Bankroll Theory Question

[MJ]

If one's goal is to grow their bankroll at a moderate pace(.5 kelley), is it better to split the bankroll into SEVERAL SMALLER bankrolls and play each one aggressively(meaning full kelley and it either doubles or goes bust, in which case he simply moves on to the next one) or is it better to just base your unit size on the ONE ENTIRE bankroll?

As an example, let us assume one has a bankroll equal to $50K. The individual can lose every last penny and it would not effect his lifestyle in the least bit. Even though he can afford to lose, the counter obviously wants to win some $$$.
Rather then play off one bank of $50K, is there ANY advantage for him to divide his BR into 5 bankrolls of $10K and play full kelley on each individual one?

Thanks for any insight,

-MJ

 

[stainless steel rat]

one answer

"kelly" is defined based on a risk of ruin for a bankroll. If you divide your BR into "chunks" and then kelly bet each chunk, the first effect is that your bet size will shrink by 1/N where N is the number of chunks. This means your win rate will also shrink by 1/N, and your risk of ruin will go to something very low as well.

If you want to bet 1/2 kelly, that's a safe approach. I've heard pros mention 1/4 kelly even, but that is with a huge BR where full kelly might blow the table max in most places.

Personally, if I were worried about 1/2 kelly ROR, rather than dividing my BR into chunks and 1/2 Kelly betting a single chunk, I would just drop to 1/4 Kelly on the whole BR.

Others might disagree of course...

Whether you sub-divide or keep it in one big BR, your ROR will _never_ reach 0.00 anyway, you can always hit one of those ugly streaks...

 

[The Mayor]

The real question...

The real question is -- what ROR do you feel comfortable with for your 50k bankroll? After you know that number, you can proceed.

Here is a number (I'm just going to approximate on this one). If you split your 50k into 5 10k bankrolls, and play full kelly on a 10k BR, then your ROR (for the full 50k) is approximately .004% (1 in 25,000 chance of losing the 50k). You could probabaly be a bit more agressive and feel comfortable.

 

[kbp]

Kelly

Perhaps I misunderstand the concept, but I believe Kelly proportional betting is based on maximizing the growth of a bankroll by betting a portion of your bankroll in proportion to your advantage. Betting in this manner, without resizing, HAS a risk of ruin, but I don't think Kelly is DEFINED based on a ROR. After all, a strict Kelly wager isn't based on what your starting bank was, only what your current bank (and advantage) is.

 

[stainless steel rat]

Kelly

Has to include ROR. For example, a full Kelly bet is (if I recall correctly, my BJ books are 25 miles from where I am right now) based on a 13.5% ROR. And yes, technically you bet advantage * .76 * bankroll, where advantage is expressed as a decimal percentage (.015 for 1.5% advantage). And yes, you technically resize your bet each hand. But in reality, that's undoable, as is exact Kelly betting, because you are not going to bet $27.52 on a hand. I believe most "kelly bettors" compute the kelley bet per TC, then round to multiples of $5 or whatever chip size they are betting, and stick with that through thick or thin until the bankroll increases or decreases significantly enough to justify re-calculating and re-memorizing the new betting ramp.

I will add that I don't do this, because I don't have a bankroll in those terms. I just go play for a particular stakes level depending on where and when and don't give much thought to going bust as I can just return again next week... I'm just a simple $5-$40 or $25-$100 DD player myself...

 

[MJ]

Thanks Professor

I understand what you did. You took .135 ^ 5 = .00004484 = .004%. You computed the probability that each fraction of the bankroll would be lost betting at full
kelley.

So, assuming I wanted to maintain the level of risk above, I can just size my unit based upon .2 Kelley for the size of my ENTIRE bankroll and your saying it would be the same thing as playing each chunk of my BR at .2 kelley?

I guess I feel comfortable with no more then 1/2 kelley.

By the way, once a player has determined kelley how would they figure out how many
units they should play with?

-MJ

 

[MJ]

Could be right

Hey SSR thanks for responding. I've seen your posts(on various boards) over the last few months and find them to be helpful.

"Personally, if I were worried about 1/2 kelly ROR, rather than dividing my BR into chunks and 1/2 Kelly betting a single chunk, I would just drop to 1/4 Kelly on the whole BR".

Well the idea is to FULL kelly bet with a single chunk of the BR at a time to maximize the EV. So you would start with 10K, full kelly bet with it and see what happens. If you lost it then you would take the next segment and full kelly bet with that...and so on and so forth.

I'm not sure if the idea would be any better then just being extra conservative and playing off the full BR.

-MJ

 

[stainless steel rat]

maybe

I worded that too poorly.

Say you have $50K and you decide to bet full-kelly, which will put your ROR in the 13% range. And will have you betting nearly $400 at a 1% advantage (.01 * .76 * 50000 if my math is close).

Now divide that bank into 5 chunks. You now will be betting about 75 bucks at a 1% advantage. With a 13% ROR on the 10K bank. But note that your overall ROR is now roughly .13 ^ 5 which is a very small number. And notice that if your counting system and game typically has a (say) 2 big bet win per hour, then with the former system, you are winning $800 per hour, with the latter, you are winning $150 per hour. Huge difference. And a very non-aggressive betting plan...

I'm not a pro, but I'd suspect that a pro is not willing to play to a 13.5% ROR, which means he is going to be doing some sort of fractional Kelly betting. I've seen teams mention 1/4 and 1/8 Kelly betting for example, but as I mentioned that is on a really huge bankroll. But the ROR is way down there as well, which is their aim as if a pro goes bust, he _really_ goes bust and has become broke.

Also, whether you play 1/4 kelly on your full bankroll, or full kelly on four separate bankrolls seems to make little difference in anything. Only when you start varying your bet less does things take a turn for the worse. I'm hardly a Kelly expert (Mayor is certainly better here) but the idea of 1/4 kelly is simply to compute a full kelly bet, then use 1/4 of it as the bet amount, which drives the ROR down into the mud, but which also reduces your hourly win rate by 1/4 as well.

 

[phinitum]

Kelly and ROR

The 13.5% ROR with full Kelly is based on the assumption of no resizing, in other words of starting with bets sized based on Kelly then ignoring everything that happens and just betting a fixed schedule.

One of the conveniences of fractional Kelly is that it isn't such a pressing need to do frequent resizing.

Another nice feature is that fractional Kelly with resizing shows a reduction in bankroll growth much less than the reduction in risk. 50% Kelly grows the bankroll at about 76% the rate that 100% Kelly does (assuming constant resizing).

 

[kbp]

Not Kelly

"Kelly....Has to include ROR. For example, a full Kelly bet is (if I recall correctly, my BJ books are 25 miles from where I am right now) based on a 13.5% ROR."

No, the opposite is true. Kelly has a theoretical ROR of zero and is BASED ON maximizing bankroll growth.

What you're talking about is setting a betting ramp using the Kelly Criterion for a particular bank and NEVER resizing. This IS NOT Kelly betting. It's fixed betting and carries a theoretical ROR (13.5%?) if you NEVER EVER resize. Throw in the occasional resizing you mention, when a bankroll increases or decreases significantly, and 13.5% is no longer the ROR.

Of course, resizing after every hand isn't practical, but it does it have to be overly complicated? Can't a solo player on his own bank recognize he's lost say 50 percent of his original bank and just divide his original betting ramp by two? Or say with a 20 percent loss doesn't a 50 dollar unit just resize to a 40 dollar unit.

 

[toddler]

BJRM should give you all the info you need.

 

[stainless steel rat]

won't argue

since as I said, I'm not a "Kelly expert".

But re-sizing can be a problem. If you play (as I do frequently) a DD game spreading 5-40, re-sizing is not going to happen. I can't actually bet less than $5 easily, without drawing a _lot_ of attention. And by the time you re-size down to $.50, you are stuck, haven't seen a casino that accepted pennies at a BJ table yet.

I always compute my bet ramp using CVCX. I tell it "unit = $5", "spread=1-8" and see what it says the optimal betting ramp is. If it seems "queer" (and it often does as it will come up with some odd bet ramps for some games) then I adjust it to something I can deal with. My favorite S17 DD game goes $5, 10, 20, 40 for TC=0, 1, 2, >= 3. I can mentally do that easy enough.

 

[jblaze]

Re: Kelly

raise your ror to the 3/2 power, ie (ROR)^1.5 to get your ROR if you resize your bets once when half your bankroll is completed. therefore, a 10% ROR with no resizing becomes ~3% ROR if you resize at .5*BR.