Standard Deviation for Multiple Hands Played Simultaneously

I obtained the following text and chart from the Wizard of Odds, Appendix 4:

[Begin quote] "Following is the standard deviation per individual hand, playing flat betting 1 to 3 hands at the same time. These numbers are courtesy of Marsha Ness, who used Blackjack Audit simulation software. The rules are 6 decks, dealer stands on soft 17, double any two cards, double after split, no surrender, split up to 4 hands, no resplitting aces, no drawing to split aces.

Multiple Hand Standard Deviation
Hands Std. Dev.
1 1.15514
2 1.34942
3 1.51957

For example, the total standard deviation of three hands of $100 each, played at the same time, would be $100 � sqr(3) � 1.51957 = $263.20. By comparison, the standard deviation of a single hand of $300 would be $300 � 1.15514 = $346.54." [End quote]

I realize the following is not a likely real-world casino scenario, but my question is what would be the total standard deviation of FOUR hands of $100 each, played at the same time, and what would be the total standard deviation of FIVE hands of $100 each, played at the same time?

The variance for any number of simultaneous hands is given in Griffin's The Theory of Blackjack, p. 142, section E.

He uses 1.26 for variance, and therefore, 1.122 for the s.d. of a single hand. Mike uses 1.155 for single-hand s.d., and, therefore, 1.33 for variance, so if we plug that into Griffin's formula, we get:

Var(n) = 1.33n + .50n(n-1) for the total variance when one unit is bet on each of n spots.

For four hands of $100 each, as per above, variance would be 11.32, so s.d., which is the square root of variance, would be 3.36, or $336.

I'll leave it to you as a "homework" assignment to do it for five hands!

Don

Operating with standard deviation as opposed to variance is always a bit tricky since square roots are involved but let's look at Shacklefords numbers.
He says that the s.d. for three simultaneous hands of 100$ is $263. Now if we calculate the equivalent one-hand-bet that would produce the same s.d. we get roughly $230 (263/1.15).
It's easy to get in 2 rounds of 1 hand during the time needed for 1 round of three hands (it's also the same card consumption. This would mean that playing only one hand would allow to get more money on the table (thus higher winrate) for the same risk than spreading to three hands. I cannot really believe that.

Francis Salmon

I have seen Griffin's formula but frankly I don't really understand it.
Now you used his formula for 4 simultaneous hands of $100 and got a variance of $1120. But 1 hand of 400$ has a variance of only 532$ (400*1.33).Does that sound right to you?

Interesting studies on covariance were made by Stanford Wong in PBJ. He gives for Var v+(n-1)c (page 203)and on page 204 he gives optimal betsizes for any number of simultaneous hands.
These numbers contradict Shackleford's calculations to a certain extent.

Francis Salmon

"I have seen Griffin's formula but frankly I don't really understand it."

But, that doesn't say anything about the accuracy of the formula, now, does it? :-)

"Now you used his formula for 4 simultaneous hands of $100 and got a variance of $1120."

$1,132.

"But 1 hand of 400$ has a variance of only 532$ (400*1.33. Does that sound right to you?"

I don't know how it's supposed to sound, but it's the right answer!

"Interesting studies on covariance were made by Stanford Wong in PBJ. He gives for Var v+(n-1)c (page 203)and on page 204 he gives optimal betsizes for any number of simultaneous hands.
These numbers contradict Shackleford's calculations to a certain extent."

Again, you have to look at the original assumption for one-hand variance, from which everything else flows. Wong uses 1.28, while Shack uses 1.334.

Don

Who ever told you that, playing alone at the table, you should play three hands to optimize SCORE or Sharpe ratio?

What you wrote above seems accurate. If they take the same amount of time to play (four hands either way), you should prefer one hand of $230, twice, to three hands of $100 once.

The numbers are correct. You may interpret them any way you choose.

Don

With all the respect for Peter Griffin I still cannot make sense of his formula for variance. Why would the variance of 4 hands of $100 be double compared to 1 hand of 400$?. I would have expected the reverse. This would actually be the case if we took Wong's definition: Var= v+(n-1)c where v is the one-hand-variance,c the covariance and n the number of simultaneous hands. Result for 4 hands of $100:$283.
Based on Wong's formula, the equivalent one-hand bet to 3 hands of $100 would be only 173$ and not 230$ as with Shacklefords data.
The question remains which is correct.

Francis Salmon

"With all the respect for Peter Griffin I still cannot make sense of his formula for variance."

Over the years, I had several occasions where I thought Peter had made a mistake. Virtually 100% of the time, I was the one who was wrong! You aren't going to get rich questioning the accuracy of Peter's work. You simply aren't understanding. You can find the same formula in Karel Janecek's work and, doubtless, several others.

"Why would the variance of 4 hands of $100 be double compared to 1 hand of 400$?"

You or I or both of us didn't SQUARE the unit size when switching to $400! When it was one hand of $100 or multiple hands of $100, the unit size was the same, so the squaring didn't enter into the equation. When switching to $400, which was an afterthought, you don't multiply 400 by 1.33; you multiply by the SQUARE of 400! So, the multiplier is actually 16 squared units.

"I would have expected the reverse. This would actually be the case if we took Wong's definition: Var= v+(n-1)c where v is the one-hand-variance,c the covariance and n the number of simultaneous hands. Result for 4 hands of $100:$283.
Based on Wong's formula, the equivalent one-hand bet to 3 hands of $100 would be only 173$ and not 230$ as with Shacklefords data.
The question remains which is correct."

You're confusing several issues by moving back and forth between optimal bet size for multiple hands and total variance. People keep making this mistake over and over again. When you bet more, as with multiple spots, the total variance increases; it doesn't remain the same. But, the e.v. also increases proportionally, so that it's the risk of ruin, which is a function of both, that remains the same.

It goes without saying that the total variance of one $400 bet is larger than the total variance of four $100 bets, so I'm sorry if there was confusion there, but that really isn't the thrust of the discussion.

Don

Now I realize that the formulas of Wong and Griffin differ only by the factor n . So Griffin meant the variance per round while Wong was only defining the variance per hand in order to determine the bet size.
I find this whole topic surrounding variance, standard deviation and risk of ruin extremely tricky and even now I'm not entirely clear in my head.So if you allow me two more questions:
1) In my answer to the original poster, I tried to establish the equivalent one-hand-bet to 3 hands of $100 on the basis of equal s.d. and I got $230. As it turns out the correct result based on equal variance and confirmed by Wongs methodology is $170.Where did I go wrong?
2)You say that risk of ruin is defined by the ratio between winrate and variance.When we bet $170 on one hand, we have the same variance as with 3 hands of $100.But obviously we still don't have the same winrate. Wouldn't that mean that the risk of ruin is higher for the one-hand strategy?
I know these are not easy questions but if you cannot answer these I don't see who else can. Thank's for taking your time.

Francis Salmon

Francis, I can answer both questions, and will be happy to do so, but, unfortunately, I have a crazy day, and will be out virtually all day (Friday).

So, I'll try to get to this as soon as I can, but it may not be until later tomorrow.

Sorry.

Don

"1) In my answer to the original poster, I tried to establish the equivalent one-hand-bet to 3 hands of $100 on the basis of equal s.d. and I got $230. As it turns out the correct result based on equal variance and confirmed by Wong's methodology is $170. Where did I go wrong?"

On p. 20 of BJA3, I outline how to reckon total variance for h simultaneous hands. If your unit is $100, and you bet that on three simultaneous hands, your total variance is 1 (unit)^2 x 3 x [1.33 + 2(.50)] = 6.99 and your total e.v. is whatever your edge is times three units.

So, the ratio of expectation/variance is 3(e.v.)/6.99. In order to keep that ratio the same for a single hand, whose bet size is, say, x, we need to solve the following equation: x(e.v.)/(x^2 x 1.33) = 3(e.v.)/6.99. This is easily solved, with 1.75 units ($175) being the correct answer.

Yet again, any difference from Wong's answer ($170) has to do uniquely with what he chooses for variance of a single hand, and even covariance, as not everyone calls that 0.50 (some call it 0.48, for example). But these differences are minor and relatively unimportant.

2)You say that risk of ruin is defined by the ratio between win rate and variance.

ROR is proportional to the ratio of win rate to variance (see the equation at the top of p. 113 of BJA3), so when you reckon optimal bets for simultaneous hands, with respect to the one-hand wager, it is this proportion that needs to be held constant. Time and time again, people make the mistake of stating that the multi-hand s.d. or variance is the same as the one-hand s.d. or variance (see your statement, below). It isn't. It's higher. But, it's higher by exactly the same multiple as the increase in win rate.

"When we bet $170 on one hand, we have the same variance as with 3 hands of $100."

No, no, no, no, no!! See above. The variance for three hands of $100 is greater.

"But obviously we still don't have the same win rate."

Right. It's higher by a factor of $300/$170, which is why the total variance for the three hands is higher than the one-hand variance by that same 300/170 factor.

"Wouldn't that mean that the risk of ruin is higher for the one-hand strategy?"

It would if the variances were identical, but they aren't.

"I know these are not easy questions"

They are if you've answered them 100 times in your life! :-)

"but if you cannot answer these I don't see who else can."

Surely, others can, but I seem to be the first, for the moment.

"Thank's for taking your time."

Get it while you can; I won't be here much longer. :-) Meanwhile, you're welcome over at Don's Domain any time.

Don

The ratio between the variances of a 300$ bet and 3 hands of 100$ being roughly 12/7, I had mistakenly concluded that the operation $300*7/12 would lead us to the one-hand betsize with equal variance.
As you pointed out, the result $175 is the bet with equal risk of ruin instead.I had forgotten that variance evolved with the square of the betsize.
The result $230 that I mentioned in my answer to the original poster is indeed the one-hand bet with the same variance (or s.d.) but that's not what we're looking for since it has a higher ror.
Thanks again for enlightening me.

Francis Salmon