pm: Random question, Don..

[pm]

Say you were to bet perfect kelly, but you never downsized, only upsized (so you bet perfect kelly only as you upsize). As the number of hands you play approaches infinity, your ROR approaches 100% (is that right?). Is there some way to figure out how that ramp-up in ROR would occur (like you know that by using this method for 1000 hands, your ROR is going to be slightly greater than 13.53%, and that by using this method for a billion hands, your ROR is going to be much greater than 13.53%)?

My basis for the question was to try and see if it would be more profitable to play with the approach of never downsizing once you've hit an upsize point (for, say, an initial 100,000 hands, just to get your bankroll off the ground) as opposed to simply choosing a higher kelly fraction that would result in the same amount of risk.

I have no idea if this question was coherent..

[Don Schlesinger]

> Say you were to bet perfect kelly, but you
> never downsized, only upsized (so you bet
> perfect kelly only as you upsize). As the
> number of hands you play approaches
> infinity, your ROR approaches 100% (is that
> right?).

I'm not sure. I have to understand what you're saying. If you're winning, you're betting proportionally to the current bank? So, you can still cut back the bet size, if an original, say, $10,000 bank goes from $15,000 to $12,000? Or, once you hit $15,000, you never decrease the wager sizes, even if you lose a couple of thousand? Your scheme isn't clear to me.

> Is there some way to figure out how
> that ramp-up in ROR would occur (like you
> know that by using this method for 1000
> hands, your ROR is going to be slightly
> greater than 13.53%, and that by using this
> method for a billion hands, your ROR is
> going to be much greater than 13.53%)?

Don't know until we clarify the above.

> My basis for the question was to try and see
> if it would be more profitable to play with
> the approach of never downsizing once you've
> hit an upsize point (for, say, an initial
> 100,000 hands, just to get your bankroll off
> the ground) as opposed to simply choosing a
> higher kelly fraction that would result in
> the same amount of risk.

Nothing is "more profitable" than Kelly, if by "profitable," you mean growing the log of your wealth in the shortest mean time. You can always make more money than Kelly by overbetting Kelly, but the ROR grows apace. Your method is simply another way to overbet Kelly. I'm not sure what the ultimate ROR would be, until we're very specific about the betting parameters.

Don

[pm]

> I'm not sure. I have to understand what
> you're saying. If you're winning, you're
> betting proportionally to the current
> bank? So, you can still cut back the bet
> size, if an original, say, $10,000 bank goes
> from $15,000 to $12,000? Or, once you hit
> $15,000, you never decrease the wager sizes,
> even if you lose a couple of thousand? Your
> scheme isn't clear to me.

If you started with $10,000 and played (say) half-kelly, you would never decrease your bet levels. As your bank increased, you would keep increasing your bet levels, and once increased, you would never decrease them again (so in reality, if you resized at $15,000, you would never decrease the wager sizes, and then you would resize again at $20,000 and never decrease, and so on).

> Nothing is "more profitable" than
> Kelly, if by "profitable," you
> mean growing the log of your wealth in the
> shortest mean time. You can always make more
> money than Kelly by overbetting Kelly, but
> the ROR grows apace. Your method is simply
> another way to overbet Kelly. I'm not sure
> what the ultimate ROR would be, until we're
> very specific about the betting parameters.

I understand that overbetting by choosing a kelly fraction that's > 1 reduces your growth rate (and increases ROR), but if you overbet by never reducing your wager size (only increasing according to your bank), wouldn't that increase your growth rate and ROR?

[Don Schlesinger]

> I understand that overbetting by choosing a
> kelly fraction that's > 1 reduces your
> growth rate (and increases ROR), but if you
> overbet by never reducing your wager size
> (only increasing according to your bank),
> wouldn't that increase your growth rate and
> ROR?

No. Kelly is optimal. Everything else isn't! So, once you increase and experience a losing streak, where you now are overbetting by, potentially, a fatal fraction of 2x or more, you're not only not increasing your growth rate, you're courting certain disaster.

You can't cheat the math. You either play Kelly or you don't; and anything that isn't Kelly, isn't optimal.

Don

[KidDangerous]

Where can I find the most simplified, thorough in depth explanation of Kelly and it's application?

Kid

[pm]

What does growth rate actually mean? I keep thinking of this extreme example: you always bet 50% of your bank, but once you've increased your bet level, you don't decrease it. So you would go from $10,000 to $15,000 to $22,500 and so on, and if at any point you lose 3 hands in a row, you're broke.

Wouldn't this provide a huge growth rate and a huge ROR (basically 100%)? Or does "growing the log of your wealth" mean something different?

Sorry to keep bugging you, just trying to understand..

[Don Schlesinger]

> Where can I find the most simplified,
> thorough in depth explanation of Kelly and
> it's application?

Probably bjmath.com.

Also, there's an exellent discussion in Ziemba and Hausch's horseracing book, "Beat the Racetrack."

Don

[Don Schlesinger]

> What does growth rate actually mean? I keep
> thinking of this extreme example: you always
> bet 50% of your bank, but once you've
> increased your bet level, you don't decrease
> it. So you would go from $10,000 to $15,000
> to $22,500 and so on, and if at any point
> you lose 3 hands in a row, you're broke.

> Wouldn't this provide a huge growth rate and
> a huge ROR (basically 100%)? Or does
> "growing the log of your wealth"
> mean something different?

No, it wouldn't provide a huge growth rate; it would leave you broke virtually all of the time. What kind of growth is that?!

With schemes such as these, one person out of a million wins all the money in the world, and the EV for the whole group looks wonderful, but, in actuality, the other 999,999 go broke. Not my concept of an ideal system.

Don

[KidDangerous]

[pm]

> No, it wouldn't provide a huge growth rate;
> it would leave you broke virtually all of
> the time. What kind of growth is that?!

> With schemes such as these, one person out
> of a million wins all the money in the
> world, and the EV for the whole group looks
> wonderful, but, in actuality, the other
> 999,999 go broke. Not my concept of an ideal
> system.

I understand; I was simply trying to use an extreme example to figure out if my understanding of growth rate was correct. Of course you wouldn't play like that because your ROR is 100%; I was just trying to point out that when you play like this in a scaled back fashion (so not actually 50% of the bank, more like half-kelly or third-kelly), your bank would necessarily grow quicker than if you play in a normal fashion (i.e. resizing after enough loss).

So I was just trying to see how risky that would be. Half-kelly is 1.83% ROR with no resize whatsoever; I was trying to see how much your ROR would increase if (for say an initial 50 or 100K hands) you resized after winning enough, but never resized after losing.

Intuitively, it seems like this would be more profitable than simply choosing a higher kelly fraction that carries the same amount of risk.

Hopefully this made sense; if you get my meaning and I'm grossly incorrect, could you let me know where I'm going wrong..

Thanks, Don.

[Don Schlesinger]

> I was just trying to point
> out that when you play like this in a scaled
> back fashion (so not actually 50% of the
> bank, more like half-kelly or third-kelly),
> your bank would necessarily grow quicker
> than if you play in a normal fashion (i.e.
> resizing after enough loss).

And I keep trying to point out that this isn't true, but you seem not to be listening. The quickest growth rate is pure Kelly. Everything else, which is sub-optimal, produces slower growth rate. You need to read a little more about Kelly. You might also check Thorp's "The Mathematics of Gambling."

> So I was just trying to see how risky that
> would be. Half-kelly is 1.83% ROR with no
> resize whatsoever; I was trying to see how
> much your ROR would increase if (for say an
> initial 50 or 100K hands) you resized after
> winning enough, but never resized after
> losing.

Not sure. But, understand that, along with an increase in ROR comes a decrease in growth rate, not an increase. It's a lose-lose proposition -- not a tradeoff.

> Intuitively, it seems like this would be
> more profitable than simply choosing a
> higher kelly fraction that carries the same
> amount of risk.

> Hopefully this made sense; if you get my
> meaning and I'm grossly incorrect, could you
> let me know where I'm going wrong.

I'm trying! :-)

Don

[pm]

I'm dense as usual, sorry.

So, if you had only one downsize point for when you drop below you initial BR level (like a $50K BR, and you resize at $25K), that would still provide a higher growth rate than never resizing if you're below the initial BR level ($50K)?

As an idle curiosity, how do you define growth rate? I keep thinking of the bet 50% of the BR example and it sounds like a high growth rate & high ROR; I guess I don't understand what "growth rate" actually means.

Also, does "The Mathematics of Gambling" involve some pretty intense math? I'd like to understand kelly betting fully, but having seen some of the stuff on bjmath.com, I get the feeling that I wouldn't be able to. Is Thorp's book generally readable?

Thanks for your patience, Don (seems like most of my threads end with me saying that).

[Don Schlesinger]

> I'm dense as usual, sorry.

> So, if you had only one downsize point for
> when you drop below you initial BR level
> (like a $50K BR, and you resize at $25K),
> that would still provide a higher growth
> rate than never resizing if you're below the
> initial BR level ($50K)?

Yes.

> As an idle curiosity, how do you define
> growth rate? I keep thinking of the bet 50%
> of the BR example and it sounds like a high
> growth rate & high ROR; I guess I don't
> understand what "growth rate"
> actually means.

You'll probably be sorry you asked, but, for even-money bets, the logarithmic growth rate of your bank is defined as:

Growth (bank) = p*ln(1 + f) + (1-p)*ln (1-f), where p is the probability of success, 1-p is the probability of failure, f is the optimal bet size, and ln is the natural logarithm to the base e. After N bets, you will have about e raised to the [N + Growth(bank)] power, times as much money as what you started with.

> Also, does "The Mathematics of
> Gambling" involve some pretty intense
> math? I'd like to understand kelly betting
> fully, but having seen some of the stuff on
> bjmath.com, I get the feeling that I
> wouldn't be able to. Is Thorp's book
> generally readable?

The math is understandable. And, I think the book is still available.

> Thanks for your patience, Don (seems like
> most of my threads end with me saying that).

No problem.

Don