David Spence: Optimal betting with unlimited bankroll

[David Spence]

Before anyone gets the wrong impression, I am not infinitely wealthy :-) However, my bankroll is easily replenished (due to, heaven forbid, a non-blackjack career), and is not the limiting factor in deciding how much I bet.

My current betting strategy is to pick a somewhat arbitrary unit size and spread with which I and the casinos are comfortable, then use BJA3 or CVCX to determine an optimal betting ramp. It dawned on me, however, that since all of this is predicated on maximizing the log of wealth given a limited bankroll, with associated concerns of ROR, I may not be betting what is truly optimal. My ROR is very close to zero, no matter what unit size I choose (within reason).

If one's bankroll is easily replenished, what is the best way to bet? Is the strategy of betting a large amount whenever one has an advantage, however small, non-optimal in this case? Within reason, ROR=0 whatever the bet size, so why not bet a large amount in all favorable situations? Granted, I probably won't follow this strategy, regardless of whether it's mathematically sound, but I'm wondering whether my current "optimal" betting method has only a psychological basis.

[Blackjack Knight AKA Paladin]

> Before anyone gets the wrong impression, I am not
> infinitely wealthy :-) However, my bankroll is easily
> replenished (due to, heaven forbid, a non-blackjack
> career), and is not the limiting factor in deciding
> how much I bet.

> My current betting strategy is to pick a somewhat
> arbitrary unit size and spread with which I and the
> casinos are comfortable, then use BJA3 or CVCX to
> determine an optimal betting ramp. It dawned on me,
> however, that since all of this is predicated on
> maximizing the log of wealth given a limited bankroll,
> with associated concerns of ROR, I may not be betting
> what is truly optimal. My ROR is very close to zero,
> no matter what unit size I choose (within reason).

> If one's bankroll is easily replenished, what is the
> best way to bet? Is the strategy of betting a large
> amount whenever one has an advantage, however small,
> non-optimal in this case? Within reason, ROR=0
> whatever the bet size, so why not bet a large amount
> in all favorable situations? Granted, I probably won't
> follow this strategy, regardless of whether it's
> mathematically sound, but I'm wondering whether my
> current "optimal" betting method has only a
> psychological basis.

If you play rather infrequently and/or have a replenishable bankroll then you would want to maximize a trip, your biggest concern should be not going broke early in a trip. You may also consider betting bigger toward the end of a trip if you can handle the swings (big variance). However, I think you would really need to ask yourself if your bankroll is really large enough to press a trip and lose half of your bank in a day and then can you replenish that when you play again?

It gets a little sticky. I think you would need to consider the amount your bankroll grows over time from adding funds. Then the variance of your actual time played. I would think once you know that then you can come up with a stable betting plan.

[Don Schlesinger]

> Before anyone gets the wrong impression, I am not
> infinitely wealthy :-)

But, you're working on it! :-)

> However, my bankroll is easily
> replenished (due to, heaven forbid, a non-blackjack
> career), and is not the limiting factor in deciding
> how much I bet.

I've been like that for ages.

> My current betting strategy is to pick a somewhat
> arbitrary unit size and spread with which I and the
> casinos are comfortable, then use BJA3 or CVCX to
> determine an optimal betting ramp.

Perfectly normal.

> It dawned on me,
> however, that since all of this is predicated on
> maximizing the log of wealth given a limited bankroll,
> with associated concerns of ROR, I may not be betting
> what is truly optimal.

Right. To bet "optimally," it has to be "optimally" with respect to a specified bank size. Just doesn't make sense otherwise. As we'll get to in a moment, if the bank is "infinite," then a) "optimal" doesn't make much sense, and b) what do you need blackjack for in the first place? :-)

> My ROR is very close to zero,
> no matter what unit size I choose (within reason).

Lucky you! :-)

> If one's bankroll is easily replenished, what is the
> best way to bet?

All you can afford, and all you can get away with -- the latter being at least as problematic as the former, and probably much more so! :-)

> Is the strategy of betting a large
> amount whenever one has an advantage, however small,
> non-optimal in this case?

No, it's fine, so long as you're OK with (relatively) huge variance, which will plague you all your living days! ;-)

> Within reason, ROR=0
> whatever the bet size, so why not bet a large amount
> in all favorable situations?

Indeed, why not?

> Granted, I probably won't
> follow this strategy, regardless of whether it's
> mathematically sound,

Coward! :-)

> but I'm wondering whether my
> current "optimal" betting method has only a
> psychological basis.

Yes. For the record, I agree with all that you have written.

Don

[David Spence]

>what do you need blackjack for in the first
> place? :-)

I think it was Fast Eddie Felson who said, "Money won is twice as sweet as money earned."

> Coward! :-)

Absolutely correct. True to my nature, I think I'm going to stick to my cravenly ways and continue to be sub-optimally :-)

> Yes. For the record, I agree with all that you have
> written.

Thanks for the quick response and at least partial validation for my cowardice, which is really all I was looking for anyway :-)

David

[Wolverine]

Interesting thread, David. I too have a nice job that would allow for an "replenishable" bankroll at reasonable betting ranges. However, I suffer from the same cowardice that makes me desire to minimize my ROR of the *current* bankroll (so as to not *need* to replenish it!). Why? Because I have other needs/wants/desires with my earned income instead of losing it to a casino with betting ramps and units that are too large for the *current* bankroll. Which could lead to a bust out due to normal variance at the inflated betting level. I prefer to grind out a small income on my committed bankroll. I then have the option of including in my bankroll (and thereby increase my unit size or lower my ROR further) or taking it as income (interest, if you will) on the cash bankroll that I am unable to invest (or use to purchase things) so it will be ready for the next session/trip.

More to the point: how much money are you willing to put into the casinos at inflated units before you cry "uncle" and walk away? When you've cleaned out your savings, are unable to afford daily living expenses, or can't purchase things that you should be able to at your income level? That indicates "problem gambling" to me, not a hobby of card counting. Oh yeah, once you are down "six-digits" and start to make a comeback, do you think the casinos are going to let you win it all back and then some without backing you off?

Good luck and good cards!

[Brick]

If your risk of ruin is 0 and you have an unlimited bankroll,you can use a small bet spread because the significance of 1 unit may mean the difference of thousands$$$.However are you quite certain you'll replenish your bankroll when you're down $80,000 or even $40,000? If not, your risk of ruin may be VERY high.

I never assume my bankroll is unlimited, and never will.
Having a good paying job does not justify a reason for risking big bucks.

[Gramazeka]

I had a discussion about full Kelly betting. One of my colleagues asked: “why to bet full Kelly and what is an unlimited bankroll?” This is a difficult question. The key here is what we consider a bankroll? Pocket-money, cash at home, real estate or our total wealth? Answer: Assuming an unlimited bankroll I’ll never bet maximum at +1, but I’ll ALWAYS bet table maximum at +4 (as well at +3 if penetration is bad). Forgot to mention – only 6-8 deck games are present in the former USSR.
Players who due to bankroll limit can’t afford table maximum at TC of +4, in my opinion shouldn’t go to a casino to earn money. Most will hardly earn their taxi fare. I think everyone can understand it. Second – the more you play in a casino, the more are chances to get backed off. Other known thing - good games tend to disappear. You must use them while they are present. No risk no money. Teams betting full Kelly will earn more than those betting half or quarter Kelly. That’s an axiom – full Kelly gives a maximal bankroll growth. I understand “bet less, be sure” approach. But that works in case of infinite time. While you were playing with small bets (less camouflage possibilities) you were getting barred. All your earnings were spent to food and transportation. Or – the casino was closed, law was changed etc. Blackjack scene changes rapidly in our times. I mean accessibility and attraction of the game. Regular job gets a fixed salary and game gets a chance to win in a arithmetical or geometrical progression. There’s a chance of losing, but it doesn’t kill. Imagine you have three casinos in your town offering good rules. Half-Kelly betting doesn’t make much space for a camouflage. It’s obvious. With this approach, we’ll get barred in all three casinos without earning descent money. On the other hand, even having lost in one of them playing full Kelly, we’ll be able to enter this and other casinos in a month or two. And it’s very actual in blackjack. It’s important to keep the good places. Access to ones is lost once and forever. Besides, the chances to get into the 87 per cent of lucky full Kelly players are not too small. In blackjack it is mandatory to remember: we have a limited time to get the result. And what if a bankroll allows three-Kelly betting for most of world’s blackjack tables? What to do?