How often will you switch the value of your units after a big win????

lets say you start with 5 grand using a max bet of 40$, min. bet of 5$ 1-8 spread. get some good cards and double your money and now have 10 grand. do proffesionalls then double the thier unit size to 10$, 80$ max. or do they just keep playing wiht the same betting spread and unit size????
thanks

If you are serious about the game, you would resize your bets. You cannot expect to make any serious money without doing so. The loss of performance from violating Kelly and not doing so is huge, you could easily be wasting a hundred thousand dollars or more over the course of a lifetime's play. It is the most important thing to learn once the basics of an advantage play technique have been mastered.

There are as many answers to that question as there are gamblers! The gambler and what he THINKETH! So shall it be!

Yes, you could make the bet (after doubling your br) $10 min. and $80 max. It is math based, and therefore you're doing the right thing!

Next LEVEL is the $15,000 br. And, it is logical and the formula tells it- min. bets of $15 and max $120... and SO ON... AND ON UNTIL... YOU MAKE A KILLING!

WHY LEAVE THE TABLES BEFORE MAKING THE BIG BUCKS? ISN'T THE PURPOSE OF GAMBLING TO MAKE BIG MONEY? THEN, WHY TO STOP? MAKE IT OR BREAK IT!

Changing bet sizes when you double your bank or lose half is very reasonable. Your growth rate is not much different from instantaneous Kelly resizing. But the more frequently you resize bets the bigger you make your long run index as well. Smaller bets mean it takes longer to recover losses and bigger bets can give back wins faster. Kelly betting requires 4X as many hands to reach each confidence level of being ahead that fixed bets do. So just take that into consideration.

Full Kelly is not too bad when you are playing at a level that is supported easily by outside income. If you bust out it doesn't take long to raise a new bank. As your bank grows into something that cannot be easily replaced by "honest work" you should become increasingly conservative. This means increasing bets at a slower rate than the bank grows... double the bank but raise bets only 50%, for example. Few pros bet bigger than 1/3 Kelly, and for those in it really for the long term the fraction is generally considerably smaller, essentially zero risk of ruin.

Which books on this site (or other sites) would be the best for learning about the Kelly betting?

I suspect MathProf would object to your statement: Few pros bet bigger than 1/3 Kelly, and for those in it really for the long term the fraction is generally considerably smaller, essentially zero risk of ruin. Full Kelly is 0% ROR (not 13.5% ROR as sometimes stated) because it entails continual resizing. Even if you only resize once, at BR/2, your ROR is greatly reduced. I know you know this, but it might be confusing to newbies when you use the word "Kelly" both ways in one post.

I like to nurture my growth (exponential and multiplicative). I don't worry too much about my N0. ;-)

ETF

I have been often flamed on this but this is exactly why I emphasized that the Wong effect on proportional betting is not a decrease in advantage, if you define advantage as the return on the total invested, but in the variance increasing. Full Kelly is the fastest way to grow a bankroll. N0 goes up, but this does not mean that the average time needed to double a bankroll does, as is the implication of using the N0 measure in issolation. The opposite occurs; it is just the time need for ev to excede the fluctuations at one sigma of negative standard deviation that goes up by 4 times.

The only thing that goes up is the uncertainty of your results, and this (the following is just a compilation of ignored parts of commonly accepted results) follows the following relationship, which is perhaps more clear than the existing Certainty Equivalent formulas:

If k is the fraction of a Kelly fraction that is bet the probability of being at a fraction F of your current bankroll is F*(k^2).

Red Aces has a series of FAQ posts, that thopper found for me, that shows an interesting assertion: that the highest ratio of overall bankroll growth to fallback probability occurs with k=1/3. The derivation of this was not included.

Perhaps the fastest bankroll growth occurs not with k=1, but with k=1 plus some mild outside additions to capital. If you go to the original J. L. Kelly Bell Labs paper you will find where the log maximization is a two term approximation to the "fanout," of possible outcomes and not the full spectrum of risks. Usually these terms are minor but they can be overcome in the simplest manner by a subliment that is at least 1/50th of your gambling expected value.

The frequency of how often you readjust is simply set by how often you are willing to tolerate betting at levels other than your targeted k fraction, and the chip sizing needed to bet those units. A precise answer would be found by how (assuming your goal was betting k=1) much deviation from optimal you were willing to tolerate. From k=2 to k=1 your N0 goes from infinity to minimum, and goes from minimum back to infinity from k=1 to k=0. The long run index equates to the number of hands needed to double the bank for an optimal bet/bank ratio series of bets, without changing your betting size. Find how far from optimal you are willing to have that number swing for the range k=1 to k=0, based on the average hands to double, and then solve for the range k=1 to k=2 that has the same N0. A number of functions will do this. A modified use of the CE would be one of them.

One thing to keep in mind is that ideally, these considerations should be problems only for the growth phase of your playing. There should be some point where you are topping off your betting levels and then drawing money out for other purposes, with a bankroll large enough to have a low risk of ruin. Heat rises with betting levels and will always cut off your growth potential.

You can even go ahead and use k^4* Nsub0 as an approximation for the above rise in time needed to double, for betting k=1 to about 1.5, compared to k=1, where time to double is Nsub0.

Overall though the exponential growth of your bankroll, is not hindered by bets in the range below k=2, but the probability of your bankroll being off peak rises to near certainty. With rounding due to chip sizes and table limits your ruin is near certain as well. Think of the growth being compacted into more and more improbable winning streaks, where the exponential growth, except for the other terms of the "fanout," being ignored, is not impacted, until where the probability of being down F is 4*F. No win streak can then overcome this. The proof can be taken from ML's proof of where betting at k=2 results in no long term growth. It is a bit harder to prove that exponential growth would still exist, for less than 2, but don't bother even thinking about it. The roller coaster ride of betting less than k=2 but significantly more than k=1 will be enough of a stress to have you losing for other reasons anyway. The probability of being F down being equal to F, is enough to send most people over the edge, except for those who are so stable that they have been actually rather insanely living on the edge for some time, and just have gotten good at managing their problems---sort of like a high functioning heroin addict (or ex-addict) (would that make Ozzy Osborne a good counter?;-)("wow iz that a CSM? Could I a look atz tha mechanizmiz of tha't thingaee thatz maucking the cardz come outz so'rt likez crap? Wellz I kindza haad to keepz at least a one addicztion after havin' t'a givez up tha't sweet draeems needle and all a thatz antifreeze forz tha noggin, soz I kinda gotz to smear mah crinkle roundz tha felt"---Ozzy at the 21 tables.).

All proportional betting schemes are theoretically zero risk of ruin. It is not a special quality of Kelly. You can never go broke putting a fraction of your money on the table. But losing 90% of the bank is essentially going broke for most people, and full Kelly will do that with a good degree of regularity. You'll lose half the bank almost 37% of the time, and then halve that again 37% of those times. Knowing that you can't go broke with infinitely divisable units is very little comfort.

Full Kelly is the fastest way to grow a bankroll. N0 goes up, but this does not mean that the average time needed to double a bankroll does,

Certainly not. But variance means as much to serious gamblers as averages do. I would gladly take 1/3 my expected win rate in exchange for zero variance. Full Kelly is very unwise and most who employ it are still betting at the same level several years down the road, if they are not broke.

You missed reading the recomendation to top-off (virtualy the same wording as you used in your other post). Unbounded full Kelly is indeed foolish. But so is an overly slow attempt at climbing too. Exposure to pit scruteny at $5 units does not vanish when you are betting $100 units. The money that results from positive fluctuations has the same character as money derived from that slow turtle climb too. Your approach just results in actually getting $100 unit heat, from past play at the $5 level, more than mine does. Don't forget my recomendations are highly cautioned as well.

Rather than throwing erasers, and us calling each other the usual names, I would like this time to ask you to examine the different theoretical approaches that have been offered for the transition stage as you have called it in your prior post:

I advocate topping off your unit size, at your goal final level.

T-hopper has advocated that one actually use a nested approach where one shoots past the top-off goal, by maintaining play at a high Kelly fraction, and then drops down to the final bet size goal with a low risk of ruin.

Your post advocates a transition from a higher to a lower Kelly fraction, and a final unit sizing at k=1/3 for possible (I don't think you have ever stated this except possibly on the Green pages etc.) further growth.

Could you also be confusing blackjack fluctuations with more personal fluctuations? I admit I have had to go all in on some very expensive to solve personal tasks several times and have had to start over. One was making sure a large bank had to re-absorb a major block of issued stock in a variation of the old signing this check indicates acceptance of the condictions referenced in document blah, blah, blah, filed on date...., bait tactic. We both know Leaving Las Vegas personaly, and where he had to start over. I expect another such situation will brewup for me pretty quick, in regard to the northern wicked witch's family, where thankfully I have a major consulting payment paying off soon to help in addition to blackjack winnings.

None of these situations should be confused with what is needed to make the blackjack side of our lives perform optimally. There is also a refusal to admit how in normal circumstances that this wild ride during the growth phase of one's playing should not go on forever. It is temporary or should be. But I already have posted on how very few people can sustain their sanity during that wild ride which should be adaquate cautioning. That is an individual choice not to be gospeled with a recomendation of k=1/3. That is why I put in the Ozzy example, where it might be true that only people with high coping skills, or who are just plain the hell crazy but high functioning might be the only ones able to do this. I actually have gone beyond your cautions if you read carefully. What a person can actually psychologically stand and what is optimal for growth often diverge, but I am not going to hobble the recomendations for those who can handle the optimal just because most cannot (that would be rather temperant, as in being a no booze fanatic, if I did.).

DD: You seem about the same as me with respect to risk. Out of curiosity, what ROR do you play these days? Thanks as usual for great posts--I throughly enjoy them.

But since the criterion is so conservative I use a simpler measure. My max is 1% of the bank spread across two hands @ +5 TC... if the casino takes that kind of action.

None of these situations should be confused with what is needed to make the blackjack side of our lives perform optimally. There is also a refusal to admit how in normal circumstances that this wild ride during the growth phase of one's playing should not go on forever. It is temporary or should be.

Yes. One needn't be overly conservative with a bankroll that could easily be replaced. Even Peter Griffin says, "when the truly degenerate gambler is wiped out of one bank he need only go back to honest work for a few months until he has another." (p131 TOB)

I think there should have been more response on my Ozzy example--perhaps I was too subtle at first?--on how those who can stand a wild ride growth phase are often the high funtioning insane. Or the unfortunately experienced due to reasons unrelated to blackjack. Call them the numb or shell-shocked maybe? Grifter probably falls into that category a bit also (it is no secret he has all of those team play ventures he trolls for because of restitution payments biting off each sucessful trip share---see his posts here and on the CCC), as well as Leaving. How did everyone but me not notice there really is an F formula, for other than k=1, unless I missed it being discussed due to it only being discussed on Green or in the Workshop? I noticed it just hanging in one of mathprof's posts on the ups and downs of betting k=1. I would think you would feel relieved to know that your probability of your bankroll being down F fraction is between F/9 and F/16, by being able to put a number on this. If discussion was not hidden away, behind the Green Doors, I would feel outraged that no one else thought it was important enough to tell you how to do this estimate.

There is even a proof available if you take the original Kelly paper, and mathprop's posting, and show how the further relative distribution, for proportional betting, goes down by the square of the number of units in your bankroll, including for ruin. There is even an alternative proof from a section of a link Leroy Nimka had for proving the normal distribution, on bjmath.com

if the casino takes that kind of action.

It seems that many high stakes players play to a low ROR. But I am wondering if that is more due to a massive bank, casino tolerance and CTR issues, than risk management.

All things considered, do you think you would play more aggressively than 1/3 to 1/4 kelly with a $25K bank?

Secondly, how do you handle casino tolerance and risk? In a simple case, if casino A can handle 1.5x max bet as casino B, do you kind of fudge the risk as an average of the two bets, "overextending" a little at the tolerant casino?

Thanks as always,

motheye

If the 1x Kelly bet is 50% of bank, then a 2x Kelly bet is 100% of bank, no?. Lose that one and you're broke.

But 1x Kelly and less, I agree you'd never go bust. (However, you may practically be bust as your bank drops below table minimums.)

All proportional betting schemes are theoretically zero risk of ruin.

This is only true if by "ruin" you mean literally hitting zero. There is another definition of more practical interest to gamblers, IMHO.

With k > 2, in time your bankroll will "almost surely" (ie. with probability as close to 1 as desired) dwindle to less then any prespecified amount. This is a common definition of ruin outined in probability texts, and by Thorp in his paper in '97 at Montreal. This can not happen at k=1. This is also touched on by Carlson in BJfB. This provides considerable comfort, IMHO. I agree it provides little comfort to know you will never hit BR = 0 if your bankroll is spiralling downward with no end in sight. That's why I think this is a better definition of "ruin."

For example, suppose you start with a bankroll of $10,000. It's little comfort to know BR will never theoretically hit zero, if you know it will become $1.00 with probability .9999 in a few thousand coin flips, and $0.00001 in a few million. OTOH, with positive growth (Kelly's g > 0) this is not true, and in fact you will reach any positive goal in time "almost surely." I think this is a very important, practical way to look at things for your average gambler.

10X Kelly ... YOWZER! Sends shivers up the spine. :>)

ETF

It seems that many high stakes players play to a low ROR. But I am wondering if that is more due to a massive bank, casino tolerance and CTR issues, than risk management.

It is all of those things. Casino tolerance is the main issue.

All things considered, do you think you would play more aggressively than 1/3 to 1/4 kelly with a $25K bank?

That depends. If I had a profession where I had a net income of $150,000/yr then I would probably play to a higher ROR with a 25K bank. If I were a struggling pro bj player down to his last 25K I certainly would not be betting bigger. I would be sticking to a max of a about 2X $150 and doing my best to live off of comps while I rebuilt my bank. Either that or I would be seeking financing from a team.

Secondly, how do you handle casino tolerance and risk? In a simple case, if casino A can handle 1.5x max bet as casino B, do you kind of fudge the risk as an average of the two bets, "overextending" a little at the tolerant casino?

No. Long term ROR is not based on hours played. If you play half your hours at each of two games, your overall ROR is that of the game where you take more risk. I don't know if I'm being completely mathematically precise here. But MathProf told me something like, "two mistakes don't average out to make things right. You just have two mistakes."