pm: Risk of Ruin

[pm]

a) Does BJRM calculate RoR assuming that you never change your betting levels from the initial optimal ones as your bank fluctuates?

b) If it does, how do you calculate RoR assuming that you always bet optimally? Wouldn't it be zero (assuming that you could bet down to fractions)? After all, if you're only betting a fraction of your bank each time, you would have to play a number of hands that approaches infinity for your bank to approach zero. Granted, in reality, you've busted if you're bank is less than a dollar, but I'm guessing the odds of that happening must be low if you start with, say, $25K and always bet optimally (and play with a positive edge, of course).

c) In the SCORE section in BJA3, the scenario of four friends playing different games assumes that after determining an optimal bet schedule, they never change the schedule as their bank fluctuates. That's not what you're actually supposed to do, is it? I'm guessing that the unchanging bet schedule was for calculating SCORES; when you actually play, you're always supposed to bet optimally, right?

Don, some more help, maybe?

[Don Schlesinger]

> a) Does BJRM calculate RoR assuming that you
> never change your betting levels from the
> initial optimal ones as your bank
> fluctuates?

Yes.

> b) If it does, how do you calculate RoR
> assuming that you always bet optimally?
> Wouldn't it be zero (assuming that you could
> bet down to fractions)? After all, if you're
> only betting a fraction of your bank each
> time, you would have to play a number of
> hands that approaches infinity for your bank
> to approach zero. Granted, in reality,
> you've busted if you're bank is less than a
> dollar, but I'm guessing the odds of that
> happening must be low if you start with,
> say, $25K and always bet optimally (and play
> with a positive edge, of course).

Right. If constant resizing were practical, which it isn't, and if fractional bets were permitted, which they aren't, then optimal betting would have zero ROR.

> c) In the SCORE section in BJA3, the
> scenario of four friends playing different
> games assumes that after determining an
> optimal bet schedule, they never change the
> schedule as their bank fluctuates.

Correct.

> That's
> not what you're actually supposed to do, is
> it? I'm guessing that the unchanging bet
> schedule was for calculating SCORES; when
> you actually play, you're always supposed to
> bet optimally, right?

No, not really. You can't "always" bet optimally. Chances are, if the optimal bet calls for $37.42, you're going to bet $35 or $40. And, chances are also that, if you've lost, say, 25% of your bankroll, you may decide to continue to play for your original stakes rather than downsize by 25%, preferring instead to wait until further losses may deplete your bankroll to, say, half of its original size.

Don

[pm]

Thanks for replying, I really appreciate it. Let me bother you about this one more time; I'm trying to get 100% clear on the fundamentals.

Say your session bank was always 10% of your current bank (and you had some win ceiling as well), and say you always recalculated your optimal bet schedule after a session. Wouldn't your growth rate take a minimal hit (as compared to keeping the same bet schedule until 50% bankroll depletion) while keeping a very low RoR? Wouldn't your positive and negative fluxes pretty much cancel each other out in the long run? To be sure, if you caught a big negative flux and dropped down to 50% of your bank, you would have to claw your way back to the break even point if you were betting optimally. But, on average, wouldn't you catch a positive 50% flux at some point to put you way ahead of schedule, thereby cancelling out the time lost from the -50% flux?

Also, if you kept the same bet schedule until 50% depletion, wouldn't your chances of reaching that point be much higher than if you recalculated after each session?

Thanks for the info.

[Don Schlesinger]

> Say your session bank was always 10% of your
> current bank (and you had some win ceiling
> as well), and say you always recalculated
> your optimal bet schedule after a session.
> Wouldn't your growth rate take a minimal hit
> (as compared to keeping the same bet
> schedule until 50% bankroll depletion) while
> keeping a very low RoR?

No. The growth rate of the log of your wealth (bankroll) is maximized by constantly resizing. That's what is assured by Kelly wagering.

> Wouldn't your
> positive and negative fluxes pretty much
> cancel each other out in the long run? To be
> sure, if you caught a big negative flux and
> dropped down to 50% of your bank, you would
> have to claw your way back to the break even
> point if you were betting optimally. But, on
> average, wouldn't you catch a positive 50%
> flux at some point to put you way ahead of
> schedule, thereby cancelling out the time
> lost from the -50% flux?

Consider this: You lose 50% of your bank. To get even, what percent do you now have to win of your current, depleted bank? 50%? I don't think so! Try 100%!!

> Also, if you kept the same bet schedule
> until 50% depletion, wouldn't your chances
> of reaching that point be much higher than
> if you recalculated after each session?

What point are you referring to?

Don

[pm]

> Also, if you kept the same bet schedule
> until 50% depletion, wouldn't your chances
> of reaching that point be much higher than
> if you recalculated after each session?

> What point are you referring to?

Right here I was referring to the 50% depletion point.

I have to say, I'm utterly confused. What I'm understanding now is that you definitely want to use Kelly wagering in all circumstances (granted that you probably can't resize till your session's over) because your bankroll will grow fastest this way. I'm also understanding that Kelly wagering has a very low RoR that would remain constant in all circumstances (provided that your session bank is a fixed % of your bankroll).

So then why are people bothering to use BJRM and such to calculate RoR on the assumption that you won't always be betting optimally? It seems like a useless figure because (if I'm understanding you right) through Kelly wagering, you maximize your growth rate and minimize your RoR anyway; it almost seems like there's nothing to worry about.

Thanks for the help.

[pm]

> I have to say, I'm utterly confused. What
> I'm understanding now is that you definitely
> want to use Kelly wagering in all
> circumstances (granted that you probably
> can't resize till your session's over)
> because your bankroll will grow fastest this
> way.

Actually, is that right? I mean, if you took the extreme example and bet 100% of your BR every time you had a positive edge, that would give you the highest growth rate and the highest RoR, wouldn't it?

I have this feeling that I'm completely missing something in this discussion here.

[pm]

Scratch that last post, I see my error now: if you were allowed to bet fractions and you always bet 99% of your BR on positive edges, then sure, you wouldn't ever go broke, but your BR would approximately = BR * 1/(100^(n/2)) after n bets.

OK, but say you start with a $50K BR, 6dk, S17 DAS, 75%, play-all, 1-12 spread, unit=$100 (just as random parameters). BJRM gives a winrate of $124/hr. If your bank dropped to $25K and you continued betting at the same levels, it would take 25000/124 = 202 hours to get back to break even (assuming all winning sessions). If you bet optimally (resizing after every hour of play), it would take 280 hours to get back to break even (according to my calculations; I'm pretty sure that that's accurate).

Now I may not know how to calculate the likelihood of reaching 50% depletion with a constant bet schedule or with kelly betting, but clearly such a situation is more likely to happen with a constant bet schedule. And I don't know how to do the math to figure out which betting method has the best trade-off between likelihood of depletion and hours to get back up, but, again, considering that you said:

a) kelly betting maximizes your growth rate and
b) kelly betting will give you a very low (near zero) RoR

I can't understand why you wouldn't resize as often as possible.

Don....let me know......(hope you had a nice vacation)..............

[paranoid android]

> I can't understand why you wouldn't resize
> as often as possible.

I think you are understanding this better than you think. You DO want to resize as often as is possible. Betting perfect Kelly on each bet will maximize your growth rate. People use BJRM so they can bet as optimally as possible (why are you thinking that people are using it to bet non-optimally?) One point to note is that the penalty for betting more than Kelly is much greater than betting less than Kelly. If memory serves, if you bet half Kelly, your bankroll grows at 75% of full Kelly, but with half the risk. Therefore, you should always error on the low side. Also, most people don't bet full Kelly because they can't resize their bets that frequently. It's not practical. Therefore, they may play at half Kelly (or less) and resize when they lose half. That way they are never betting more than full Kelly. Also, betting less than Kelly compensates for possible errors you make in your play or overly optimistic estimation of penetration. There's a good paper on Kelly on bjmath.com, but the math was pretty intense for my ability.

[pm]

Ahhhh... I see. What threw me is when Don said

> ...chances are also that, if you've lost, say, > 25% of your bankroll, you may decide to
> continue to play for your original stakes
> rather than downsize by 25%, preferring instead > to wait until further losses may deplete your
> bankroll to, say, half of its original size."

Did he mean that if you start betting at half-kelly, you can continue that bet level until 50% depletion (because you'd theoretically never be overbetting in that 50% depletion period)?

Also, I noticed that in chapter 8, Don says (paraphrasing) "We never bet perfect Kelly for a variety of reasons," (don't have the book with me right now so I don't know the page number). Did he mean that you don't bet perfect Kelly because a) it's not possible (you can't constantly resize) and b) it's better to bet less than full Kelly?

> People use BJRM so they can bet as optimally
> as possible (why are you thinking that
> people are using it to bet non-optimally?).

The optimal bet schedule is fine; what I was trying to say about BJRM was that the RoR number it generates seems useless. As Don said, in order to calculate the RoR, it assumes that you NEVER deviate from your original bet level. If that's not what we're doing, what's the point of looking at that RoR figure?

I see (as you pointed out) that bjmath has some an article or two on RoR formalas for proportional betting. I'll have to take a look at that and get back to this discussion if it doesn't make any sense.

[pm]

I'm indebted to you, paranoid android, for pointing me to the bjmath articles.

Don, reading Bill C. and Marc Ingenosos "Risk Formulas For Proportional Betting" has cleared up all my previous questions; I'll probably have a slew of new RoR questions now.

[G Man]

> If memory serves,
> if you bet half Kelly, your bankroll grows
> at 75% of full Kelly, but with half the
> risk.

Wrong ! You bankroll grows at 75% of full Kelly is correct BUT your RoR isn`t halves, in fact it is square, or 1.83% instead of 13.5 You risk is a little over 7 times less betting half Kelly.

[paranoid android]

> Did he mean that if you start betting at
> half-kelly, you can continue that bet level
> until 50% depletion (because you'd
> theoretically never be overbetting in that
> 50% depletion period)?

I can't speak for Don, but that *may* have been the reason. However, there could be reasons for betting more than Kelly also (however maximizing the growth rate of your bank roll isn't one of them).

> The optimal bet schedule is fine; what I was
> trying to say about BJRM was that the RoR
> number it generates seems useless. As Don
> said, in order to calculate the RoR, it
> assumes that you NEVER deviate from your
> original bet level. If that's not what we're
> doing, what's the point of looking at that
> RoR figure?

Not everyone attacks the game from the same angle. If you have a replenishable bankroll, and you want to play at a unit level such that you have a certain hourly rate (perhaps you consider it a waste of time to play for less), you may consider playing for more than Kelly for that bankroll, knowing that you can eventually replenish that bankroll if you lose it all. Knowing the ROR for that bankroll is certainly of interest.

[pm]

> Not everyone attacks the game from the same
> angle. If you have a replenishable bankroll,
> and you want to play at a unit level such
> that you have a certain hourly rate (perhaps
> you consider it a waste of time to play for
> less), you may consider playing for more
> than Kelly for that bankroll, knowing that
> you can eventually replenish that bankroll
> if you lose it all. Knowing the ROR for that
> bankroll is certainly of interest.

Thanks for this clarification. I like to get all my doubts cleared up, even if it turns out that the answer isn't applicable to me (the BJRM RoR was preying on my mind!).