Sun Runner: Lifetime Bank

[Sun Runner]

(This question was spawned from a post on Parker's Pages. Any thoughts would be appreciated.)

My question goes to how to handle a 'lifetime bank.'

If I had a BR in hand of $50K, I could set up a game plan for that and it might crank out a max bet of say $300 and a respective ROR of say 1% (I'm guessing about the 1%.)

If I had a known BR, to be collected at the rate of $2,000 every two weeks for the next twelve and a half months -my 'lifetime bank' is also $50K.

If I said my ROR is 1% on the $2,000 stipends, I would never get off the ground (to conservative.) If I increased my max bet to $300 while playing my stipends (to risky), chances are excellent that the flux wipes me out every week I show up to play.

So the question, how do I calc the optimal bet for a series of $2,000 stipends which eventually total $50K and keep the same ROR as above?

If an answer is forthcoming I'm hoping it is in the form of a transportable formula as I am definitely not a math guy!

Thanks for any input.

[Don Schlesinger]

To catch everyone up, I'm reposting two responses that I made on the Free pages:

1) I think this question more properly belongs on the Theory and Math page of Don's Domain. I've never thought about it before, and, frankly, I've never seen the question asked, either. I don't think it has a trivial, or intuitive, answer.

The reason is that the correct way to play clearly lies in between playing each week just as if your bank were no more than the money you actually have at the time (too conservative) and playing as if you had all the future, promised, money right now (too risky). The right answer is not apparent to me, so I'll try to give it some thought.

2) I have no doubt that the BJ problem is quite solvable, as it is posed. I just don't know the right answer at the moment.

There is actually a kind of related problem that Johnny Chang asked me about and that he hadn't yet solved. It concerned having a large bank at home and bringing only a portion of it with you on a trip. The question is: How do you play on that trip? As if you had just the money that's with you (too conservative), or as if you really had all the rest of the money, but which you can't access on this trip (too risky)?

The answer is clearly a function of how long the trip is going to last. If it's very short, you play just about as if you had all the money with you, because, should you tap out, you once again have access to the full bank rather quickly. But, suppose the trip is going to last a year? Then, clearly, you have to play as if the money that's with you is all the money that you have, because you can't afford to tap out and not be able to play for so long a period of time.

So, the question is: How do you play for intermediate periods of time, such as, say, two weeks or a month? Again, I don't know the answer right now.

Don

[Zenfighter]

The reason is that the correct way to play clearly lies in between playing each week just as if your bank were no more than the money you actually have at the time (too conservative).

Actually, the conservative approach looks like the rationale solution to this intriguing problem. How far the player can go increasing his/her risk and survive is not an easy question and would need a lot of research and work, to answer it properly.

and playing as if you had all the future, promised, money right now (too risky)

Here too risky and being continuously wiped out seems to be synonyms. With the aid of Auston?s BJRM, a practical example can be worked through.

Rules: 6dks, s17, das, 4.5/6 dealt out, Hilo count w/IL18
Spread: 25$ to 300$ (max bet at TC=4), leaving any table at TC <= -1
Initial bankroll: 2000$ (80 units)
Hours played every two weeks: 20

1) Hours: 0 ? 20
Avg bet = $ 58
Sd/hnd = $107
Win/100 =$ 51.50
Expected win = $ 1030
RoR = 80.81%
Cum. Bankroll = 3030$

The survival player can be somewhere around his expectation. Above, below, it doesn?t matter. We?ll take here the long run figure, to simplify the problem.

2) Hours 20 ? 40 (with p = 19.19%)
Initial bankroll= 3030 + 2000 = $ 5030 (201 units)
RoR = 58.55%
Cum. Bankroll = $ 6060

3) Hours 40 ?60 (with p = 19.19 * 41.45 = 7.95%)
Initial bankroll = $ 8060 (322 units)
RoR = 42.42%
Cum. Bankroll = $ 9090

4) Hours 60 ? 80 (with p = 19.19 * 41.45 * 57.58 = 4.58%)
Initial bankroll = $ 11090 (443 units)
RoR = 30.73%
Cum. Bankroll = $ 12120

5) Hours 80 ? 100 (with p = 3.17%)
Initial bankroll = $ 14120 (564 units)
RoR = 22.26%
Cum. Bankroll = $ 15150

6) Hours 100 ?120 (with p = 2.47%)
Initial bankroll = $ 17150 (686 units)
RoR = 16.09%
Cum. Bankroll = $ 18180

7) Hours 120 ?140 (with p = 2.07%) and so on.

A fortunate soul among 50 players!

A problem with emotional implications for me. One of my best friends, an European counter, seems to be the father of this ?Virtual Banks?, as he used to call them. A depressive story full of debts and false illusions. He?s actually trying his luck at poker. No longer a card counter anymore. Sad. :-(

Hope this helps somehow.

Regards

Zenfighter

[Don Schlesinger]

Very interesting approach. Of course, with it, 4 out of 5 such players are wiped out on their very first try. So, the problem becomes one of trying to balance ROR with e.v., and, in that regard, perhaps there is no single "optimal" approach. But, I wonder if there isn't an approach that is the equivalent of the full-Kelly approach, when the bank is fixed. In other words, if I wanted to play the equivalent of the risk that a full-Kelly player takes under "ordinary" circumstances, what would I do?

Don

[Sun Runner]

> Very interesting approach. Of course, with
> it, 4 out of 5 such players are wiped out on
> their very first try.

Yes, however, you have $2,000 coming in two weeks, and $2,000 coming in two weeks more, etc.

So you get wiped out 4:5 times. 'Random Poster' eluded to this but felt (intuitively) that the 'fifth' breakout session would send you down the trail to N0 and beyond.

Any thoughts?

How about this -how far should the 'max bet' be moved backwards, in each example, so as to bring ROR equal in both cases ($50K in hand or $2K stipends for 12.5 months .. looking for an equal max bet and an equal ROR for the same game?)

[Sorry, its the best I can do; I am hopeless in determining the path to the answer .. if it exists at all.)

[Don Schlesinger]

> Yes, however, you have $2,000 coming in two
> weeks, and $2,000 coming in two weeks more,
> etc.

That's included in Zen's example. He added the extra $2,000 every two weeks.

> So you get wiped out 4:5 times. 'Random
> Poster' eluded to this but felt
> (intuitively) that the 'fifth' breakout
> session would send you down the trail to N0
> and beyond.

Doesn't seem to work that way. See Zen's example.

> How about this -how far should the 'max bet'
> be moved backwards, in each example, so as
> to bring ROR equal in both cases ($50K in
> hand or $2K stipends for 12.5 months ..
> looking for an equal max bet and an equal
> ROR for the same game?)

No, not quite the solution.

> [Sorry, its the best I can do; I am hopeless
> in determining the path to the answer .. if
> it exists at all.)

I have no doubt there's an answer, once we agree on what we're trying to achieve. But that answer is neither intuitive nor simple, I'm quite sure.

Patience. We'll get there.

Don

[Zenfighter]

if I wanted to play the equivalent of the risk that a full-Kelly player takes under "ordinary" circumstances, what would I do?

For our player with his initial bank of $2000, there is no way he can mimic the above given example, where the table is left, after seeing the count dropping below TC = -1., while keeping a fixed 13.55% RoR. Another question is Wonging at Tc1. For example:

Tc = 1 $ 5
Tc = 2 $ 14
Tc =>3 $ 33

W/100 hnd = $ 7.10
RoR= 13.55%

Assuming here, that our counter will survive and realize his expectation, the second plateau with the new dollars supply added, should be carefully adjusted again, so as to have an optimal (full-Kelly) betting ramp. A matter of time, before he can sit at the table, mimicking the White - Rabbit player.

As a general rule full-Kelly looks wonderful on paper, but getting away with it is another question. What makes the whole difference in the World here is that our player will receive a continuous supply of money every two weeks. From a psychological point of view, the benefits are unquestionable. Full-Kelly is the upper boundary of what rationale betting should be. Needless to say.

Zenfighter

[Don Schlesinger]

> if I wanted to play the equivalent of the
> risk that a full-Kelly player takes under
> "ordinary" circumstances, what
> would I do? For our player with his initial
> bank of $2000, there is no way he can mimic
> the above given example, where the table is
> left, after seeing the count dropping below
> TC = -1., while keeping a fixed 13.55% RoR.
> Another question is Wonging at Tc1. For
> example:

> Tc = 1 $ 5
> Tc = 2 $ 14
> Tc =>3 $ 33

> W/100 hnd = $ 7.10
> RoR= 13.55%

> Assuming here, that our counter will survive
> and realize his expectation, the second
> plateau with the new dollars supply added,
> should be carefully adjusted again, so as to
> have an optimal (full-Kelly) betting ramp. A
> matter of time, before he can sit at the
> table, mimicking the White - Rabbit player.

> As a general rule full-Kelly looks wonderful
> on paper, but getting away with it is
> another question. What makes the whole
> difference in the World here is that our
> player will receive a continuous supply of
> money every two weeks. From a psychological
> point of view, the benefits are
> unquestionable. Full-Kelly is the upper
> boundary of what rationale betting should
> be. Needless to say.

Not only from a psychological view, but from a mathematical one, as well. If you are certain to have a somewhat replenishable bank, then playing full-Kelly simply with the money that you currently have on hand seems too conservative to me. You should be playing for more than the person who has just that amount of money with no hope for an "infusion" of more capital in the immediate future.

Or, so it seems to me.

Don

[Zenfighter]

The reader?s main goal seems to be, to achieve and/or to consolidate a 50K bankroll.

If I had a known BR, to be collected at the rate of $2,000 every two weeks for the next twelve and a half months -my 'lifetime bank' is also $50K.

After doing this he will be able to:

If I had a BR in hand of $50K, I could set up a game plan for that and it might crank out a max bet of say $300 and a respective ROR of say 1% (I'm guessing about the 1%.)

So let us do not lose perspective on this, his main goal being to arrive at the desired level.

So far we have considered the following:

1) Staying quietly at home without playing a single hand. There is a 100% probability of having 50K after one year and a half-month.

2) Starting to spread $25 to $300 like as if he had the promised money in his pocket, actually. An announced failure, surely.

3) Flirting with full-Kelly and/or 1.5 or double-Kelly. Who knows?

While building his bankroll, let?s assume the following:

1) The player will play 20 hours every two weeks with a fixed RoR = 13.55%. He will never adjust or reduce his bets during any given session. Here a session is, obviously, playing 20 hours straight.

2) Money won plus the added one is always risked as stated, until reaching the 50K level.

3) What if we consider the 25 sessions (more or less, as a function of the gain/losses) as independent trials? (While building the bankroll). With a fixed p = .8645 (success) and a q = .1355 (failure), we can a apply a binomial test, in order to evaluate our chances.

Here we go.

 

Succ.Failures  Probability Cum.Probability 

0 	25 	0.000%  	 0.000%  

1 	24 	0.000% 	         0.000%  

2 	23 	0.000% 	         0.000%  

3 	22 	0.000%  	 0.000%  

4 	21 	0.000% 	         0.000%  

5 	20 	0.000% 	         0.000%  

6 	19 	0.000% 	         0.000%  

7 	18 	0.000% 	         0.000%  

8 	17 	0.000%  	 0.000%  

9 	16 	0.000% 	         0.000%  

10 	15 	0.000% 	         0.000%  

11 	14 	0.000% 	         0.000%  

12 	13 	0.000%  	 0.001%  

13 	12 	0.003%  	 0.004%  

14 	11 	0.016% 	         0.020%  

15 	10 	0.077% 	         0.097%  

16 	9 	0.306% 	         0.403%  

17 	8 	1.034% 	         1.437%  

18 	7 	2.932% 	         4.369%  

19 	6 	6.893% 	        11.262%  

20 	5      13.193% 	        24.455%  

21 	4      20.041% 	        44.496%  

22 	3      23.247% 	        67.743%  

23 	2      19.346% 	        87.089%  

24 	1      10.286% 	        97.375%  

25 	0 	2.625%         100.000%  

 

 

A random walk through 25 full-Kelly sessions (no bankroll adjustments, remember) without a single failure is not an easy thing to achieve!

To be continued, I guess.

Zenfighter

[Don Schlesinger]

> A random walk through 25 full-Kelly sessions
> (no bankroll adjustments, remember) without
> a single failure is not an easy thing to
> achieve!

One of the problems with your approach is that it treats all successes the same -- regardless of the magnitude of the win during the period (session). If, at some point, we reach $50,000, we would not continue to play out the experiment, as we would have reached our goal prematurely -- a little bit like my barrier problems in the book.

So, because of variance of outcomes, we need to consider the random walks not only for "success" and "failure" but for the magnitude of those items, especially the successes (when we fail, we lose everything and are back at zero).

Don

[Zenfighter]

we need to consider the random walks not only for "success" and "failure" but for the magnitude of those items, especially the successes

Absolutely, it didn?t escape to me this fact. As usual, when something is very difficult to asses with simplified math formulas, there is still the possibility to arrive at a more satisfactory solution with the aid of computer simulations. My bet is, that this is the way to go.

when we fail, we lose everything and are back at zero

Yes, but being back to zero at the early stages will have not the same effect that returning to zero at the late ones. We can count only with 25 extra supplies. It will be harder then to achieve our goal.

A disciplined and somehow lucky player can achieve his goal, a possibility. But in general, I?m frankly pessimistic about the chances of success using this path. Personal opinions mixed with fear. The dreaded equation high standard deviation versus the small expectation at our disposal. A never-ending story!

Zenfighter

[Sun Runner]

> The reader?s main goal seems to be, to
> achieve and/or to consolidate a 50K
> bankroll.

No; but mine and Johnny C.'s are different, I'm thinking.

Let's change the parameter a little and define a game.

HiLo, Ill 18, 6D, 75% pen, DA2, DAS, no surrender. I'll tolerate 5% ROR. Let's assume that works out to a 1:12 spread on a $5 unit and a $6K BR. (If you would like to check the math and adjust the BR as needed -be my guest!)

This individual is playing to a $6K BR ceiling because, as he has been taught, that is his BR.

If the 'lifetime bank' concept has any credibility, if this individual knew for certain he not only had the $6K he possess now, but $6K a month to add to his BR over the next 12 months ...

... he should, it would seem, be able to play -today -to either a higher max bet, or a more agressive bet spread, or a higher unit value (or some combination thereof) and STILL MAINTAIN that same 5% ROR.

If not, then I'm wondering if it is possible at all to put defineable actionable feet to the concept of a lifetime bank.

Granted, this individual's game plan would not be as strong in total EV dollars at the end of the year as the individual that started with $50K in hand ... but it should be far and away better than the individual that limits his game plan to a $6K BR in month one, a $12K BR (plus month one addittions, if any) in month two, etc, etc, to year end.

Would seem there would be an optimal unit value and spread he could use -today -and over the next twelve months -rather than reassessing his BR on the first of each month when his stipend comes in.

If not, then the conversation around the net I hear of a lifetime bank is pointless. If not, even though I'm willing to put $72K at risk in my life at BJ, it's pointless if I only have $6K today.

And maybe the lifetime bank concpet is pointless. I thought I have heard Wong speak of it (could be wrong.) I know I have heard DD' and bigplayer speak of it. If an executable plan can not be defined and implemented, then what is the point of discussing it!?

That is my question.

[Cacarulo]

Haven't read all the thread but here are some comments. Let me know what I am missing.

> Let's change the parameter a little and
> define a game.

> HiLo, Ill 18, 6D, 75% pen, DA2, DAS, no
> surrender. I'll tolerate 5% ROR. Let's
> assume that works out to a 1:12 spread on a
> $5 unit and a $6K BR. (If you would like to
> check the math and adjust the BR as needed
> -be my guest!)

> This individual is playing to a $6K BR
> ceiling because, as he has been taught, that
> is his BR.

> If the 'lifetime bank' concept has any
> credibility, if this individual knew for
> certain he not only had the $6K he possess
> now, but $6K a month to add to his BR over
> the next 12 months ...

> ... he should, it would seem, be able to
> play -today -to either a higher max bet, or
> a more agressive bet spread, or a higher
> unit value (or some combination thereof) and
> STILL MAINTAIN that same 5% ROR.

I don't think of this as the equivalent of having $72K before playing. What if you lose $6K on the first day? You can't go home and pick some more until next week. That's not the case if you already have the money.
Think that if you bet more than the determined bet for a $6K BR your ROR would be higher and so your probability of going broke. At least for that week.

> If not, then I'm wondering if it is possible
> at all to put defineable actionable feet to
> the concept of a lifetime bank.

> Granted, this individual's game plan would
> not be as strong in total EV dollars at the
> end of the year as the individual that
> started with $50K in hand ... but it should
> be far and away better than the individual
> that limits his game plan to a $6K BR in
> month one, a $12K BR (plus month one
> addittions, if any) in month two, etc, etc,
> to year end.

I like this plan better although I think there should be an answer to the previous. Perhaps we should think of an year-BR of less than $72K but more than $6K. What is the optimal number? I don't know yet.

Just some thoughts.

Sincerely,
Cac