Bankroll Theory Question for BP, MathProf, Stealth, DD...

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 the counter to divide his BR into 5 bankrolls of $10K and play full kelley on each individual one? In the long run would there even be a difference?

Thanks for any insight guys,

-MJ

The way I see it is; playing full Kelley on 50K that is stored in one place is the same as playing full Kelley on 5 - 10K rolls stored separetly and played one at a time.

Is there something I'm missing here?

In practice things are always different than in theory. Nobody would take his full 50k bankroll to the casino. Dividing it into 10k session bankrolls sounds reasonnable. But now you should still play to the full 50k bankroll may be with half Kelly as you say.You want to avoid resizing which would be inevitable if you played to the session bankroll.
The Kelly principle always applies to the full bankroll. Not to any artificial divisions thereof.Note that losing a session bankroll is not a tragedy. You strike back another day.

Francis Salmon

Your question seems to make a common false assumption. The Kelly optimum is not a static value. It changes as your bankroll changes. Assume you start with a BK of $50,000. You win your first hand of $25. Your bankroll is now $50,025 and your Kelly optimum bet has increased.

This is one of the reasons you can never lose your entire BK if playing Kelly. As your BK declines, your bet size declines as well.

This is one of the reasons you can never lose your entire BK if playing Kelly. As your BK declines, your bet size declines as well.

This is in fact not a special property of Kelly, but common to all proportional wagering systems. If you bet 99.99% of your bankroll every hand, you would never go bust, were money infinitely indivisible. Unfortunately money is not infinitely indivisable. New players shouldn't be confused about the fact that it is perfectly possible to go broke using Kelly.

I think the other posters have addressed your question fairly well. However sicne you directed your question toward me, I will add my comments.

You wrote:

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 the counter to divide his BR into 5 bankrolls of $10K and play full kelley on each individual one? In the long run would there even be a difference?

First, I want to comment on the definition of bankroll. You said �The individual can lose every last penny and it would not effect his lifestyle in the least bit.�

You are using a recreational player�s definition of bankroll: Money I can afford to lose. For us, bankroll is something we can�t afford to lose.

Do you have a 401 K plan? Would you describe it by saying �I can afford to lose every penny and still no effect my lifestyle?� Most would say No. You should think of bankroll the way you think of your 401 K.

Now regarding your issues of playing full Kelly to each fifth of the bank. I imagined you were asking about sending a team of 5 players out with a piece of the bank, and having each of them play full Kelly to their part of the bank. Now this would mean that their initial betting levels would be at the 0.2 Kelly level relative to the whole bank. I don�t that was your original goal?

However, evem if your goal was to play 0.2 Kelly overall, this is a very poor approach. Because some of he players would double or triple their sub-banks, and would then double or triple their betting levels. They would be over-betting relative to the team. And some would lose part of their stake, and start under-betting relative to the team. This inconsistent betting is bad on several levels.

I can elaborate more on this, but I don�t know if it is necessary. I think I will wait for reply, and then follow-up on any issues or questions raised there.

There is a real sense in which betting more than double Kelly has a Risk of Ruin of 100%. For such betting,

Given any epsilon >0, the Probability that Bn < epsilon approaches 100% as n goes to infinity, where Bn is the Bank after n bets.
Thorp discuses this in his Kelly paper.

Thanks MathProf and others for responding.

There is no team play involved in my scenerio. The counter is playing solo. He can either play from the ENTIRE bankroll of 50K betting ONE HALF kelley(roughly 1.822% ROR),

-OR-

he can divide his bankroll in fifths. He would start off with the first 10K, play it at FULL kelley(13.5% ROR), and see where it goes. If he doubles the 10K great!! Then he will resize his unit and play to a 20K bank betting FULL kelley once more. The counter will not touch a penny of the remaining 40K which he has placed to the side.

-BUT-

If he goes broke on the initial fifth of the bank, then and only then will he use the second segment of the bankroll...and so on and so forth...

Basically this question is posed so I can get a feel for whether it is advantageous to play from a BR in segments(playing one segment until it doubles or busts) with higher ROR or is it better to play from the entire BR with lower ROR. Francis's response seems to indicates to just play from one BR.

Thanks for any further clarification.

-MJ

-MJ

The counter will not play from his bank of $50K at full kelley(ROR is too high). Rather, he will bet to ONE HALF kelley.

On the other hand, if he splits it into fifths, then he will play each fifth(one at a time) to full kelley.

I dont think it is the same thing...but I could be wrong. Please see my
post below.

-MJ

It seems that if you bet full kelley on 1/5 of the bankroll, you're actually just betting a portion of kelly on the entire bank of $50K. Thus, you're somewhere in between .5 kelly on the entire thing and full kelly on 1/5. Just take the whole $50K and figure out the weighted average for your ROR. Right? I'm green so take this with a grain of salt.