Optimal bets in simultaneous spots with different advantages

I've read that the optimal bet at a particular true is given by:
OB=(advantage x bankroll)/variance

Also, if you play multiple spots you divide by the sum of variance and (n-1) x covariance.

Let's say that I want to play two spots simultaneously, but the second spot does not have the same advantage as the first one. For example, I play my own spot but I also back-bet a ploppy who is not playing perfect BS. As a result, on average, at a particular true, the second spot will have less advantage than the first one (my own).

My question is, how do I calculate the optimal bets on each of the spots, given the advantages and the variances for each spot at a particular true? Also, how do I compute the variance of such a betting pattern?

Takis

The advantage difference is dependant on how bad the ploppy plays his hand. Such a thing is a parameter that's too hard to calculate. I've seen some ploppys play the same hand differently due to some supersticious reason (like they lost the hand last time they hit the 15 vs 10, so now they stand on it...etc).

I am not asking for the average ploppy advantage, this is another problem.
I'm asking that given this advantage (this is my problem how to calculate it) what is the optimal bet on the ploppy's spot, when simultaneously I play my own spot.

.. the variance cutting you get by going to two hands is far, far compensated for the MINIMUM -1% loss in EV by back-betting the ploppy. Plus, they won't make any juicy +EV strategy variations based on the count either - rest assured you won't get to double that A8 v 5..

So to answer your question, your optimal bet involves putting every chip you can into your square.

M.

What you are ignoring is that the "advantage" figures that we quote for blackjack, is dependent on the player knowing perfect BS, and also how to correctly deviate from BS based on the count.

You just want to know how much advantage a ploppy has compared to your own advantage, in a given round. Well, the simple answer is: the ploppy hand has a negative expectation. So avoid betting on his box.

A ploppy's advantage can be measured if he always played perfect BS. But then again, if he plays perfect BS, then he would no longer be a ploppy, but rather a BS player.

don't try to evaluate the potential gains of my approach. Although I appreciate you even got to trouble to post, I know what I'm doing.

I would even more appreciate if you could answer my ORIGINAL question. If you did not like the example I gave, just imagine, from a the theoretical point of view, that the first spot has 0.7% advantage and the second spot has 0.5% advantage at a purticular true.

As an information, backbetting ploppies, under the proper conditions could be the best approach to a game from a risk point of view or even a SCORE point of view.
I do not intend to go into details, but consider these:
-ploppies do have advantage at positive true counts
-backbetting ploppies you do not have to play a single negative hand at all
-less standard deviation, so less risk, so less need for a big bankroll
-yes, jumping into the game at positive trues, is the superior approach of all. But I don't play in Vegas, rather in casinos where opening boxes is not the normal procedure at all, although it is absolutely permitted by the rules of the house. After all, I do not feel comfortable arguing with half the ploppies of the casino because I ruined their "flow".

But, again, please do not reply about my approach. Instead, the only I ask, is an answer to my original question.

Thanks,
Takis

"I do not intend to go into details, but consider these:
-ploppies do have advantage at positive true counts "

No, without basic strategy you may not have an edge at positive counts, but it depends on the ploppy.

"-backbetting ploppies you do not have to play a single negative hand at all"

Backcount a table with 1 player, you don't have to play a single hand of negative ev.

"less standard deviation, so less risk, so less need for a big bankroll"

Yep, but the way ploppies play will add to the risk. Also backcounting a full table will give you less rounds of +ev and slower game.

Backcount a guy who is doing the Uston APC, he'll even do the correct BS deviations for you!!

I do not want to go through this, but here are some data from SBA. The player is flat betting, playing BS except of the following:

Always stand on 16 v. 8,9,10,A
always stand on 15 v. 9,10,A
never doubling on 11 v. 8,9
never doubling on 10 v. 8,9
never doubling A,7
always stand A,7 v. 9,10,A
never split 9,9

TC PERCEN AC PERCEN ADVANT SDROUND ACTION

2 6,3163 2 49,19349 0,208 1,0985 1,0846
3 3,264 3 25,42105 0,722 1,0942 1,1035
4 1,7065 4 13,29104 1,308 1,086 1,1168
5 0,8358 5 6,50989 1,902 1,0807 1,1151
6 0,4024 6 3,13428 2,43 1,076 1,1125
7 0,1814 7 1,41301 2,852 1,0708 1,11
8 0,081 8 0,6312 3,434 1,0668 1,1079
9 0,0324 9 0,25204 4,05 1,0623 1,1059
10 0,0198 10 0,15402 4,683 1,0565 1,1035

(In bold are the advantages at the corresponding true counts)

This is a common "BS" of a ploppy I backbet. But I always find better ones. And yes, that Asian ploppy did not hit the 14 v. 7 (!) (I saw it one time out of 200 hours of play) but the good ploppy than hit his 16 v. A, and 15 v. A several times in the night gave me back all the lost money and even more.

And, yes this is a very dangerous approach but I choose the ploppies I backbet and "clock" them before the count goes high and it's time for me to play. If he makes a more serious mistake I avoid him in the future. I go for averages. Plus that some times I can open my own spot and play perfectly.

But, THIS IS NOT THE SUBJECT.
Please, can a good soul answer my original question?

Takis

Just don't argue with them, start playing and if they have a problem with you betting, they can leave if they want.

so the optimal bet on his spot would be ZERO.Why would you want to bet on him? Is he holding a gun on you?

ic

This will tell you how to compute your total Variance:

If you bet B1 on spot 1, B2 on spot 2, B3 on spot 3, etc, then you can estimate your Variance by:

{ B1^2 + B2^2 +B3^2 + CC * [2 B1*B2 + 2 B1*B3 + �. ] } * Var

Var is the Variance of a single had of BJ, which is in the ballpark of 1.25,depending on the game and the rules. The sum in the brackets is taken over all pairs B1, B2.

This formula would be exact if the same strategy were used in each spot. Your premise is that there are different strategies. So there would be different Variances on each spot. For example, the ploppy who doubles too much would have higher variance; the one who doubles too little would have lower variance. However, Variance doesn't change that much with strategy (unlike EV) so this will give us an estimate for betting.

There was a very interesting POM last year which dealt with this issue. I will include a link; you need a GC password to open it.

You have asked about optimal bets, and that is more complicated. I will think about that and maybe post something later. It obviously depends on the EV of the other spots. I think that as long as there is some EV in another spot, there will be some gain in diversifying.

I do not agree with your assessment of the EV for ploppies. I am not sure that your strategy is representative. But this is really another issue, and one we can discuss at another time/place.

If you do this, you may want to carry a BS card, and try to talk the other player into following it. It could not hurt.

Just to clarify my thought on the issue that I said I wouldn't address �

I agree that sufficiently high counts a ploppy will have an advantage, and there is money to made by back-betting on them. But I think it comes at a higher count that +2. Like maybe +5. As I say, this particular question can be discussed in other thread.

You should also factor in the possibility fo being cheated on payouts. That has happened to me on more than one occasion. Throughout much of the Midwest, back-betting is illegal, and while the casinos may tolerate it, I would not count on them to adjudicate disputes.

I broke my planned silence over this, because I read some of the other posts and I thought the nay-sayers were piling on a bit too much in the other direction.

You keep saying your "original" question hasn't been answered, even after several people explained why it was un-answerable. We haven't digressed, you just keep ignoring logic.

Your question can be answered if we set up a player that always plays with the exact same BS errors - 100% of the time.

You seem to have the tools to work this out yourself, so why bother posting the question here? With the deviations you give, just figure out how much those wrong plays are costing the player for every true count (say up to +6), then you'll have your ploppy advantage on a given true count, and tehen you can bet accordingly with the multiple hand betting formula (forgot what that is right now). Word of caution though...you're skating on very thin ice here.

A big problem with this is that there is no such thing as a consistent playing ploppy. Therefore, the BS deviations you give for your ploppy player is useless because (as I said in my second post in this thread) a ploppy will play the same hand differently for some supersticious reason (i.e. like he busted last time he hit that 15 vs 10, so now he decides to stand instead, or vice-versa). This is the whole nature of being a ploppy. They are guided by supersticion and "gut feelings", and you want to set up a mathematical paramater to this kind of player? Just 1 different decision per hour will nullify your ploppy information that you "clocked" them at.

Just accept the answer that you keep ignoring - A PLOPPY BOX HAS A NEGATIVE EXPECTATION REGARDLESS OF THE COUNT. They only win when they get lucky (much like a Roulette player).

I'm sure you'll now post another frustrated response to this, saying that we still have not answered your original question...right?

I posted a more detailed answer on BJ-Math. Link is below.

Where have you been all this time! I was really expecting you to answer my original question, as the only person quilifying.

About the other subject there is much analysis to be done. It isn't simple at all. Of course, it is sensitive to how much badly the ploppies play, so you have to be careful. Just a few more things:
-the regular ploppies generally are afraid to hit their stiffs, and it becomes more correct in high counts.
-the split option for the backrider.
-the card eating effect. I think this is not simple at all! Spread to second spot at the high counts, or backbet the ploppy?

But, maybe some other time/place...

Thanks again for the answer!

In "Beating the races with a computer - Seven L. Brecher" page 8-6,
there is a section about "multiple bets".
It claims that sometimes betting on a negative EV is a good move.

First post I have seen where it is properly pointed out that variance and covariance, for precise answers, are differential equation, requiring matrix solutions.

Now whether the extra precision will make a difference in any future speanding plans is another question.