Us­ing the Kel­ly Cri­te­ri­on for Op­ti­mal Bet Siz­ing

Sup­pose a gam­bler is play­ing a game in which he has a sta­tis­ti­cal ad­van­tage. And let’s as­sume he can quan­ti­fy his ad­van­tage with a fair amount of ac­cu­ra­cy. If the gam­bler plays this game over and over again, what per­cent­age of his bankroll should he bet on each round if he wants to max­i­mize his win­nings? In this post, I ex­plore this ques­tion us­ing a se­ries of ex­am­ples il­lus­trat­ing the ap­pli­ca­tion of the Kel­ly cri­te­ri­on. Many of the ideas pre­sent­ed here are in­spired by ma­te­ri­als writ­ten by Ed Seyko­ta, Ed­ward O. Thor­p, and J. L. Kel­ly.

Re­peat­ed Coin Toss Game

For our first ex­am­ple, let’s con­sid­er a re­peat­ed coin toss game. The gam­bler bets on the out­come of a coin toss. If he pre­dicts the out­come cor­rect­ly, he wins an amount equiv­a­lent to the size of his bet. If he is wrong, he los­es the amount he wa­gered. In this ex­am­ple, let’s as­sume the coin is bi­ased in such a way that it lands on heads 60% of the time and tails 40% of the time. We can rep­re­sent this us­ing the fol­low­ing no­ta­tion:

Let’s as­sume the gam­bler is aware of this bi­as. Of course, the gam­bler will al­ways bet on head­s. But how much should he bet? Should he bet 10% of his stake? Should he bet 20%? Per­haps he should bet 30%? Maybe more? Maybe less? This is the ques­tion we want to an­swer. If the gam­bler bets too high, he risks de­plet­ing his cap­i­tal too quick­ly dur­ing a los­ing streak. If he bets too low, his wealth may not grow as quick­ly as it might oth­er­wise. We can be­gin to an­swer this ques­tion by con­sid­er­ing an ex­am­ple and then con­struct­ing a mod­el. Sup­pose the gam­bler plays 200 rounds of the game and bets 10% on each play:

And sup­pose the gam­bler starts with a bankroll of $100:

Based on these num­ber­s, we can use the fol­low­ing for­mu­la to cal­cu­late the size of the gam­bler’s hold­ings af­ter play­ing 200 rounds of the game:

Be­cause there is an el­e­ment of ran­dom­ness in­volved, there is no guar­an­tee that we’ll get the same re­sult every time we com­pute this val­ue. One se­ries of 200 coin toss­es might have a dif­fer­ent num­ber of heads and a dif­fer­ent num­ber of tails than an­oth­er se­ries. How­ev­er, the num­ber of heads and the num­ber of tails will al­ways add up to the same amoun­t:

To get an idea of what kind of re­sults we can ex­pec­t, we can run some sim­u­la­tions of the coin toss game us­ing a ran­dom num­ber gen­er­a­tor to mim­ic the coin toss. Here is a graph­i­cal rep­re­sen­ta­tion of three such sim­u­la­tion­s:

As you can see, these three dif­fer­ent sim­u­la­tions yield three very dif­fer­ent out­comes. The fol­low­ing ta­ble sum­ma­rizes the re­sult­s:

To es­ti­mate the ex­pect­ed val­ue of the gam­bler’s hold­ings af­ter 200 rounds of the coin toss game, we might run this sim­u­la­tion many more times and then take the mean of the re­sult­s. That would tell us the ex­pect­ed val­ue for a bet size of 10%. We would then have to re­peat this process again for every pos­si­ble bet size to de­ter­mine which bet size val­ue gives us the high­est ex­pect­ed re­sult. That might be a com­pu­ta­tion­al­ly ex­pen­sive thing to do. A bet­ter ap­proach might be to look at this prob­lem an­a­lyt­i­cal­ly. Keep in mind that what we re­al­ly care about here is not the gam­bler’s wealth af­ter a spe­cif­ic num­ber of rounds of the coin toss game. What re­al­ly mat­ters is the geo­met­ric mean of all the gains and loss­es tak­en in ag­gre­gate:

This gives us a growth rate fac­tor in­de­pen­dent of the num­ber of plays and the in­de­pen­dent of the gam­bler’s start­ing eq­ui­ty. This is the val­ue we want to max­i­mize. And specif­i­cal­ly, what we’re re­al­ly in­ter­est­ed in is the ex­pect­ed val­ue of the growth rate as a func­tion of the bet size:

As we will see short­ly, max­i­miz­ing the log­a­rithm of the growth rate is go­ing to be more con­ve­nient than max­i­miz­ing the growth rate it­self. For our pur­pos­es here, we will use the nat­ural log­a­rith­m. Con­sid­er the fol­low­ing:

Since the nat­ural log­a­rithm is a monoton­i­cal­ly in­creas­ing func­tion, max­i­miz­ing the growth rate is the same as max­i­miz­ing its log­a­rith­m. As men­tioned, we are in­ter­est­ed in the ex­pect­ed val­ue of the growth rate. But in this case, it is the log­a­rithm of the growth rate:

You can think about the ex­pect­ed val­ue in terms of the num­ber of heads and the num­ber of tail­s. If you play 200 round­s, the ex­pec­ta­tion is that the game will pro­duce 120 heads and 80 tail­s. This can vary based on ran­dom­ness, but these are the ex­pect­ed val­ues. We can plot the func­tion above to see what it looks like:

Us­ing vi­su­al in­spec­tion, we can ob­serve that the curve peaks where the bet size is 20%. But how do we solve this an­a­lyt­i­cal­ly? We want to be able to com­pute the most op­ti­mal bet size. Let’s use the fol­low­ing no­ta­tion to rep­re­sent the most op­ti­mal bet size:

The de­riv­a­tive of the growth rate func­tion is ze­ro at the max­i­mum, so we can say this:

Ex­pand­ing out the de­riv­a­tive of the growth rate func­tion, we get this:

Set­ting the de­riv­a­tive to ze­ro, we can solve for the op­ti­mal bet size:

We can sim­pli­fy this fur­ther know­ing that the sum of the prob­a­bil­i­ties adds up to one:

Here is the sim­pli­fied so­lu­tion:

And now, plug­ging in the prob­a­bil­i­ty val­ues that quan­ti­fy the bias of the coin, we can com­pute the an­swer to our orig­i­nal ques­tion:

Thus, our gam­bler would max­i­mize his win­nings by bet­ting 20% of his cur­rent bankroll for each round of the coin toss game. We can even take it a step fur­ther and com­pute his ex­pect­ed growth for rate for a sin­gle round of the game. Con­sid­er the re­la­tion­ship be­tween the orig­i­nal growth rate func­tion and its log­a­rith­mic equiv­a­len­t:

Us­ing the op­ti­mal bet size, we can cal­cu­late the op­ti­mal growth rate val­ues:

So now, if our gam­bler bets 20% of his stake each round, he can ex­pect to in­crease his wealth by 2.03% every time he plays. Of course, this is just an ex­pec­ta­tion. The ac­tu­al re­sult will vary every round based on the ran­dom­ness of the game.

Dif­fer­ent Pay­offs

In the pre­vi­ous ex­am­ple, the win­ning and los­ing amounts were the same. In this ex­am­ple, we are go­ing to see what hap­pens when there is an asym­met­ric pay­off rate. To make things in­ter­est­ing, we will start with a coin that on­ly lands on heads 30% of the time and tails for the re­main­ing 70% of the time:

As with the pre­vi­ous ex­am­ple, the gam­bler al­ways bets on head­s. You might think this would be a los­ing propo­si­tion since there will be more los­ing rounds than win­ning round­s. But if the pay­off for a win­ning round is high­er than the loss suf­fered on a los­ing round, the gam­bler still might have an ad­van­tage when bet­ting on head­s. In this case, let’s say the gain is four times the loss:

If the gam­bler choos­es a bet size of $1, he will on­ly lose $1 if the coin lands on tail­s. But if the coin lands on head­s, he will gain $4. This is enough to off­set the high­er loss rate. We can now up­date our mod­el of the game to ac­count for this type of asym­met­ric pay­of­f:

So now the ques­tion is, how much should the gam­bler bet? What per­cent­age of his hold­ings should he wa­ger if he wants to max­i­mize his win­nings? We can do the same analy­sis as be­fore to come up with the log­a­rith­mic growth rate as a func­tion of the bet size:

This is sim­i­lar to the growth rate func­tion in the pre­vi­ous ex­am­ple. Ex­cept now, we have tak­en in­to con­sid­er­a­tion the asym­met­ric pay­off­s. Here is a plot of this func­tion:

Like we did in the pre­vi­ous ex­am­ple, we can find the max­i­mum growth rate by find­ing the point at which the de­riv­a­tive of the growth rate func­tion equals ze­ro. Here is the de­riv­a­tive of the growth rate func­tion:

Set­ting the de­riv­a­tive to ze­ro and solv­ing for the bet size, we can ar­rive at the op­ti­mal bet size that max­i­mizes the growth rate:

Plug­ging in the val­ues, we get the con­crete so­lu­tion:

And now we can eval­u­ate the growth rate func­tion at the op­ti­mal bet size:

Ac­cord­ing to these cal­cu­la­tion­s, if the gam­bler us­es the op­ti­mal bet size of 12.5%, he can ex­pect to gain 2.86% every time he plays. De­spite los­ing more of­ten than he win­s, the gam­bler can still make a prof­it un­der the con­di­tions of this par­tic­u­lar game.

Mul­ti­ple Out­comes

This ex­am­ple is a gen­er­al­iza­tion of the pre­vi­ous ones. In­stead of us­ing a coin toss, here we con­sid­er a bet­ting game in which there are six pos­si­ble out­comes. Each out­come has a dif­fer­ent prob­a­bil­i­ty. And each out­come has a dif­fer­ent pay­off rate. The fol­low­ing ta­ble shows the pay­offs and prob­a­bil­i­ties of each out­come:

For our analy­sis, let’s use the fol­low­ing no­ta­tion to rep­re­sent the pay­off ma­trix above:

No­tice that, like in the pre­vi­ous ex­am­ples, the sum of the prob­a­bil­i­ties adds up to one:

Since there is a 10% chance that we can lose twice the bet size, there is an im­plic­it hard lim­it on the max­i­mum pos­si­ble bet size at 50%. We will ad­dress these kinds of lim­i­ta­tions in a lat­er sec­tion. For now, here is the growth rate func­tion that we want to max­i­mize:

While there are six pos­si­ble out­comes in this ex­am­ple, there can be an ar­bi­trary num­ber of pos­si­ble out­comes in the gen­er­al case. Be­cause of this, it is best to rep­re­sent the growth rate func­tion as a sum­ma­tion. Here is a plot of the growth rate func­tion:

Now at this point, we might try the same ap­proach that we did be­fore in the last two ex­am­ples. We can take the de­riv­a­tive of the growth rate func­tion and then find the point at which the de­riv­a­tive is equal to ze­ro. But look at this de­riv­a­tive:

Can you solve this? Go ahead and try it. But it might just be eas­i­er to use a nu­mer­i­cal root-find­ing method. And if you’re go­ing to use a nu­mer­i­cal method, you might al­so con­sid­er us­ing a nu­mer­i­cal op­ti­miza­tion method on the growth rate func­tion it­self in­stead of both­er­ing with the de­riv­a­tive. I de­cid­ed to use the Nelder–Mead method im­ple­ment­ed by a third-par­ty li­brary to find the bet size val­ue that max­i­mizes the growth rate func­tion. Here is the re­sult:

And now, we can plug this val­ue in­to the growth rate func­tion:

For this game, the op­ti­mal bet size is 15.71%. The gam­bler can ex­pect to grow his wealth by 2.35% every round if he plays the op­ti­mal bet size. No­tice that, in this case, un­like the last two ex­am­ples, the bet size val­ue does not rep­re­sent the max­i­mum amount the gam­bler stands to lose. In­stead, it is sim­ply a mul­ti­pli­er or a scal­ing fac­tor. Based on the pay­off ma­trix for this par­tic­u­lar ex­am­ple, the gam­bler can lose up with twice the bet size on any giv­en round.

Stock Trad­ing

Now let’s see how to ap­ply this op­ti­mal bet siz­ing tech­nique to stock trad­ing. You can think of stock mar­ket trad­ing as a form of gam­bling. That is the best anal­o­gy. Sup­pose the gam­bler—or the trader, if you want to call him that—spec­u­lates that the price of a stock will close at one of six pos­si­ble price points af­ter a spe­cif­ic amount of time:

The price points range from as low as $18 per share to as high as $23 per share, and the gam­bler’s hy­poth­e­sis as­so­ciates dif­fer­ent prob­a­bil­i­ties with each out­come. Let’s as­sume that the stock has a cur­rent mar­ket val­ue of $20 per share:

This is the price point at which the trad­er can en­ter in­to a po­si­tion. To de­ter­mine the pay­off rate for each pos­si­ble out­come, we can use a pay­off func­tion that takes the en­try price in­to con­sid­er­a­tion. Here is the pay­off func­tion:

Here are the com­put­ed pay­off val­ues for each one of the pos­si­ble out­comes:

Here is the growth rate func­tion that we want to max­i­mize:

No­tice how the growth rate func­tion us­es the pay­off func­tion in­side the sum­ma­tion. Plot­ting this func­tion on a chart, here is what it looks like:

As with the pre­vi­ous ex­am­ple, we can use a nu­mer­i­cal op­ti­miza­tion method to find the most op­ti­mal bet size that max­i­mizes the growth rate. Here is the re­sult:

This val­ue is greater than one. What does this mean? It means that our trad­er would have to use lever­age if he wants to take the most op­ti­mal po­si­tion in this stock. A trad­er with a $10,000 ac­count would have to buy $31,422 worth of this stock­—about 1,571 shares. Us­ing this amount of lever­age may or may not be pos­si­ble. As­sum­ing it is, here is the ex­pect­ed growth rate:

No­tice the sim­i­lar­i­ty to the pre­vi­ous ex­am­ple. In fac­t, this is a re­peat of the pre­vi­ous ex­am­ple. The on­ly dif­fer­ence is that the pay­off val­ues are scaled down by a fac­tor of twen­ty. And this re­sults in the op­ti­mal bet size be­ing scaled up by a fac­tor of twen­ty. But the op­ti­mal growth rate re­mains the same for both ex­am­ples.

Con­tin­u­ous Dis­tri­b­u­tions

The pre­vi­ous ex­am­ple used what is ba­si­cal­ly a dis­crete prob­a­bil­i­ty mass func­tion to mod­el the pro­ject­ed be­hav­ior of stock price move­ments. Since stock prices can move along a con­tin­u­ous range of val­ues, it might be more re­al­is­tic to use a con­tin­u­ous mod­el in­stead of a dis­crete mod­el. In this ex­am­ple, we will use a log-nor­mal prob­a­bil­i­ty den­si­ty func­tion:

This func­tion ranges from ze­ro to in­fin­i­ty, and the area un­der the curve is one:

For a log-nor­mal prob­a­bil­i­ty den­si­ty func­tion, we need to pro­vide ap­pro­pri­ate val­ues for the mean and the stan­dard de­vi­a­tion. How to come up with these val­ues is be­yond the scope of this dis­cus­sion, so we will just make up some fic­ti­tious num­bers for the sake of ex­am­ple:

Us­ing these val­ues for the mean and stan­dard de­vi­a­tion, we can plot the prob­a­bil­i­ty den­si­ty func­tion to see what it looks like. Here is the plot:

This is our spec­u­la­tive fore­cast of what the stock price might be af­ter a spe­cif­ic amount of time has passed. As you can see from look­ing at the chart, the price is most like­ly to fall some­where be­tween $15 per share and $25 per share. But there is a small chance that it could fall out­side this range as well. For this ex­am­ple, we will as­sume the cur­rent price at which we can en­ter in­to a po­si­tion is $20 per share, and we will bound the ex­it point to some­where be­tween $17 per share and $23 per share:

We are as­sum­ing ide­al­ized ex­it point bound­aries here. We can nev­er gain or lose more than $3 per share. In re­al­i­ty, en­force­ment of this bound­ary con­di­tion might take the form of a stop-loss or­der at $17 and a take-prof­it or­der at $23. Or it might al­so take the form of a long po­si­tion in a put op­tion with a strike price of $17 and a short po­si­tion in a call op­tion with a strike price of $23, dis­re­gard­ing any pre­mi­ums paid or re­ceived. Us­ing these val­ues, here is our pay­off func­tion:

This pay­off func­tion is a piece­wise func­tion that lim­its the loss and caps the gain at the ex­it point bound­aries. While this func­tion is not nec­es­sar­i­ly dif­fer­en­tiable at all points, note that it is con­tin­u­ous at all points. Here is a graph­i­cal rep­re­sen­ta­tion of this func­tion:

Giv­en the con­tin­u­ous prob­a­bil­i­ty den­si­ty func­tion and the con­tin­u­ous pay­off func­tion, we can now com­pute the ex­pect­ed growth rate as a func­tion of the bet size. In­stead of us­ing a sum­ma­tion as we did pre­vi­ous­ly, in this ex­am­ple, we will use an in­te­gral:

This is the growth rate func­tion that we want to max­i­mize. Since the pay­off func­tion used here is a piece­wise func­tion, we might want to break this in­te­gral up in­to three dif­fer­ent par­ti­tion­s:

I have not fig­ured out how to boil this in­te­gral down to an ex­pres­sion in an­a­lyt­ic for­m. In par­tic­u­lar, it is the mid­dle par­ti­tion I have not been able to find the so­lu­tion for. How­ev­er, this prob­lem can be solved us­ing nu­mer­i­cal in­te­gra­tion. In this case, I am us­ing the Gauss–Le­gendre method to find an ap­prox­i­ma­tion. Here is what this func­tion looks like on a chart:

Us­ing nu­mer­i­cal op­ti­miza­tion on top of nu­mer­i­cal in­te­gra­tion, we can com­pute the op­ti­mal bet size that max­i­mizes the growth rate func­tion. Here is the re­sult:

Just like the pre­vi­ous ex­am­ple, this is a lever­aged po­si­tion. Here is the max­i­mum growth rate us­ing the op­ti­mal bet size:

Ad­mit­ted­ly, this ex­am­ple is a bit ide­al­is­tic. It as­sumes that the trad­er can re­peat the same bet again and again. But in re­al life, mar­ket con­di­tions can change over time. The price point where the trad­er can en­ter a po­si­tion can change as the tide of the mar­ket goes up and down, chang­ing the pay­off func­tion. And the prob­a­bil­i­ty dis­tri­b­u­tion of the pro­ject­ed price move­ments can change as well. In fac­t, the price move­ments might be dif­fi­cult to mod­el ac­cu­rate­ly in the first place.

Short Po­si­tions

The op­ti­mal bet size does not al­ways have to be a pos­i­tive num­ber. Some­times it can be a neg­a­tive num­ber. A neg­a­tive num­ber can in­di­cate that the bet on of­fer is a los­ing bet that is not worth tak­ing. It might al­so rep­re­sent an op­por­tu­ni­ty for prof­it if the gam­bler is able to take the oth­er side of the bet. For a stock trader, a neg­a­tive val­ue in­di­cates that the most ad­van­ta­geous thing to do would be to take a short po­si­tion. Let’s re­peat the pre­vi­ous ex­am­ple us­ing a slight­ly dif­fer­ent prob­a­bil­i­ty den­si­ty func­tion. We’ll use the fol­low­ing mean and stan­dard de­vi­a­tion val­ues:

Here is a plot of the log-nor­mal prob­a­bil­i­ty den­si­ty func­tion us­ing these val­ues:

The peak of the curve is shift­ed a bit to the left com­pared to the den­si­ty func­tion used in the pre­vi­ous ex­am­ple. For this ex­am­ple, we are go­ing to use the same en­try price and ex­it point bound­aries that we used in the pre­vi­ous ex­am­ple:

As you might ex­pec­t, the pay­off func­tion is ex­act­ly the same as well:

The ex­pect­ed val­ue of the growth rate as a func­tion of the bet size looks like this:

No­tice the max­i­mum val­ue ap­pears on the neg­a­tive side of the ver­ti­cal ax­is this time around. Com­put­ing the op­ti­mal bet size nu­mer­i­cal­ly, this is the re­sult:

It is a neg­a­tive num­ber, in­di­cat­ing that the trad­er should sell the stock short. Here is the ex­pect­ed growth rate cal­cu­la­tion if the trad­er takes the op­ti­mal­ly sized short po­si­tion:

I think it’s worth men­tion­ing here that this re­sult is the po­lar op­po­site of the op­ti­mal bet size val­ue found in the pre­vi­ous ex­am­ple. This is sur­pris­ing be­cause these two ex­am­ples are near­ly iden­ti­cal. The on­ly dif­fer­ence is in the mean and stan­dard de­vi­a­tion val­ues used in the prob­a­bil­i­ty den­si­ty func­tion­s. And the dif­fer­ence is­n’t that great. The two prob­a­bil­i­ty den­si­ties are not that much dif­fer­ent from one an­oth­er. I think this il­lus­trates how sen­si­tive this bet size op­ti­miza­tion tech­nique is to the ac­cu­ra­cy of the mod­el used to quan­ti­fy the game.

Con­straints and Lim­i­ta­tions

As al­lud­ed to ear­li­er when dis­cussing mul­ti­ple out­comes, there are hard lim­its to how much a gam­bler can bet or how much lever­age a trad­er can use. When us­ing the Kel­ly cri­te­ri­on, if there is any chance at al­l, no mat­ter how small it might be, that the gam­bler can lose his en­tire stake on any one round of the game, then the ex­pect­ed growth rate is ze­ro. This is the ex­treme case of the gam­bler bet­ting an amount high­er in mag­ni­tude than the op­ti­mal bet size. In the stock trad­ing mod­el il­lus­trat­ed in the last two ex­am­ples, we must take these lim­its in­to con­sid­er­a­tion when al­low­ing the trad­er to take a lever­aged po­si­tion. Sup­pose a trad­er takes a po­si­tion in a stock at an en­try price of $20 per share:

Set­ting the low­er ex­it point val­ue to $15 per share would con­strain the max­i­mum bet size for a long po­si­tion to four times the trad­er’s ac­count val­ue. If the trader were to lose more than $5 per share with this amount of lever­age, his ac­count would go in­to neg­a­tive ter­ri­to­ry. Like­wise, tak­ing a short po­si­tion with this amount of lever­age would con­strain the up­per ex­it point val­ue to no more than $25 per share. Here is how to cal­cu­late the ex­it point lim­its in re­la­tion to the bet size:

For a long po­si­tion, there is ef­fec­tive­ly no lim­it to the up­per ex­it point. It can go to in­fin­i­ty. And for a short po­si­tion, the low­er ex­it point can go all the way to ze­ro with­out putting any re­stric­tions on the bet size. In the case of an un­lever­aged long po­si­tion, both the low­er and the up­per ex­it points can be un­lim­it­ed. Here is a plot of the for­mu­las above to give a vi­su­al il­lus­tra­tion:

The sol­id lines show the hard lim­its that con­strain the up­per and low­er ex­it points. The shad­ed re­gions show the valid ar­eas where the ex­it points can be placed. But this does­n’t tell the whole sto­ry. The stock trader’s bro­ker might have mar­gin re­quire­ments that im­pose ad­di­tion­al con­straints on the above. Sup­pose the bro­ker has a 50% ini­tial mar­gin re­quire­ment to en­ter in­to a po­si­tion and a 25% main­te­nance mar­gin re­quire­men­t:

The ini­tial mar­gin re­quire­ment puts a cap on the mag­ni­tude of the bet size. The al­low­able range of bet size val­ues is in­verse­ly pro­por­tion­al to the ini­tial mar­gin re­quire­men­t:

We can al­so use the fol­low­ing no­ta­tion to de­scribe the ex­tremes of the bet size val­ues:

Tak­ing the main­te­nance mar­gin re­quire­ment in­to con­sid­er­a­tion, we can now re­vise the for­mu­las for the ex­it point lim­it­s:

The mar­gin re­quire­ments add ad­di­tion­al con­straints both hor­i­zon­tal­ly and ver­ti­cal­ly. Here is the re­vised vi­su­al­iza­tion:

As you can see, the range of ex­it point val­ues that would al­low the trad­er to avoid a mar­gin call is a sub­set of the range of val­ues that can be con­sid­ered un­der the Kel­ly cri­te­ri­on. To per­form the op­ti­mal bet size analy­sis un­der these con­straints, there are dif­fer­ent ap­proach­es you can take. You might try set­ting the ex­it point val­ues to the up­per and low­er lim­its for the most ex­treme bet size val­ues and then per­form­ing the op­ti­miza­tion pro­ce­dure to find the most op­ti­mal bet size across all pos­si­ble bet size val­ues. Al­ter­na­tive­ly, you might choose the ex­it point val­ues first based on some cri­te­ria and then lim­it the range of pos­si­ble bet size val­ues to those that al­low for the cho­sen range of ex­it points.

Psy­chol­o­gy and Vari­ances

In ad­di­tion to the hard lim­its de­scribed above, there may al­so be some soft lim­its to how big the bet size can be. The fo­cus of this study has been on find­ing the most op­ti­mal bet size, but we did not ad­dress the vari­ance of pos­si­ble out­comes for a re­peat­ed bet­ting game. Even when there is a pos­i­tive ex­pec­ta­tion, the gam­bler may ex­pe­ri­ence los­ing streaks that cause him to sec­ond guess him­self. Tak­ing the most op­ti­mal bet size might be a psy­cho­log­i­cal­ly dif­fi­cult thing to do, and the gam­bler might be more com­fort­able tak­ing a small­er bet. It might be worth do­ing a fur­ther study to un­der­stand how much vari­a­tion the gam­bler might ex­pect in the pos­si­ble out­comes of play­ing a re­peat­ed game of chance.

Ac­com­pa­ny­ing source code is avail­able on GitHub.