Takisgaias: Variance question

[Takisgaias]

How can I compute the variance of playing multiple spots with different bet sizes on each of the spot. For example, I put 4 units on the first spot and 6 units on the second.

What is the general formula for n spots and b(n) different bet sizes?

Thanks,
Takis

[Clarke Cant]

I put my answer on bj21.com BJ misc. This is not recomended camoflage!

[Takisgaias]

It is not that simple as you told me. And it cannot be derived from the formula for the same bet sizes. It is more complicated (at least for me) as it requires very good knowledge of statistics. To be honest I was hoping MathProf being around as the expert at these issues. He had given me the answer some time ago, but I didn't understand exactly some variables he included in the formula he had posted. I also emailed him but he didn't respond.

This is not recomended camoflage!

I didn't understand why you said that. What do you mean?

[Clarke Cant]

I checked some of the files at bjmath.com and if you didn't understand that you should stick to just the formula that applies to the same bet sizes for starters.
Actually it can. The total per round variance is proportional to the sum of the squares of the amounts bet. The minimum variance occurs when the same amount is bet on each original hand that you play. Then total amount of variance for a round, compared to betting the same in each spot is a ratio of: b1 to bn amounts bet: (sum [1 to n] b(n)^2)/(n*([sum b(1) to b(n)]/n)^2). (See below for Don's listing of his BJAII typos too, before procedeing).

Now go to the Formulas in Professional Blackjack. Both the total variance and covariance are altered by this ratio. Your total variance per round goes up by this ratio, AND the self hedging that normally occurs between simultanious hands is cut by this ratio. With different bet sizes, the outcomes are less likely to cancel themselves out, and THAT squared difference in outcomes, versus having the same size bets, also adds to your effective variance per round.

Try the ratio formula again:

Compare betting $5 on each of 2 spots to betting $4 on one and $6 on the other. The ratio of the sum of squares is (36+16)/(25+25). Not so bad you think, but...now go plug in the rise in variance and covariance. The usual formula for bet sizing involves n*variance and (n-1)*covariance. You just grew both by betting different amounts.

Now compare the ratio for $3 and $7 (low amounts are just to avoid complications). (9+49)/(50).

OK you are damaged a bit but livable for two hands. But that is JUST the ratios. You plug in the same change via this ratio into both the variance and covariance (I said look to one number being good, one bad just to get you to look at the number of hands this works with the least damage).

And examine what happens with 3 hands, 4 hands, 5 hands, ...all 7 spots.

Those can get very ugly. It frankly does not take a very high ratio between bets to drastically alter your overall var./ev, which is the formula for how many units you need for a kelly optimal bet sized bankroll. The changes in the long run index, var./(ev^2) or (sd/ev)^2 are even more drastic.

The only reason I can see is cover, but your cover requires you to work more hours, proportional to a small power of that ratio (determined also by the number of spots, which varries how the also effected covariance changes things), to double a bankroll. Just how much heat are you facing such that you make the time you need to put in so much greater? The Burning the Tables... chapters, that Ian Anderson wrote, on his, Ultimate Gambit, don't result is such rises. That is (still) rather effective cover. But how effective can it be to change bet sizes? Multiple hands will already peg you as at least knowing the usual procedures in a 21 game. Unless you are going for some depth charging tactics, by really changing your bet amounts for each hand, you cannot have much impact, and most of that impact will be negative!

Then you are asking for information on how to optimize such low cover value cover, after it should be clear that such will already have blasted away your return on your bankroll, and lengthened your long run index/hands to double an optimal bankroll. To me, and sorry to seem so rude, this is trying to optimize out of something that has already screwed you up!

I hope you can see how you are going back and forth on wanting to know the value of rather ineffective cover on the one hand, and then wanting to optimize the use of that bad cover, on the other hand.

[Takisgaias]

I didn't understand what to do with the ratio you were refering to. I'll give you an example, can you compute the variance? Say I bet on 3 spots 1,2 and 3 units respectively instead of betting 2 on each of the 3 spots.

You have a formula in your post above:
(sum [1 to n] b(n)^2)/(n*([sum b(1) to b(n)]/n)^2)

Substituting b(1)=1, b(2)=2, b(3)=3

we get: (1^2 + 2^2 + 3^2) / 3*( (1 + 2 + 3)/3 )^2 ) = (1 + 4 + 9) / 3*2^2 = 14/12

Now, what to do with this ratio?
I know that the variance of betting 2 units on each of the 3 spots is: (let var=1.26 and cov=0.5)
3*2^2*(1.26 + 2*0.5)= 3*4*2.26 = 27.12
(from the general formula n*b^2*(var+(n-1)*cov)

What do I do now? Multiply the 27.12 figure times the 14/12 ratio?

PS1:Where exactly at the bjmath site did you read about this issue?
PS2: I do not want this information for cover purposes...

[Clarke Cant]

Multiply the ratio I gave by the variance and covariance and replugin. Just in case I left a typo in too, the ratio I gave is designed to show how much your actual betting distribution raises both the variance and covariance. You can bet more per round generaly because with variance and covaraince, the total amount you can bet is more per round.

PS1:

There was a similar question to yours in a posting to mathprof.

PS2:

Are you trying to go for depth charging in a single deck game and trying to optimize your bankroll growth with such tactics? That is the only rational motive I can see is possible.

[Clarke Cant]

Eventually you will realize that distibuting unequal bet sizes for the total amount bet during a round raises your risks, such that you have to cut your unit sizes to still bet the same number of units, each round, that the same sized bets allow.

The ratio formula, is how much you have to raise the variance and covariance, to get the general formula to give you the variance that that same total of bets, results in. You go to the general formula, entering (total bet)/n number of spots, for each spot, but but raising the variance and covariance by that ratio.

In your example you then have var. not =1.26, but =1.26*14/12, and covariance not =.5, but .5*14/12, and enter a bet per hand of (b1+b2+b3)/3 or 2.

This method I used is round about, but instead of the direct route: which would be adding b1^2*((n*var.+ (n-1)*covar.)/n)+b2^2*...etc. for b1 to bn, you have a bit more accuracy for how the hedging drops, and the overall variance increases. The impact in increased covariance lags a bit in both formulas, but the impact of hedging dropping applies to boost overall variance for all n hands, and not just the sum of of the normal terms. It is a more distributed rise in overall variance, than the normal formula indicates.

So you are resigning yourself, by betting different amounts to always having long run indexes be up to twice the long run indexes for 2 hands, 3 times for 3....Unless this is a deeply single deck game, where you are depth charging. Then you may have something, but I doubt depth charging is that good a cover today.

[Takisgaias]

So variance is always increased, isn't it?

And, no I do not want it for depth charging(I am from Europe, no pitch games). I want to backbet a ploppy in high counts apart from playing my own spot too. But please do not tell me not to backbet ploppies! This is another story which requires lot's of analysis and I have done lots myself.

Thanks for your responses.

[Clarke Cant]

> So variance is always increased, isn't it?

Yes! The minimum distribution of squares in a sum is always with equal partitions---kind of really complicated to prove though.
> I want to backbet a ploppy in high counts
> apart from playing my own spot too.

Ahah! The variable rules on the larger bet controling the spot.
> please do not tell me not to backbet
> ploppies!

Wouldn't ever do that. That can be a very powerful play indeed. Analysis of the possibilities would start with AWS and some of his team plays in Austrailia, where he was having players spreadout, betting low though, just to exploit the rider's optional double and split strategy, covered by Wong in Basic Blackjack.

> Thanks for your responses.

Pleased! I knew something was up. I am glad you stayed with me while I turned the usual formula for unequal bets inside out. I was guessing first that you had some sort of progression type hedging in mind. Then I was guessing that you might be attempting a depth charging play.

I am very glad you saw at the end why I took the long way arround; you didn't seem to just be a ploppie etc.

You can even refine these figures for how well the player plays basic strategy or even counts, just a bit, if you are trying to figure how much you might be willing to bet to control a hand in some casinos. I don't think that will be much difference but it might prove interesting.