:+
7771:)
7767:]
7763:1
7756:)
7750:1
7746:W
7739:f
7736:+
7732:)
7726:2
7720:2
7716:)
7709:f
7706:(
7694:f
7690:(
7685:(
7674:[
7670:(
7664:E
7660:=
7655:)
7651:r
7648:)
7645:f
7639:1
7636:(
7633:+
7629:]
7625:1
7612:0
7608:S
7599:1
7595:S
7588:[
7583:(
7577:E
7549:)
7543:t
7539:W
7532:f
7529:+
7526:t
7522:)
7516:2
7510:2
7506:)
7499:f
7496:(
7484:f
7480:(
7475:(
7460:0
7456:S
7452:=
7444:t
7440:S
7414:t
7410:S
7406:d
7403:f
7400:=
7392:t
7388:S
7384:d
7362:t
7358:S
7354:d
7334:)
7331:f
7325:1
7322:(
7302:S
7282:f
7262:r
7242:S
7219:.
7215:)
7208:2
7203:2
7186:(
7182:=
7177:s
7173:R
7147:s
7143:R
7119:.
7116:t
7112:)
7106:2
7101:2
7084:(
7080:+
7077:)
7072:0
7068:S
7064:(
7055:=
7052:)
7047:t
7043:S
7039:(
7029:E
6959:t
6955:W
6930:)
6924:t
6920:W
6913:+
6910:t
6906:)
6900:2
6895:2
6878:(
6873:(
6861:0
6857:S
6853:=
6848:t
6844:S
6815:t
6811:W
6807:d
6801:+
6798:t
6795:d
6789:=
6784:t
6780:S
6775:/
6769:t
6765:S
6761:d
6736:t
6732:S
6723:(
6711:t
6691:S
6654:.
6642:1
6633:i
6622:i
6595:,
6592:1
6583:i
6579:p
6572:i
6542:k
6538:p
6514:.
6507:o
6503:G
6499:1
6494:=
6489:d
6485:T
6458:,
6455:)
6452:)
6447:o
6443:S
6439:(
6436:R
6433:(
6425:2
6416:)
6409:i
6405:p
6398:S
6392:i
6381:1
6377:(
6373:+
6369:)
6364:i
6360:r
6356:e
6353:(
6345:2
6335:i
6331:p
6324:S
6318:i
6310:=
6305:o
6301:G
6277:k
6257:k
6237:k
6217:k
6195:o
6190:k
6186:f
6163:o
6159:S
6138:k
6118:)
6115:S
6112:(
6109:R
6101:k
6097:p
6089:k
6081:D
6076:=
6071:k
6067:r
6063:e
6037:o
6032:i
6028:f
6019:o
6015:S
6008:i
5997:1
5994:=
5991:)
5986:o
5982:S
5978:(
5975:R
5949:.
5943:)
5937:k
5927:S
5921:k
5910:D
5906:(
5899:k
5895:p
5889:S
5883:k
5870:i
5857:i
5853:p
5849:=
5844:i
5840:f
5816:k
5794:o
5789:k
5785:f
5762:o
5758:S
5735:o
5731:S
5704:S
5701:=
5696:o
5692:S
5671:1
5668:+
5665:k
5662:=
5659:k
5634:k
5624:S
5618:k
5607:D
5600:k
5596:p
5590:S
5584:k
5576:D
5570:=
5567:)
5564:S
5561:(
5558:R
5538:)
5535:S
5532:(
5529:R
5509:}
5506:k
5503:{
5497:S
5494:=
5491:S
5471:k
5451:)
5448:S
5445:(
5442:R
5434:k
5430:p
5422:k
5414:D
5409:=
5404:k
5400:r
5396:e
5368:1
5364:r
5360:e
5357:=
5352:k
5348:r
5344:e
5324:1
5321:=
5318:)
5315:S
5312:(
5309:R
5289:1
5286:=
5283:k
5260:=
5257:S
5232:1
5228:r
5224:e
5202:k
5198:r
5194:e
5172:)
5169:1
5166:+
5161:i
5157:Q
5153:(
5148:i
5144:p
5140:=
5133:i
5122:i
5118:p
5114:D
5108:=
5103:i
5099:r
5095:e
5069:o
5065:S
5042:o
5037:k
5033:f
5010:o
5006:S
4983:k
4979:f
4958:k
4938:k
4914:k
4906:D
4884:t
4881:t
4861:t
4858:t
4852:1
4849:=
4846:D
4824:k
4820:Q
4796:,
4788:k
4784:Q
4780:+
4777:1
4773:D
4768:=
4760:i
4756:B
4750:i
4739:k
4735:B
4729:=
4724:k
4694:k
4690:B
4669:k
4647:k
4643:p
4622:k
4594:N
4574:N
4571:q
4551:N
4548:p
4527:b
4507:p
4481:f
4460:p
4440:p
4414:.
4411:b
4407:/
4403:q
4397:a
4393:/
4389:p
4386:=
4377:f
4353:f
4331:N
4327:C
4306:N
4302:/
4298:)
4293:N
4289:C
4285:(
4256:.
4251:N
4248:q
4244:)
4240:a
4237:f
4231:1
4228:(
4223:N
4220:p
4216:)
4212:b
4209:f
4206:+
4203:1
4200:(
4197:=
4192:N
4188:C
4164:N
4161:q
4141:N
4138:p
4118:N
4098:a
4095:f
4089:1
4069:b
4066:f
4063:+
4060:1
4057:=
4054:)
4051:b
4048:+
4045:1
4042:(
4039:f
4036:+
4033:f
4027:1
4007:f
3987:N
3984:p
3978:K
3958:N
3955:p
3949:K
3929:N
3926:p
3906:K
3886:N
3866:N
3843:W
3839:)
3835:1
3827:N
3824:K
3819:2
3815:(
3763:p
3760:=
3755:N
3752:K
3742:+
3736:N
3700:)
3696:p
3688:N
3685:K
3679:(
3675:2
3672:=
3643:)
3637:+
3634:p
3631:2
3628:(
3625:)
3622:K
3616:N
3613:(
3610:=
3607:]
3598:)
3595:p
3589:1
3586:(
3583:2
3580:[
3577:K
3551:W
3546:1
3540:K
3534:N
3530:]
3520:)
3517:p
3511:1
3508:(
3505:2
3502:[
3497:K
3493:)
3486:+
3483:p
3480:2
3477:(
3474:)
3471:K
3465:N
3462:(
3456:W
3451:K
3445:N
3441:]
3431:)
3428:p
3422:1
3419:(
3416:2
3413:[
3408:1
3402:K
3398:)
3391:+
3388:p
3385:2
3382:(
3379:K
3333:W
3328:K
3322:N
3318:]
3308:)
3305:p
3299:1
3296:(
3293:2
3290:[
3285:K
3281:)
3274:+
3271:p
3268:2
3265:(
3242:W
3239:]
3230:)
3227:p
3221:1
3218:(
3215:2
3212:[
3192:W
3189:)
3183:+
3180:p
3177:2
3174:(
3114:W
3111:)
3105:+
3102:1
3096:p
3093:2
3090:(
3067:.
3063:W
3058:K
3052:N
3048:)
3044:p
3038:1
3035:(
3030:K
3026:p
3020:N
3016:2
2992:N
2972:K
2952:N
2932:W
2929:)
2926:p
2920:1
2917:(
2914:2
2894:W
2891:p
2888:2
2868:W
2848:1
2842:p
2839:2
2819:1
2816:=
2813:b
2793:1
2790:=
2787:b
2741:=
2732:f
2708:1
2705:=
2702:b
2699:=
2696:a
2693:,
2684:=
2681:p
2655:f
2631:4
2628:=
2622:4
2578:r
2552:2
2501:f
2474:2
2465:r
2453:=
2444:f
2393:=
2390:1
2387:=
2384:a
2359:b
2356:q
2346:a
2343:p
2338:=
2329:f
2299:f
2275:0
2272:=
2266:a
2257:f
2250:1
2245:a
2242:q
2233:+
2227:b
2218:f
2214:+
2211:1
2206:b
2203:p
2197:=
2186:f
2182:=
2179:f
2174:|
2168:f
2165:d
2160:E
2157:d
2121:f
2100:f
2080:E
2057:)
2054:a
2051:f
2045:1
2042:(
2033:q
2030:+
2027:)
2024:b
2021:f
2018:+
2015:1
2012:(
2003:p
2000:=
1997:)
1994:r
1991:(
1982:=
1979:E
1954:f
1950:r
1928:q
1924:)
1920:a
1917:f
1911:1
1908:(
1900:p
1896:)
1892:b
1889:f
1886:+
1883:1
1880:(
1877:=
1874:r
1851:r
1831:a
1828:f
1822:1
1802:a
1782:p
1776:1
1773:=
1770:q
1750:b
1747:f
1744:+
1741:1
1721:p
1701:b
1681:p
1661:f
1618:)
1611:(
1602:=
1547:P
1544:L
1541:W
1521:R
1518:L
1515:W
1495:1
1489:l
1466:1
1460:R
1457:L
1454:W
1448:P
1445:L
1442:W
1422:1
1416:P
1413:L
1410:W
1390:0
1378:f
1357:R
1354:L
1351:W
1331:P
1328:L
1325:W
1313:.
1295:l
1292:g
1287:=
1284:R
1281:L
1278:W
1253:p
1247:1
1243:p
1238:=
1235:P
1232:L
1229:W
1202:)
1195:R
1192:L
1189:W
1185:1
1177:P
1174:L
1171:W
1167:1
1159:1
1155:(
1149:l
1146:p
1141:=
1137:)
1131:g
1128:l
1121:p
1117:p
1111:1
1102:1
1098:(
1092:l
1089:p
1084:=
1075:f
1034:=
1029:1
1019:1
1013:=
988:=
985:l
965:l
955:.
940:=
935:1
931:1
919:=
894:=
891:g
871:g
849:p
843:1
840:=
837:q
817:q
795:p
767:f
738:g
735:q
725:l
722:p
717:=
708:f
664:/
660:1
633:/
626:=
623:p
603:1
600:=
597:b
573:p
569:/
565:q
559:b
536:p
532:/
528:q
525:=
522:b
480:=
475:1
461:=
452:f
431:1
428:=
425:b
402:=
399:q
376:=
373:p
352:.
337:=
327:/
317:=
314:b
290:b
268:p
262:1
259:=
256:q
234:p
206:f
177:b
173:p
167:1
158:p
155:=
150:b
147:q
139:p
136:=
127:f
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