Kelly criterion

[Milkin' it Slowly]

In plain english ... Can someone practically explain what it is. I read read an this article

http://en.wikipedia.org/wiki/Kelly_criterion

and quickly figured out that this is too advanced for my math level, without being explained . While the Martindale, Anti-Martidale, the"1-3-2-6" and others will leave you broke, is this in same catagory of "system". How would someone figure out how much of an percentage you have to win as opposed to a lose(ie 60%/40%).

So say I bet 6% of may roll on +165 dog how would that translate.

[Art Vandeleigh]

I think you need to know a lot about logarithms before you can talk about Kelly.

Go look at a guitar. See the distance between frets on the guitar neck? See how the distance of each fret gets smaller and smaller as you move further and further away from the neck?

That, is the natural logarithm at work. The distance between frets is reduced in proportion to the natural logarithm e.

Also charge across a capacitor decays in proportion to 1/e. I think.

Anyway, learn your logarithms real good before asking this type of question here.


Also, the real issue you need to deal with when sports gambling is not so much the Kelly Criterion. Evertyone knows it's Murphy's law that rules this jungle.

Anything that can go wrong will. The chances of the side of toast with the jam on it landing on the carpet is proportional to the cost of the carpet. That's the sort of thing you should be studying and concerned with.

Just my 2 cents.

[Milkin' it Slowly]

Thanx Art... You're right ... It just sounded interesting...

[Art Vandeleigh]

As I understand it, which isn't very well, this is my take on the Kelly, if you're having a bad streak.

Say you had an apple pie straight out of the oven, and you cut it into eight pieces. You have made a linear dissection of this pie.

Then you ate the 8 pieces (time taken to consume is irrelevant)

This is the equivalent of eating through your bankroll.

Now with Kelly, you don't cut your pie immediately into 8 pieces. You measure the size of the new piece you will cut only after eating the previous piece(s). This new cut will be calculated using logarithms, causing a decrease in the size of each piece.

You can cut the pie an infinite amount of times using this method and never finish the pie. This is one reason Kelly is so good.

You will never go hungry again.

[Dark Horse]

Ganch is the expert on this. He's also a mathematician. That is not an unrelated fact.

[Hulu]

Kelly's system is good math and has many practical applications in finance and gambling but for my purposes, there are 2 problems.

1st it is much more difficult in sports betting to quantify your edge. A relatively accurate analysis of your edge on a given event is needed to calculate the proper stake using Kelly. The farther off you are, the less your results will be optimal.

2nd, although Kelly will guarantee that you will never go 'technically' broke, it will allow you to go 'practically' broke. To use Art's example, if you continually cut smaller pieces of the pie, you will always have some pie, however, once you get down to the crumbs, for all intents and purposes you have no pie. So if you whittle a bankroll down from $1000 to $10 technically you haven't gone broke, but practically you are broke.

[Arilou]

Kelly for those who hate math:

Bet in proportion to your edge and in proportion to your bankroll. In particular, bet a percentage of your bankroll equal to your advantage. So if you have a 5% edge, try to win 5% of your bankroll. If you bet 6% of your roll on a +165, that would mean that you are trying to win 10% of your bankroll, so you think that bet has a 10% edge.

Hulu's first point is Kelly's huge issue: No one knows what their edge really is, and when they guess they guess too high. The second I don't agree with, because any bankroll you're not prepared to lose virtually all of is false bankroll: If I have $10,000, but I'm not prepared to ever lose more than $5000 of it, then I have a $5000 bankroll and an extra $5000 I can use to make sure I have money where I need it. I don't have a $10,000 bankroll.

[trustbutverify]

As far as speculation/gambling/investment is concerned:

Kelly is a formula which maximizes total return for a given advantage(pct)- over the course of a set of discrete(seperate) events. It is progressive- like the other strategies you named, so the calculation is made on each event based on total bankroll available.

In other words, if you KNOW your exact advantage for a given bet kelly tells you exactly what pct of your bankroll to wager on that event for max long term results. Betting any more or less will not result in max returns IFF you meet your projected winning pct exactly.

The kelly approach is designed to bet more on events(games) where you have a bigger long term advantage and less when you have a smaller advantage. And, obviously, you bet bigger when your bankroll is bigger and vice-versa. The power of the calculation is that it gives you the exact bet amount to Extract max value for a given win pct and bankroll size.

For what it's worth I think kelly is a bad idea for sports investing. When dealing with outcomes of dice and cards, exact advantages can be known. In sports your are modeling history and hoping an apparent advantage over the line continues in the future.



IMO- the prime function of money management is to reduce the break even point to its lowest possible level. Varying bet size increases the break even point- in general.

[Hulu]

The second I don't agree with, because any bankroll you're not prepared to lose virtually all of is false bankroll: If I have $10,000, but I'm not prepared to ever lose more than $5000 of it, then I have a $5000 bankroll and an extra $5000 I can use to make sure I have money where I need it. I don't have a $10,000 bankroll.

You make a good point. My second point is only a response to those who oversimplify the Kelly strategy by saying "Use Kelly and you can never go broke!". While literally true, this is a dangerous thing for someone who doesn't understand the math behind it.

I guess the question is...how does one quantify his/her edge? I have historical data for my plays that numbers into the several thousands of bets but the math for that one is a bit beyond me.

[Ganchrow]

You've gotten some really good answers here, so I don't have all that much to add.

Now with Kelly ... this new cut will be calculated using logarithms, causing a decrease in the size of each piece.

While logarithms are part of the derivation of Kelly, you don't need to use logarithms to calculate the simple one-asset solution.

You can cut the pie an infinite amount of times using this method and never finish the pie. This is one reason Kelly is so good.

This feature is common to all percentage betting systems and not just Kelly. Furthermore, as other have already pointed out, you might eventually reach betting granularity using Kelly. This means that at some point the optimal bet recommended by Kelly could be lower than the minimum accepted wager at any given book. At this point you could effectively be considered broke.

1st it is much more difficult in sports betting to quantify your edge. A relatively accurate analysis of your edge on a given event is needed to calculate the proper stake using Kelly. The farther off you are, the less your results will be optimal.

Determining one's edge in sports betting is a skill like anything else. I've always advocated that when handicapping, bettors should get into the habit of setting their own lines with a view towards coming up with ever-improving estimates of edge.

Does it really make sense to claim, "I know my win probability is above 52.38% on this -110. I just don't know how much more." As TLD, a rather sharp poster on this forum, once said (and I paraphrase), "When flat betting, your implicit answer to the question of 'How likely do I think this play is to win?' is essentially 'Every game I bet has an equal chance of winning.' That's not neutral. When you flat bet, you aren’t being agnostic as to the degree of your edge; you’re taking a stand that the edge is equal for all bets." (See http://forum.sbrforum.com/players-ta...our-stand.html for some of my minor criticisms of this hard-line stance. Still, to be clear, I do largely agree with TLD here.)

On the other hand, the situation is actually somewhat more dire than Hulu says. Kelly is a maximally effective strategy. This means if you bet even a little more than suggested by Kelly you run the risk of your expected bankroll growth (average bankroll growth per bet over an arbitrarily large number of bets) going negative.

For example given a -110 and a 55% probability of winning, your Kelly stake would be 5.4950% of your bankroll. This translates into a mean bankroll growth of 0.2745% per bet and a median bankroll growth of 0.1376% (because Kelly is a percentage-based system, the median bankroll growth is always lower than the mean).

Now, if you had actually misjudged your odds of winning this bet and thought they were actually 57%, you'd be betting 9.6953% of bankroll per bet. This implies a mean bankroll growth of 0.4843% (which is great) but a median bankroll growth of -1.8902% (which isn't so great). This means that after 250 bets the most likely bankroll level would be a paltry 0.8474% of initial. (Of course your expected bankroll would be 334.6207% of initial; but that's only because a handful of really large outcome skew the distribution -- once again this is typical for a percentage-based system). See the Kelly calculator here.

So the moral of the story is this: When betting Kelly, NEVER NEVER NEVER overestimate your edge -- if you do err try to always err on the side of underestimation.

Bet in proportion to your edge and in proportion to your bankroll. In particular, bet a percentage of your bankroll equal to your advantage. So if you have a 5% edge, try to win 5% of your bankroll. If you bet 6% of your roll on a +165, that would mean that you are trying to win 10% of your bankroll, so you think that bet has a 10% edge.

Exactly. But to be perfectly clear, the way I might phrase it would be, "bet such that you if you win, you win a percentage of bankroll equal to your percent edge." So just as on your example. If you have an edge of 5% and are betting at +165, you'd bet 3.03% of bankroll to win 5%. Similarly, if you have an edge of 5% and are betting at -200, you'd bet 10% of your bankroll to win 5%. And of course this only applies to the single asset model for win-only (no push) bets.

Hulu's first point is Kelly's huge issue: No one knows what their edge really is,

I think kelly is a bad idea for sports investing. When dealing with outcomes of dice and cards, exact advantages can be known. In sports your are modeling history and hoping an apparent advantage over the line continues in the future.

I'd argue that there's no reason why one shouldn't be able to determine unbiased, statistically meaningful forecasts of the outcome probabilities of a given sporting event. If one believes this can be done in quantitative finance (which I spent 10 years doing) there's no reason to doubt it can be done just as accurately in quantitative sports betting. Just because someone doesn't understand how to properly determine robust forecasts within a quantitative modelling framework, means neither that others can't do it, nor that that person can't eventually learn how to do it.

The second I don't agree with, because any bankroll you're not prepared to lose virtually all of is false bankroll: If I have $10,000, but I'm not prepared to ever lose more than $5000 of it, then I have a $5000 bankroll and an extra $5000 I can use to make sure I have money where I need it. I don't have a $10,000 bankroll.

This is actually indicative of probably the single biggest "smart-person misconception" about Kelly -- specifically that one's bankroll only includes that which the bettor can afford to lose. This is in fact untrue. One's Kelly bankroll is actually one's entire marked-to-market cash balance (properly discounted of course). That means your bankroll would consist of the value of not just your offshore betting account, but also the value of your checking account, the value of your savings account, the equity in your house, the maximum cash advance level on each of your credit cards, the maximum amount you could borrow from your family and friends, the maximum amount you could borrow from your loan shark, the $3,000 cash your elderly neighbor keeps under her mattress, etc., etc. Of course each of these sums would need to be properly discounted to reflect the cost of obtaining them (a cost which could potentially be so great as to make the sources of cash essentially valueless, but that's beside the point), but as far as Kelly is concerned your bankroll should represent the dollar figure such that if you lost it your life would be as good as over. Another way to look at it is like this, let's say you had an even odds bet that you knew a priori would win with 100% likelihood -- how much would you bet? The logical answer would of course be, "every dollar you could safely get your hands on."

Now people might very well object when they read this, saying that this bankroll valuation just doesn't make any sense, and that no one would want to bet in this matter, etc. etc. And you know what? You'd probably be right. Kelly assumes logarithmic preferences and as I've mentioned many times before most human just don't have log prefs. So to get around this issue, people often claim (in fact I don't know anyone who doesn't) that a Kelly bankroll is only what the bettor would feel comfortable losing. That's all well and good -- but to be perfectly clear that's a compromise position and doesn't represent "true" Kelly.

In conclusion, the Kelly stake represents the optimal bet size as percentage of total bankroll that should be bet if the bettor's goal were to maximize the expected growth rate of that bankroll. (In fact, this is equivalent to saying that the bettor has log preferences.) Were that bit your goal, and it probably isn't, then strictly, strictly speaking Kelly's not for you. (The ambitious might consider implementing Kelly using a amore appropriate utility function. This actually isn't too difficult to figure computationally for well-behaved, convex preferences.) But that doesn't mean that you couldn't use a version of it with which you're sufficiently comfortable.

[Willie Bee]

One of these days I'm going to learn to speak Ganchrow

[Ganchrow]

One of these days I'm going to learn to speak Ganchrow

If I can't explain concepts clearly enough for others to understand, well that's my problem -- not anyone else's.

Please, if anyone wants clarification of anything I've written here, just ask.

[Willie Bee]

I was just being my usual smartass (dumbass?) self, Ganch.

[The HG]

Ok I'll take a crack at a plain English explanation

Kelly simply tells you what the best amount of your entire bankroll to bet is on a certain bet, assuming certain odds and a certain chance of winning the bet.

For instance, let's say someone offers you 2-1 odds on a coin flip. So you know your chances of winning are 50%, and the odds you are getting are +200. So that's a great bet, and you're definitely going to bet money on it. But how much? Let's say you've got $10,000 to your name (for practical purposes we'll ignore the value of your house, the amount you could borrow, etc etc). Obviously you're going to bet way more than $1, since the bet is so good. But of course you're also not going to bet all $10,000, because you know you could easily lose the bet. So how much will you bet? You might say OK, I'll bet $2,000. Then you might say no screw it, this is a great bet, I'll bet $5,000! Well the Kelly equation tells you exactly how much you "should" bet. Anything more or less, and you are betting "sub-optimally". The Kelly percentage tells you what percentage of your bankroll to bet that will grow your expected bankroll most quickly over time. The Kelly equation is actually pretty simple, and in this case, it spits out 25%. So your optimal bet would be $2,500.

Kelly is most useful when you are making repeated bets/investments, with varying edges. In the coin flip example, you would want to bet 25% of your bankroll each time, regardless of whether you won or lost your previous flips.

There is one huge caveat when using Kelly percentages in real-world betting situations. People have mentioned it before, but it bears repeating. The essential number in the Kelly equation is your chance of winning the bet. Obviously with a fair coin flip, you know your chance exactly. In sports betting, you are making an educated guess (anywhere on the spectrum from "reasonably well-educated" to "not at all educated"). With Kelly, if you overestimate your edge, you are, in plain English, massively screwing yourself. If you assume you have a 57% chance of winning each of 100 bets at -110 odds, but your actual chance of winning each game is only 54%, that is, in plain English, really really bad. The fact that you do have a 54% edge means little, you are likely to wind up losing a lot if you bet according to Kelly as though you have a 57% chance of winning those bets.

In other words, if you can handicap games well enough to come out ahead, that's great. The next question is, what should your betting strategy be to maximize your profits on the edge you have from handicapping? In theory, the answer is Kelly. BUT, if you overestimate how well you can handicap, and you use the Kelly strategy, your most likely outcome will be to lose money, even if you can in fact handicap well enough to have an edge.

OK that's my stab at a plain English contribution to the Kelly explanations. Hope it helps, I have, unfortunately, been speaking plain English for many years now.

[Dark Horse]

Very nicely put.

The problem I have with it is simple. I could easily see a 58-60% bettor, i.e. someone who bets a large chunk of his bankroll with the Kelly criterion, experience a month during which his percentage dropped to 40%. If he made a hundred bets during this period, the result of this fluctuation, which has little or no effect on his long term winning percentage, would be ... not pretty.

While not mathematically precise, 'not pretty' could mean jumping off a bridge after having build up a bankroll for many years, only to (effectively) wipe it out during one short stretch, due to the power of short term fluctuations (streaks).

[Ganchrow]

The problem I have with it is simple. I could easily see a 58-60% bettor, i.e. someone who bets a large chunk of his bankroll with the Kelly criterion, experience a month during which his percentage dropped to 40%. If he made a hundred bets during this period, the result of this fluctuation, which has little or no effect on his long term winning percentage, would be ... not pretty.

I'd just point out that a real 60% handicapper has only a 0.004247% probability of picking 40% or less over 100 picks (that's odds of about 23,547:1 against). If this occured it would then take you about 4.29 months of average performance to get your bankroll back to where it started.

[Dark Horse]

Streaks are real. I learned this in high school when playing double or even flipping a coin with a friend after a math class in statistical probability. We started out with a small bet. I ended up winning twelve times in a row. By the eight time I didn't want to bet anymore, but my friend insisted that he flip the coin one more time. And so on. The chance of this happening, I just calculated, was 0.00024414%.

He did pay me. And it took a while to smoke.

[Dark Horse]

Ganch, I really admire your faith in math, and I hope it's making you stinking rich. Purely from a romantic's perspective, however, it may even be better if, after a prosperous life, you ended up poor, but with a completely undisturbed faith in math. That would be a movie. (Rich would only be a 'story').

I realize there's a contradiction between faith and math, but that's the point.

Going back to my space shuttles, what would be the chance of losing two space shuttles within 20 years? If they launched five shuttles a year (I'm just guessing here), that would be 100 shuttles, out of which 2 were lost. That's a 0.02% chance! I'm quite certain that astronauts may have thought twice about those odds. But they wouldn't have had to, because the mathematical formulas would have presented a far more favorable picture. Maybe something in the order of 0.0002. What interfered with math/science? Real life.

[Hulu]

Just because someone doesn't understand how to properly determine robust forecasts within a quantitative modelling framework, means neither that others can't do it, nor that that person can't eventually learn how to do it.

I find this absolutely fascinating. Can you recommend where someone like myself, who has never been good at or paid much attention to mathematical theory, get started learning? Is there a 'Forecasting Within a Quantitative Modelling Framework for Dummies' book out there?

PS Thanks for taking the time to explain these advanced (to me anyway) concepts here. It isn't always understood, but it is always appreciated.

[Ganchrow]

I find this absolutely fascinating. Can you recommend where someone like myself, who has never been good at or paid much attention to mathematical theory, get started learning? Is there a 'Forecasting Within a Quantitative Modelling Framework for Dummies' book out there?

Probably the two books that best demonstrate basic quantitiave analysis technique in sportsbetting are Fixed Odds Sports Betting: Statistical Forecasting and Risk Management by Jospeh Buchdahl and Sharp Sports Betting by Stanford Wong.

[Ganchrow]

Streaks are real. I learned this in high school when playing double or even flipping a coin with a friend after a math class in statistical probability. We started out with a small bet. I ended up winning twelve times in a row. By the eight time I didn't want to bet anymore, but my friend insisted that he flip the coin one more time. And so on. The chance of this happening, I just calculated, was 0.00024414%.

I assume you meant 0.024414%.

But are you claiming that the odds of this occurring with a fair coin are under certain circumstances more than 0.024414%?

If so, do you have any evidence to back up this assertion? You could have a million dollars coming to you courtesy of the James Randi Educational Foundation's $1 Million Paranormal Challenge.

[Ganchrow]

Ganch, I really admire your faith in math, and I hope it's making you stinking rich. Purely from a romantic's perspective, however, it may even be better if, after a prosperous life, you ended up poor, but with a completely undisturbed faith in math. That would be a movie. (Rich would only be a 'story').

It's not about faith in math, it's about comprehension.

Going back to my space shuttles, what would be the chance of losing two space shuttles within 20 years? If they launched five shuttles a year (I'm just guessing here), that would be 100 shuttles, out of which 2 were lost. That's a 0.02% chance! I'm quite certain that astronauts may have thought twice about those odds. But they wouldn't have had to, because the mathematical formulas would have presented a far more favorable picture. Maybe something in the order of 0.0002. What interfered with math/science? Real life.

I don't understand what you're saying here at all.

I would point out that according to Wikipedia, "the original disaster potential, was estimated during Shuttle development at one every 75 missions." That's pretty close to the actual figure -- which was 2 disasters over 115 missions. However, even if we were to stick with the original 1/75 probability estimate, the chances of seeing two or more disasters over 115 missions would be 45.45%. Hardly an unlikely event.

[Dark Horse]

0.02% is one in fifty.

To flip heads twelve straight times is not 0.02%
Six times is less than that.

[Ganchrow]

0.02% is one in fifty.

Nope. 0.02% is 1 out of 5,000.

To flip heads twelve straight times is not 0.02%

Indeed it is. it's 2^-12 ≈ 0.000244141 = 0.0244141%

[Dark Horse]

They estimated shuttle disaster at once every 75 missions?!

I didn't know that.

Was that decided before the disasters or after? Always easy to rewrite history after the fact, so just making sure that these astronauts were knowingly taking that type of a chance with their lives.

Imagine being on the 75th flight.
"Relax. No need to sweat it. We're on a 74-0 streak, buddy. Our chances as good as ever."
"Errr... Did you read the small print?"

[Dark Horse]

OK. My math language is rustier than an old shuttle.

1 in 5000 sounds right.

[Ganchrow]

They estimated shuttle disaster at once every 75 missions?!

I didn't know that.

Was that decided before the disasters or after? Always easy to rewrite history after the fact, so just making sure that these astronauts were knowingly taking that type of a chance with their lives.

According to Wikipedia these were ex-ante estimates.

But even if that were completely untrue what would that prove other than the fact that people sometimes make mistakes? And that conclusion violates no mathematical principles of which I'm aware.

Imagine being on the 75th flight.

And if you were, and if the 1/75 estimate were correct, your probability of disaster would be .. guess what? ... 1.33% or 1/75.

[trustbutverify]

0.02% is one in fifty.

To flip heads twelve straight times is not 0.02%
Six times is less than that.

2% is 1 in 50

.02% is 1 in 5000

[trustbutverify]

sorry- got in late

[Dark Horse]

And if you were, and if the 1/75 estimate were correct, your probability of disaster would be .. guess what? ... 1.33% or 1/75.

I knew you would say that.

But the question is: would you step on that flight just as easily?


By the way, are you saying that over 1000 flights, the chance of disaster remains 1/75th per flight? Or does that chance begin to approach 1 as the flight number increases?

[trustbutverify]

I knew you would say that.

But the question is: would you step on that flight just as easily?


By the way, are you saying that over 1000 flights, the chance of disaster remains 1/75th per flight? Or does that chance begin to approach 1 as the flight number increases?

remains the same

[Ganchrow]

I knew you would say that.

Am I really that predictable?

But the question is: would you step on that flight just as easily?

Assuming my confidence in the estimates of disaster probability were held constant, I'd be no more or less less likely to get on the 75^th flight than I would on any of the 1^st through 74^th flights.

By the way, are you saying that over 1000 flights, the chance of disaster remains 1/75th per flight? Or does that chance begin to approach 1 as the flight number increases?

All else being equal (assuming no wear and tear, for example), the probability would remain constant.

[trustbutverify]

Ganch- As you have stated- the entire issue really revolves around the issue of predictive accuracy of the market model.

Does the fact that the stock market and the sports wagering market are- to certain degrees- fluid and self correcting have any impact on the predictability of such models?

[Dark Horse]

I think we're back to sequences (aka as streaks).

I'm talking about projecting a sequence of events in advance (different from evaluating each single event). If a chance of failure is 1/75, that failure will occur with certainty. It's just a matter of time.

In simple math. The chance of failure is 1.
It just may not be this time.

[Dark Horse]

So from that perspective, an astronaut stepping on a shuttle flight would first have to embrace his own death. Only then could he operate without fear. Only then, in real life, would the end result no longer matter. (as it doesn't in the abstract world of math).

If I translate this (things may be lost in translation) to what Ganch said about what one's true Kelly bankroll really is (everything!), then a gambler using Kelly must either embrace bankruptcy upfront or live in a state of constant fear (controlled by math based assurances).