Sun Runner: Lifetime Bank

[Don Schlesinger]

I agree with everything that both Sun Runner and Cacarulo have written in the above two posts. All of your comments are correct.

At the risk of sounding a bit cocky, if this problem were easy to solve, I would have given you the answer already! :-) The point is, it is neither easy to solve nor intuitive as to what the correct answer is.

Again, I know this much: Lifetime ROR formulas assume that you establish a bank, play in a certain manner, never alter the bet size, and either win all the money in the world or go broke trying. So, if you infuse $2,000 additional dollars into the bank every two weeks, you are clearly going to stave off some situations where, had those dollars not been forthcoming, you would have gone bankrupt.

But, the promise of future dollars can help only so much today, because, if I overbet my current bankroll by too much, I may never get to play for the two weeks, before I get the new money. And, losing that playing time, and the e.v. associated with it, might be worse than playing lower stakes while waiting for the new funds to arrive.

To solve such a problem, we need to be able to state very clearly what it is we are trying to maximize, or optimize. We need a clear vision of what we're trying to accomplish before we can even hope to flesh out an answer. Mind you, even when we have such vision, I'm not saying that the solution will be simple; it won't be.

I guess we're trying to maximize our risk-adjusted return for a given period (say the year), which is equivalent to saying that we'd like to play in a manner that will maximize our SCORE. This, as you know, involves a tradeoff between e.v. and variance, or ROR.

I think that this and the Johnny Chang problem are both quite fascinating, and I look forward to using this forum as the venue for the eventual solution to the problem(s).

Don

[Sun Runner]

> Let me know what I am missing.

> I don't think of this as the equivalent of
> having $72K before playing.

You are right; it is not. The concept of a lifetime bank being if you only have $6K to put in play you better work out a game plan for it and not overbet your BR. However if you have $6K today and know you have the ability to add to it at a semi-fixed rate with a series of inflows (up to a stated maximum) then you are limiting your potential today by holding fast to the previously mentioned playing limitations imposed on your $6K BR.

> 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.

Of course.

> 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.

Of course.

> I like this plan better ...

If I had the $72K in hand now, I'd like it a lot better; but I don't.

> ... although I think
> there should be an answer to the previous.

Me too!

Seems I should not have to limit my play today with $6K in hand if I know that I'll have $72K in hand over the next twelve months.

That's what it seems -but I can't quantify how much I can relax (expand) my current play based on that knowledge.

> 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.

Now your getting it. Somewhere between $6K and $72K lies the optimal playing condition (unit size, spread, max bet.)

BTW, one year is fine because it easy to talk about. Truthfully, if only talking about a 12 month wait, the prudent (certainly easier) path would be to be patient and wait to collect the full amount before proceeding. But what I am really looking at here is the concept of a 'lifetime.'

What if I have $6K today and can only replace it at the rate of $6K per annum for ten years?

Again, this may be an archaic question with little practical value but it is talked about occassionally like it is a concept that has merit.

> Just some thoughts.

I appreciate your responding. Your concise answer a day or two ago as to how to quantify the difference in PE values was great.

[Sun Runner]

> At the risk of sounding a bit cocky, if this
> problem were easy to solve, I would have
> given you the answer already! :-)

I know you would have!

And I don't pretend to even be lining up with ya'll if this ever get's worked out because I am just not capable. If left to my own; I'd be working on something else because 'every man needs to know his own limitations' and I know mine; I'll just be happy enough to be standing around for the answer!

My only purpose in continuing was to as best I can describe the problem I'd like to see solved. Not being able to even know where to begin, my requests are simply that, requests. Kinda like Christmas. If/when someone stumbles across the answer, that will be great. If not, it's still all good.

I think I have described what I was looking for and now will probably stop contributing much more and wait patiently for the work someone else will have to do if it gets done at all.

Meanwhile, I'll keep working the 37 cents I have in hand as best I can.

[Zenfighter]

What about using Table 8.4 for his selected 5% RoR and treating his initial $6000 as a Trip Bankroll. Taping out early will mean going back home until next month, with the wasted hours obviously, but I do not see another solution to add more light here, than switching to Trip Bankroll probabilities. Leaving home pretty soon, so I do not have the time to elaborate more in depth my current thoughts regarding this.

Regards

Zenfighter

[bfbagain]

To be fair, (and hopefully I won't embarrass myself by not doing so) I haven't as yet read the free pages posts regarding this topic, so I may have missed a lot here, but I just returned home, so please accept my apologies beforehand. I'm also pretty tired, so that may contribute to some errors as well, but here goes.

After reading all of the posts in this thread, it appears that there has been something that has been missed, and that's the genus of SR's posts. Plese correct me if I'm wrong SR, but I too have read "others" posts about "lifetime" banks, and I think what you're asking is, is there really such a thing as a "lifetime bank" as it relates to "real world blackjack play." And more specifically, playing blackjack now, in the real world!

In other words, how does a player, who has the expectation of achieving an "in hand" bankroll of X dollars over a period of time, play to that larger bank....NOW. And the reason that a player wants to be able to do that, is that playing to a small bank, is like watching paint dry in today's real world of advantage blackjack play.

So while Don and Zen and Cac, and anyone else attempts to come up with some real world math type answers, I'll make an attempt to come up with some plain real world answers.

Real world, as in high stakes blackjack in today's world, and how bankrolls need to be assessed.

The concept of a lifetime bank is very misleading. The reason that it's misleading, is that a lifetime bank is not really a lifetime bank, in that it's a not a magical number that will be attained at some point in time in the future. It's a "total" bank that can be accessed....today, if necessary.

What SR's interpretation of a lifetime bank seems to be, is a bank that is replenishable. A totally different concept. A replenishable bank merely allows one to "restart" a new trip so to speak. It is not, what some interpret, as part of a lifetime bank.

An example is in order:

Assume that someone wants to play at the hundred a hand level. Determining what the required bankroll requirements are, are no longer a mystery to anyone who seeks the information. Merely buying Don's wonderful book will provide that information. Or Norm's unbelievable software.

But what if you want to play at that level, but you only have half of the money in available cash to you at the moment. Do you play at a smaller level?

This is where the concept of lifetime bank comes into play. Not replenishable money, but money that can be accessed immediately. And what you're willing to risk to put it into play. Assume that you are going to play DD with the Mirage rules, i.e., at the Mirage. You will use HiLo, I18, Fab4, heads-up (not my system, but it is for many). To play a 1-8 spread, at 2 hands, you need a BR of 65K at full kelly.

However, you only have 35K in available cash. (almost like a trip bank) But....you have 100K in a portfolio account, a house worth 250K (with 150K available for a line of credit), a 401K worth 200K...I think you get the picture. What a lifetime bank provides you are real assets that can be accessed if you need it, and quickly. Stipends aren't going to make it.

A stipend is replenishable, and that makes it more like a trip br.

The problem with these two approaches is that they mask what an actual bank requirement is.

A trip bank is designed to allow play at predetermined levels to withstand extreme fluctuation without requiring additional dollars for the required trip hours of play. It is not foolproof, clearly. But it is usually accurate. A trip bank is a subset of total bank, meaning that there are additional dollars that can be had immediately to continue play without waiting if your trip bank was tapped out at the end of a trip. (hopefully, at the end, if it happens. )

In the end, although minimizing risk while maximizing potential return is the goal, the operative word here is risk.

A lot is focused on risk, as well it should. However, there is a requirement of risk, if there is to be a worthy reward, thus the reason to play at higher levels.

And that is, I believe, the reason that SR has broached this question.

Unfortunately, IMO, there has to be a certain level of dollars available in order to play at reasonably higher levels that make the game both more enjoyable, and more rewarding, none of which is really the case at low limits, if advantage play....real advantage play, is the goal.

There really is no substitute for having the hard cold cash needed to play effectively. If that means you need to not play, and accumulate the dollars to risk at some point in time, so be it. The reason is, that that is the time that your replenishable "stipend" (adding to a virtual bank so to speak) can play a role, allowing a slightly higher playing level, i.e., maybe going from a $50 minimum, to a $75 min. to be "risked" in the form of that higher bet level.

Just some late night thoughts. Am going to bed now.

cheers
bfb

[Sun Runner]

> After reading all of the posts in this
> thread, it appears that there has been
> something that has been missed, and that's
> the genus of SR's posts. Plese correct me if
> I'm wrong SR, ...

No, you hit the mark.

While I think there must be (may be) some coorelation between the two scenarios I keep using as examples -I am coming to wonder if what I believe the commonly held notion of a lifetime bank to be may be simply in error; non existent if you will.

For now, for me, probably I need to spend some more personal time in trip BR theory. No doubt I'll start in BJA3!

Trip BR theory, heretofore, has been something not particularly important to me as my BR fits on my person fairly easily and I'm not that much of a believer in session BRs. My sessions are generally limited by time (60 minutes MOL), not losses. I have plenty of time to review my play while moving from place to place -I don't need a session BR for that.

Thanks as always.

[bfbagain]

To be fair, (and hopefully I won't embarrass myself by not doing so) I haven't as yet read the free pages posts regarding this topic, so I may have missed a lot here.....

Well, now that I've read the posts on the free pages this morning, I should have looked at that them a little before I posted. Oh well. That's what flying and alcohol will do to you. Maybe I'll be forgiven. And then again, maybe not.

As to the topic at hand, the intermediate BR requirements are the issue at hand. I think that Johnny Chang's issue is the key to SR's question being successfully answered.

Don had a (I'm sure -- maybe) tongue in cheek answer to having a bet spread and being kicked out in ten minutes (if I recall this correctly), and its along these lines that I may be able to offer some assistance.

Unlike the game of chess, in that you are allowed to play a "perfect" game without being excluded, blackjack played in casinos is a totally different animal.

Granted, this "doh" type statement is well known, and doesn't have an appearance of answering or solving real bankroll type issues, but it is something that needs to be considered.

Being too "conservative", or too "risky" as it relates to betting levels, is why understanding the need to have available cash on hand is an important requirement in solving this question.

In fact, this question may not have an answer that's applicable to the real world of casino play for small bankrolls.

In my Mirage DD example (with a 64% pen.), even at the $25 min bet level, you need 16K at full kelly, and 23K for a 5.8 RoR (.7 kelly.) This is for a win rate of $149 /hr, a DI of 9.6, c-score of 92.17, a CE of $74, and an N0 of 10,850.

Now we can go through a 6 deck illustration, at smaller bet levels, and it becomes apparent that blackjack either is entertainment or a passing of time activity, but growing bankrolls it is not.

This is the unvarnished truth! Not in todays world anyway. And that's what makes this such a dilemma for new and/or underfunded players who want to make their time in casinos worth something.

I have no doubt that there is an acceptable solution to this, albeit a higher degree of risk assigned to it, and that is where I believe the solution will reside, unfortunately.

Even adequately financed players have severe variance, and that in and of itself will test both your resolve, and your very soul, not to mention your finances, so I believe that Don and other's who will attempt to solve -- mathematically, this problem, are approaching it with the gravity it deserves. But for everyone else, the mathematical numbers alone, are not the only aspects that need to be considered. Your tolerance of risk, not your BR RoR, will ultimately decide how you view a viable playing bankroll.

When DD discusses lifetime bank (and I'm certainly not talking for him as I don't know him) the essence of what he means, IMO, is that he will tap into whatever assets he has available to continue playing at his predetermined RoR that he's comfortable with, as do I.

FWIW, a $50K BR, with a continuing stipend of $4K/mo will allow anyone to play at an acceptable bet level, with a low enough RoR, to realize a fair return on their investment dollars, which is how I look at the game of blackjack. And I hope future serious players will look at it as an investment as well. That's what separates winning counters, from losing counters. I do agree with AS (as Norm pointed out) that you will lose, but only if you're a gambler. Become an investor, and the chances that you'll succeed, have just increased.

cheers
bfb

[Karel Janecek]

In order to find the optimal strategy it is correct to maximize the expected utility of some farter out time horizon. For this kind of option-related problems it is not correct to maximize SCORE or similar measures. The reason is that when the agent is low on cash and is forced to make small bets, the remaining risk and wealth fluctuation (equivalently standard deviation) are not significant, and all that matters is the expected profit.

First, let me note that under the assumption of the so-called 'complete' market one should calculate the sum of present value of futures cash inflows and include those into the personal investment wealth, or bankrol. Complete market would mean that the agent can freely borrow funds. However, this assumption is not realistic. Let us assume next that the agent cannot borrow anything. (The analysis would be even more complicated if one could borrow some funds, or perhaps borrow for higher interest rate.)

As has been correctly conjectured in other posts, the optimal strategy of the player should be much more aggressive due to future income, however, not too aggressive since when tapping out temporarily, the agent experiences the opportunity cost of not beeing able to play until the next cash inflow. The correct strategy is clearly to increase the bets as the next cash inflow is approaching and/or the player has made a profit. On the other hand, it is rational to decrease the bet size temporarily when the player loses.

I ran exactly into this problem with my wife during our last trip to Las Vegas in May of this year. Our available cash-on-hand was limited, and so we started with some constrained bet size. After having experienced few lucky swings we were ahead and so we decided to increase are unit size. Unfortunately, bad swings happened and we were forced to scale down even more. Towards the end of the trip we were almost out of cash, however, we did not scale down any more since the trip was about to end and so we did not mind going broke. (In fact, we were very lucky last day and won the entire loss back.)

The qualitative description of the optimal strategy is clear:
1) For a given fixed amount of cash-on-hand, increase the bet size as time passes.
2) Increase, resp. decrease, the bet size noticably, as soon as one experiences a larger positive, resp. negative swing.

The precise calculation is probably quite a bit more complicated, and I conjecture that an explicit formula might not be available. As mentioned above, the correct objective is to maximize expected utility. One could certainly perform some kind of complicated Monte-Carlo simulations to find the optimal bet as a function of time remaining and current cash-on-hand.

The problems seems very interesting and I might try to do some more research in this area. A related question is the optimal investment in a risky asset (say a stock market) for an agent with a given wealth but low cash-on-hand, who cannot borrow. In case I obtain any quantitatively useful results I will post them on these pages.

[Don Schlesinger]

Happy to see you posting here, Karel. Thanks for your valued thoughts. I have a quick question and then a comment or two. With interest rates so very low at the moment (although just raised for the fifth time this year by the Fed, to 2.25%), when one takes the present value of the bi-weekly inflows of $2,000 for, say, one year, it's almost like having the $52,000 right now. So, is that what you would call the current bank? That sounds much too risky to me. What am I missing?

> The problems seems very interesting and I
> might try to do some more research in this
> area.

I would love you to. And, I'm glad that your assessment is that the answer isn't something trivial, which would have been embarrassing, as I stuck my neck out conjecturing that this is not easily solved.

> A related question is the optimal
> investment in a risky asset (say a stock
> market) for an agent with a given wealth but
> low cash-on-hand, who cannot borrow.

That is precisely the Johnny Chang problem, described above. You have a large bank at home, but you go on a trip with only a small portion of the cash, and you don't have access to the rest of the money. What is your optimal strategy? Chang surmised (I'm sure correctly) that, if the trip is very short, you are quite bold, since if you tap out, you will once again have access to new funds rather quickly. As the trip length stretches out, you begin to play more and more as if the funds with you are all that you possess, since you have to make the current money last -- perhaps for a very long time. Lose it, and you're out of the game, sitting on the sidelines.

It's the middle ground -- a trip of intermediate length -- that poses the biggest problem.

> In case
> I obtain any quantitatively useful results I
> will post them on these pages.

That would be great!!

Don

[Karel Janecek]

> With
> interest rates so very low at the moment
> (although just raised for the fifth time
> this year by the Fed, to 2.25%), when one
> takes the present value of the bi-weekly
> inflows of $2,000 for, say, one year, it's
> almost like having the $52,000 right now.
> So, is that what you would call the current
> bank? That sounds much too risky to me. What
> am I missing?

Yes, in fact that would be the correct calculation assuming that you can *borrow* for this interest rate. However, even if the market was complete, the interest rate for borrowing would be larger. For farther out cash inflows one should use the long interest rate, which is yet higher. For example, the 5 year forward is almost 5%.

> It's the middle ground -- a trip of
> intermediate length -- that poses the
> biggest problem.

In fact, I would think that even a long trip poses a problem. It is not clear how much more one should be conservative. A very short trip, with small probability of tapping out, is not a problem.

Karel

[Don Schlesinger]

> Yes, in fact that would be the correct
> calculation assuming that you can *borrow*
> for this interest rate. However, even if the
> market was complete, the interest rate for
> borrowing would be larger. For farther out
> cash inflows one should use the long
> interest rate, which is yet higher. For
> example, the 5 year forward is almost 5%.

Understood. But, say I bring $2,000 with me on a trip and I'm expecting a total of $52,000 ($2,000 bi-weekly) for the next year. Furthermore, to simplify matters, interest rates are zero. Does it really make sense to play that very first session as if I had $52,000 and overbet my current bankroll by a factor of 26? I'm having trouble accepting this.

> In fact, I would think that even a long trip
> poses a problem. It is not clear how much
> more one should be conservative. A very
> short trip, with small probability of
> tapping out, is not a problem.

Well, one thing is clear to me. If there is a formulaic approach, that formula is a function of the ratio of the cash on hand to the total somewhat, or completely, unavailable bank, and it is also a function of the length of time that the additional funds will be unavailable -- naturally, with longer leading to more conservative play.

But, I've stated the obvious, I'm afraid.

Don

[Karel Janecek]

> Understood. But, say I bring $2,000 with me
> on a trip and I'm expecting a total of
> $52,000 ($2,000 bi-weekly) for the next
> year. Furthermore, to simplify matters,
> interest rates are zero. Does it really make
> sense to play that very first session as if
> I had $52,000 and overbet my current
> bankroll by a factor of 26? I'm having
> trouble accepting this.

Why are you having trouble with this? The important (and unrealistic) assumption is that you can borrow the cash. If the casino is willing to repeatedly give you a loan as soon as you lose your current cash, then it does not matter if you already have the $52,000 or not. Or take it this way: You know that you'll have $52,000 in one year, you take a loan of $52,000 from the casino right now, for zero interest rate, and pay in a year. This way, assuming zero interest rate, you effectively have the $52,000 available right now, so there is no difference.

I did a bit search for work done in the spirit of our problem. (Unfortunately) it seems that the theoretical work has already been done. I found a paper by Claus Munk, 'Optimal Consumption/investment policies with undiversifiable income risk and liquidity constraints', Journal of Economic Dynamics & Control, 1999. The author resolves even more general problem when the future income is stochastic.

I am afraid, however, that this paper does not provide immediate quantitative results. In order to get hard numbers one would have to do numerical solutions of complicated differential equations (the Hamilton-Jakobi-Bellman equation). This is a lot of work indeed.

Best regards,

Karel

[Don Schlesinger]

> Why are you having trouble with this? The
> important (and unrealistic) assumption is
> that you can borrow the cash. If the casino
> is willing to repeatedly give you a loan as
> soon as you lose your current cash, then it
> does not matter if you already have the
> $52,000 or not. Or take it this way: You
> know that you'll have $52,000 in one year,
> you take a loan of $52,000 from the casino
> right now, for zero interest rate, and pay
> in a year. This way, assuming zero interest
> rate, you effectively have the $52,000
> available right now, so there is no
> difference.

Your way is clear and no problem, but the original question, as stated, mentioned nothing of a loan nor of casino credit, knowing that you would pay it back later. So, I'm not sure that we should proceed on that assumption. In fact, I'm sure that we should not. Rather, we might have to proceed on the basis that the money we have with us is ALL the money that we can play with, until future funds arrive. So, now, I assume you see what a big difference that makes. In essence, the future funds simply guarantee that we won't be bankrupt for any long period of time; they do not, however, imply that that money is available to us today. Clear? (By the way, the Johnny Chang problem would make this same assumption, I'm quite sure.)

> I did a bit search for work done in the
> spirit of our problem. (Unfortunately) it
> seems that the theoretical work has already
> been done. I found a paper by Claus Munk,
> 'Optimal Consumption/investment policies
> with undiversifiable income risk and
> liquidity constraints', Journal of Economic
> Dynamics & Control, 1999. The author
> resolves even more general problem when the
> future income is stochastic.

> I am afraid, however, that this paper does
> not provide immediate quantitative results.
> In order to get hard numbers one would have
> to do numerical solutions of complicated
> differential equations (the
> Hamilton-Jakobi-Bellman equation). This is a
> lot of work indeed.

Well, it's a start. You've located a reference, and you've confirmed that we're dealing with a non-trivial problem! Maybe one day, you'll want to try to tackle this, and we'll publish the research here or -- dare I think it! -- BJA4?! :-)

Don