Myooligan: Kelly in plain English

[Myooligan]

I have repeatedly come across the explanation that Kelly betting "maximizes the logarithmic growth rate of your bankroll." I understand that for some reason Kelly betting offers the "optimal" risk/reward ratio, but I have never understood why, or what it means to "maximize the logarithmic growth rate of your bankroll." Can't visualize it.

Would somebody please provide the English version?

thanks in advance,
Myoo

[Sonny]

> I understand that for some reason Kelly betting offers
> the "optimal" risk/reward ratio, but I have never
> understood why, or what it means to "maximize the
> logarithmic growth rate of your bankroll." Can't
> visualize it.
>
> Would somebody please provide the English version?

Sure!

Using the Kelly Criterion (or, at least, the Blackjack equivalent of it) will have you resizing your bets after large fluctuations as opposed to always betting the same amount for the rest of your life.

Imagine that you and your friend Kelly each deposit $1,000 into an account with a 2% monthly return. After the first month you withdraw your $20 profit while Kelly leaves the $20 in the account. The following month you earn another $20, but Kelly earns $20.40. The month after that Kelly will earn $20.81.

After 12 months you will have earned $240 while Kelly earns $268.24. After 5 years Kelly will have earned $2,281.03 compared to your measly $1,200.

Your growth rate is linear since you are always earning the same amount in relation to your original stake. However, Kelly?s growth rate is exponential since it increases by a given factor each month. As you can see, Kelly not only earns much more than the standard approach, it also earns it much faster. That is the magic of Kelly.

The drawback is that is takes longer to recover from a large loss, but the security is often worth it.

-Sonny-

[jblaze]

> Sure!

> Using the Kelly Criterion (or, at least, the Blackjack
> equivalent of it) will have you resizing your bets
> after large fluctuations as opposed to always betting
> the same amount for the rest of your life.

> Imagine that you and your friend Kelly each deposit
> $1,000 into an account with a 2% monthly return. After
> the first month you withdraw your $20 profit while
> Kelly leaves the $20 in the account. The following
> month you earn another $20, but Kelly earns $20.40.
> The month after that Kelly will earn $20.81.

> After 12 months you will have earned $240 while Kelly
> earns $268.24. After 5 years Kelly will have earned
> $2,281.03 compared to your measly $1,200.

> Your growth rate is linear since you are always
> earning the same amount in relation to your original
> stake. However, Kelly?s growth rate is exponential
> since it increases by a given factor each month. As
> you can see, Kelly not only earns much more than the
> standard approach, it also earns it much faster. That
> is the magic of Kelly.

> The drawback is that is takes longer to recover from a
> large loss, but the security is often worth it.

> -Sonny-

That's a good definition of compounding, which I'm sure Myooligan understands.

Suppose you have a biased coin which you know gives 51% heads 49% tails, and you have 1000 dollars and you want sustainably make as much money as possible. How much should you bet? Clearly, betting the farm on heads will make your bankroll grow fastest in that situation if you are right, but you are broke if you are wrong. You wish to maximize profit/risk. If you graphed profit/risk vs. bankroll%/bet (which adjusts automatically for bankroll fluctuation since it is a relative measure) you would get a parabolic graph. on the extremes, betting very little would minimize risk but minimize growth. on the right extreme, betting near your whole bankroll repeatedly would almost ensure ruin but in the rare event you don't go broke you would grow your bankroll most rapidly. at the maximum value of the parabolic curve you would find that you are betting ev/variance... the kelly criterion.

[Myooligan]

> That's a good definition of compounding, which I'm
> sure Myooligan understands.

Yup.

> You wish to maximize profit/risk.

In blackjack terms, advantage/ror?

> at the
> maximum value of the parabolic curve you would find
> that you are betting ev/variance... the kelly
> criterion.

This helps, thankyou. But I'm left with another version of the same question: Why should we conclude that the ideal way to bet is the one that maximizes profit/risk? This measurement presumes that a profit of 50 (not sure what units we're using) at a 50% ror is equal in value to a profit of 1 at 1% ror. Right? It seems arbitrary to assign this 1:1 relationship to profit and risk.

[Sonny]

> This helps, thank you. But I'm left with another
> version of the same question: Why should we conclude
> that the ideal way to bet is the one that maximizes
> profit/risk?

Because we want to get the most bang (profit) for our buck (money at risk). Sure, you could take less risk and try to earn less money, or you could take more risk and try to earn more money. The term "optimal" is very subjective, as you mentioned in your first post.

The reason we consider it "ideal" is because it maximizes our return while minimizing our risk. We want to achieve the highest EV with the smallest ROR. As jblaze showed, the fastest way to double your BR is to bet it all, but that is also the fastest way to lose your entire bankroll. As smart gamblers we are always looking for the balance between risk and reward. On top of the compounding effect that I mentioned before, bending Kelly's rules to apply to Blackjack helps us to manage that balance at the tables.

-Sonny-

[Don Schlesinger]

The discussion here has been well done. Here are a few other qualities of Kelly betting that make it "optimal":

1. ROR is, theoretically, zero.

2. Your bankroll will grow faster than anyone else's, with the same bankroll, but with a different bet scheme.

3. You reach a specified goal in the shortest average time. So, if you have $5,000 and wish to turn it into $50,000, using Kelly will get you there in the minimum number of hands, compared to using any other wagering system.

Don

[Myooligan]

> 2. Your bankroll will grow faster than anyone else's,
> with the same bankroll, but with a different bet
> scheme.

Maybe this is where I'm confused. I thought a person who is overbetting Kelly would have a faster rate of bankroll growth, but also a higher risk of ruin.

> 3. You reach a specified goal in the shortest average
> time. So, if you have $5,000 and wish to turn it into
> $50,000, using Kelly will get you there in the minimum
> number of hands, compared to using any other wagering
> system.

Same question: Wouldn't betting it all get you there in the shortest average time?

Or, should I be adding the caveat "with the constraint that you will only play at a theoretical risk of ruin of 0%" to #2 and #3?

> Don

[Sonny]

[pm]

>> 2. Your bankroll will grow faster than anyone else's,
>> with the same bankroll, but with a different bet
>> scheme.

> Maybe this is where I'm confused. I thought a person
> who is overbetting Kelly would have a faster rate of
> bankroll growth, but also a higher risk of ruin.

LOL!! I asked this exact same question like a year ago (beginner's page). Funny thing is that I'm still not sure I understand it either. If you could bet fractions, you could still say "Well, I'll bet 99% of my bank every time" and still have a theoretical RoR of 0%. It seems that something like that would let you increase your bank at the fastest rate if you're insanely lucky.

I was pondering this for a while after that and I always wondered: I guess in laymen's terms, do we say that a Kelly bettor has the highest growth rate because they will end up with the biggest bank on average after any number of bets? The guy that bets 99% of his bank every time will average a ridiculous negative growth rate; is that how we should look at the term "growth rate" in general?

[Myooligan]

> LOL!! I asked this exact same question like a year ago
> (beginner's page). Funny thing is that I'm still not
> sure I understand it either. If you could bet
> fractions, you could still say "Well, I'll bet
> 99% of my bank every time" and still have a
> theoretical RoR of 0%. It seems that something like
> that would let you increase your bank at the fastest
> rate if you're insanely lucky.

I had the same thought, immediately after posting my last post! But then I thought, maybe it has something to do with this: While it's true that always betting 99% of our bankroll would leave us with a 0% risk of ruin (in theory), the problem is that every time we lose a hand, we have to start over. So our $10,000 bank becomes $100. And soon afterwards, it'll become $1, or $2, or $4. It's easy to see that it'll almost never get back up to $10,000 using this strategy. On the other hand, betting some very small percentage of your bankroll will definitely get you up to your goal, given enough time. But now I can see that there must be a happy medium that results in the highest average gain per hand, as you say below.

One thing I learned from this: The often mentioned Kelly-betting property of having a 0% ROR is really a property of any betting scheme which always calls for a max bet of less than 100% of your bankroll!

> I was pondering this for a while after that and I
> always wondered: I guess in laymen's terms, do we say
> that a Kelly bettor has the highest growth rate
> because they will end up with the biggest bank on
> average after any number of bets? The guy that bets
> 99% of his bank every time will average a ridiculous
> negative growth rate; is that how we should look at
> the term "growth rate" in general?

[Sonny]

> While it's true that always betting 99% of our bankroll
> would leave us with a 0% risk of ruin (in theory), the
> problem is that every time we lose a hand, we have to
> start over.

That is not true. If you always bet 99% of your bankroll then your ROR is 100%, not 0%. In many cases ?starting over? will require a new bankroll.

For example, why would you bet 99% of your bankroll off the top of the shoe? The house has the edge and you know that you are more likely to lose the hand than to win it. In addition, you will be unable to split, double, or take insurance if you need to. That puts you at even more of a disadvantage. If you consistently overbet your bankroll it is just a matter of time before you go broke.

Let?s take this example a step further. You obviously know that the strategy above will not work so you decide to only play during positive counts when you have the advantage. Your ROR drops back down to 0%, right? Wrong! At a TC of +2 you probably have around a 0.5% advantage. Why would you bet 99% of your bankroll on a 0.5% edge? That is grossly overbetting and will keep you at a 100% ROR even though you are only playing with an advantage (except for not being able to follow proper basic strategy as I mentioned above).

Obviously you were using 99% as an extreme hypothetical example, but overbetting by any amount will severely increase your ROR, possible to 100%.

The basis of Kelly betting is to bet a proportion of your bankroll equal to your current advantage. If you have a 1% advantage, you should bet 1% of your total bankroll. Anything smaller is barely worth putting money on the table for. That way you are making the most of your money by minimizing risk and maximizing growth.

Unfortunately, this approach doesn?t exactly work for blackjack. If your advantage is -1% (which it often is) you cannot bet -$1 (a $1 bet that you are going to lose the next hand, which is an interesting proposition to try and make, but probably not a practical one). You may be forced to make minimum bets during negative situations that will eat away at your bankroll. Also, it is very difficult to know your exact bankroll before every hand you play, and even more difficult to bet an exact proportion of that (such as $132.57). Furthermore, it does not take into consideration the additional money bet due to doubles and splits (about an extra 1.32 units). Those reasons are just a few of the ways that pure Kelly ends up with a 13% ROR. Since that is much too high for any serious player, many prefer to bet a fraction of Kelly. Because of the logarithmic nature of Kelly betting, if you decide to bet only half-Kelly then you will still retain 75% of the pure-Kelly growth rate.

> So our $10,000 bank becomes $100. And soon afterwards,
> it'll become $1, or $2, or $4. It's easy to see that
> it'll almost never get back up to $10,000 using this
> strategy.

Especially if the table minimum is $5 or more. You will be unable to play another hand and therefore, effectively ?broke.?

> One thing I learned from this: The often mentioned
> Kelly-betting property of having a 0% ROR is really a
> property of any betting scheme which always calls for
> a max bet of less than 100% of your bankroll!

Again, that is simply not true and it is very important that you understand that. If you are overbetting your bankroll by more than a factor of 2 then your ROR becomes 100%. Even just a few oversized bets in super-high counts can increase your ROR dramatically.

That is not to say that anything other than Kelly is wrong. There are many betting strategies that will safely win you money, but the ones based on some form of the Kelly Criterion will do it faster and more effectively.

I'm not sure how coherent that was because it is WAAAAY past my bedtime, but hopefully it help. Please let me know if you have any questions about any of it.

-Sonny-

[pm]

> That is not true. If you always bet 99% of your
> bankroll then your ROR is 100%, not 0%. In many cases
> ?starting over? will require a new bankroll.

Actually we were saying that if you could bet fractions (less than a penny even), then your bankroll would keep getting smaller and smaller, but you would technically never go broke. Of course, in reality you'd have an RoR of 100% (hell, in reality, if you start with a reasonable 5 figure bankroll, you're probably screwed if somehow end up in the $5-10K range). Again, just a dumb extreme example to help understand what the term "growth rate" means.

[pm]

> Furthermore, it does not take into
> consideration the additional money bet due to doubles
> and splits (about an extra 1.32 units). Those reasons
> are just a few of the ways that pure Kelly ends up
> with a 13% ROR.

Are you sure that's the case? I thought that any positive-edge proposition that's wagered at full-kelly has a 13.5% RoR, not just blackjack.

The splits, doubles etc. are accounted for when calculating the optimal percentage of your bankroll to risk. If you're betting an optimal percentage of your bank (i.e. full-kelly), I thought you end up with a 13.5% RoR regardless of what advantage game you're playing.

Full Kelly % to risk = win% - (loss%/payback%)

That accounts for everything and yields an optimal risk%; I think an optimal risk% (i.e. full Kelly) then always yields a 13.5% RoR.