7up: How much should I bet

[Don Schlesinger]

> The second thing to keep in mind is if the
> first card dealt to us is not an ace, we
> have roughly a 4.4% to 4.7% disadvantage
> depending on the house edge. I'll go with
> 4.5%. So, with only a 3% chance of an ace
> coming out on the first hand, we have a 97%
> chance of being at a 4.5% disadvantage and a
> 3% chance of having a 50% edge for and
> overall edge of -2.9% for the first hand.

Don't know what would make you think this, but it isn't correct. In, say, a 6-deck shoe, there are 24 aces. The fact that you've tracked one, and "know" that it will come out in the next six cards certainly doesn't preclude your getting any one of the other 23 aces in any of the other hands (including the one where "your" ace is, thus making a pair).

Your assumption as to your disadvantage for the other hands is, therefore, incorrect.

Don

[Don Schlesinger]

"That would be true if the ace is free to hit the dealer's hand. But if you look carefully at 7up's hypo, you'll see the ace in question is GUARANTEED to hit the player, and miss the dealer. So the
disadvantage is only a about 0.6%/(# decks)."

It wouldn't be true under either circumstance. See my post. Saying that the card won't be THE ace is not at all the same as saying it won't be AN ace. I'm surprised you both missed that.

Don

[Mr. Lucky]

You're quite right. I misread that little ol' article. This changes the classroom analysis quite a bit. But the real casino situation is still pretty much the same: bet as big as you can get away with.

[MathProf]

Yes, my post was answering Karel, and dealt with the specific issue of his post, being guaranteed the Ace on your next card on one hand. I hadn't paid attention to the comments above. (Sometimes I do that, start by reading a post from a known handle someplace in mid-thread.)

Now regarding the problem of getting an Ace on one specific hand. We have analyzed this situation before. Several years ago (maybe more than 5) I computed the exact probability distribution and posted the results. The simplified numbers that I gave above represent a rough approximation to this situation. That is, they are close enough to the situation to capture the mathematical issues involved. You see that in my hypo, the EV was 50% and not 40%.

My hypo involved no splitting and doubling to keep the hypo simpler to work with. In fact, when I looked at the problem before it looked like optimal (RA) strategy was not to soft double, with the huge bets that were out on one hand. This would be different if you had to spread over 6 hands, because your bets on each hand would be smaller.

I don't how you can say "the variance of a hand of blackjack, knowing that the ace is coming, is considerably greater than the standard 1.26-1.33 for an ordinary hand of blackjack;". As you can see, in the hypo I gave, the Variance was only 1.17. This was a 33% chance of getting a BJ. The Sum of Squares was much higher (1.4), but variance is not the same as sum of squares. We have to subtract the EV^2. In small edge games, this is a negligible effect, but here the effect is substantial.

To try to understand what is going on here, note that you are more likely to win the round in this situation. If your are more likely to win, then you have less variance, all things being equal. All things are not equal here, but this effect is certainly present.

Now, to take up the situation where the Ace will fall on one of size hands: There is one important point to be considered here. The Covariance in this situation is considerably less than the covariance for ordinary BJ. This is because the "Catching the Ace" business is negatively correlated between the hands. If one hand catches it, then the other hand will not.

Much of this could be settled with a very simple Sim. I am sure Cac., Karrel, or Norm could do it very easily. I could do it; it would me a couple of hours to get some programs out of mouth-balls and customize them, and I am not sure when I can get to it.

In case I get around to doing a Sims, what are the parameters of this game. Number of decks and rules? I would prefer to do 6 decks, but other could be done as well.

> A few comments, for both Karel and MathProf:

> 1. The edge for knowing an ace is coming is
> just over 50%, not 40%;

> 2. The variance of a hand of blackjack,
> knowing that the ace is coming, is
> considerably greater than the standard
> 1.26-1.33 for an ordinary hand of blackjack;

> 3. The question remains unanswered, because
> there is a HUGE difference in the optimal
> bet for one hand -- guaranteed the ace --
> and for each of six hands, where five will
> be played with a negative EV, while one will
> receive the ace.

> So, to me, the question as to what to bet on
> EACH of the six hands, with no knowledge of
> which receives the ace, is still very much
> unanswered.

> Don

[Don Schlesinger]

> Yes, my post was answering Karel, and dealt
> with the specific issue of his post, being
> guaranteed the Ace on your next card on one
> hand. I hadn't paid attention to the
> comments above. (Sometimes I do that, start
> by reading a post from a known handle
> someplace in mid-thread.)

Not a problem, but your remarks below bring up some interesting points, about which I have some questions.

> Now regarding the problem of getting an Ace
> on one specific hand. We have analyzed this
> situation before. Several years ago (maybe
> more than 5) I computed the exact
> probability distribution and posted the
> results. The simplified numbers that I gave
> above represent a rough approximation to
> this situation. That is, they are close
> enough to the situation to capture the
> mathematical issues involved. You see that
> in my hypo, the EV was 50% and not 40%.

No dispute here. There are several published results, in more than one book, regarding the value of an ace for the first card of the hand. Wong's "Basic BJ," p. 112, gives 50.5%.

> My hypo involved no splitting and doubling
> to keep the hypo simpler to work with. In
> fact, when I looked at the problem before it
> looked like optimal (RA) strategy was not to
> soft double, with the huge bets that were
> out on one hand. This would be different if
> you had to spread over 6 hands, because your
> bets on each hand would be smaller.

Yes, agreed. But, the real-world prospect of having to split aces, which I don't think you could afford to pass up, might very well affect the correct wager size. You ignored this, but, in real-life, I don't think you could. If you bet a huge percentage of your bank, how could you pass up splitting aces to begin a hand with A,A instead?

> I don't how you can say "the variance
> of a hand of blackjack, knowing that the ace
> is coming, is considerably greater than the
> standard 1.26-1.33 for an ordinary hand of
> blackjack."

It's simple to say; it just may not be right! :-)

> As you can see, in the
> hypo I gave, the Variance was only 1.17.
> This was a 33% chance of getting a BJ. The
> Sum of Squares was much higher (1.4), but
> variance is not the same as sum of squares.
> We have to subtract the EV^2. In small edge
> games, this is a negligible effect, but here
> the effect is substantial.

This is a very important point, but one that virtually never gets made. Whenever I have seen a discussion of this problem, participants ALWAYS agree that a wager of substantially LESS than 50% is called for, because so much of the 50% expectation is due to the increased likelihood of the natural, which, in turn increases the variance, (or so it is always claimed).

In fact, the average of the sum of the squares increases considerably, as you point out, but since we must subtract 0.5^2 = 0.25 from that value, you are claiming that the variance of outcomes for this hand is actually less than that of a normal hand of blackjack!

So, the question is: when calculating the optimal bet size, do we divide by the variance or do we divide by the sum of the squares? It is here that I think you may be in error.

Recall that, when payoffs are close to even money, dividing by the variance is the way to go to get f*, or optimal bet size. But, as soon as the ratio of the payoff of a winning wager to that of a losing one begins to drift upward from one, dividing by the pure variance is no longer the correct methodology. Thorp made this crystal clear in "The Mathematics of Gambling," with his roulette example. In such instances, we divide NOT by the variance, but rather by the ratio of the winning payoff to the losing one. This is why f* for positive-EV horse-racing systems based on betting longshots would be much smaller than one with an identical EV but that bet on favorites.

> To try to understand what is going on here,
> note that you are more likely to win the
> round in this situation. If your are more
> likely to win, then you have less variance,
> all things being equal. All things are not
> equal here, but this effect is certainly
> present.

The question is, despite the presence of the effect (extra EV), which is more important: the extra EV or the higher sum of the squares (ratio of a winning payoff to a losing one)?

> Now, to take up the situation where the Ace
> will fall on one of size hands: There is one
> important point to be considered here. The
> Covariance in this situation is considerably
> less than the covariance for ordinary BJ.
> This is because the "Catching the
> Ace" business is negatively correlated
> between the hands. If one hand catches it,
> then the other hand will not.

This introduces a new question: adding multiple hands. And, I understand the point. But, I don't think we've answered the question for the optimal bet size for ONE hand -- at least not yet to my satisfaction. And, as I've pointed out in two posts above, with 24 aces in a 6-deck shoe, not getting THIS ace in any of the other hands far from voids the possibility of getting one of the other 23!

> Much of this could be settled with a very
> simple Sim. I am sure Cac., Karrel, or Norm
> could do it very easily. I could do it; it
> would me a couple of hours to get some
> programs out of mouth-balls and customize
> them, and I am not sure when I can get to
> it.

Agreed. I also think this discussion might be more suitable for the Theory & Math page of DD than for here.

> In case I get around to doing a Sims, what
> are the parameters of this game. Number of
> decks and rules? I would prefer to do 6
> decks, but other could be done as well.

Not up to me. I'd imagine it was six decks, but I think that the hypotheticals for estimating in which hand the ace would fall are completely arbitrary. I don't think any player could make that kind of determination "on the fly," during the play of the next hand.

Don

[7up]

> Not up to me. I'd imagine it was six decks,
> but I think that the hypotheticals for
> estimating in which hand the ace would fall
> are completely arbitrary. I don't think any
> player could make that kind of determination
> "on the fly," during the play of
> the next hand.

S17, DOA, DAS, ES10, ENHC, Resplit eternity including Aces

I usually can catch around 5 aces a day for 10 hours work.
The percentages of falling of the ace to each hand are in "real cases". Cannot post the details on how to do it here, sorry for that.

Since the casino is not in US, conditions are different. Some CC's jumped from 1 to 350 for years before kicking out.

7up

[ML]

I have done some calculations but do not trust them because intuitively too much money gets put on the table each time. I am assuming you mean to bet the same on each square. I am sharing the calculations to see if one finds an error.

I am using the basic formula outlined in this thread for actual growth of a bankroll and used successive approximations.

First I assumed a 50% advantage for a hand with the known ace and a -.0035 advantage for the other hands. Twenty-six per cent of the time it will be in the middle hands so overall expectation is .12734 for each of these hands. Expectation for the others are .09723 and .02671 using a calculation .5*.26 - .0035*.74.

Then I simplified a little and did a little program for 300 plays.

10 Input f
20 k=((1+f)^56.367)*((1-f)^43.663) + ((1+f)^54.862)*((1-f)^45.139) + ((1+f)^51.386)*((1-f)^48.165)-3
30 print k

Successive approximations peak at .099 which means putting sixty percent of your bankroll on the table each time which intuitively is too much but that is what this method shows.

Criticisms of the methodology would be appreciated.

[7up]

>I am assuming you mean to bet the same on each square.

>> 6% the ace will be dealt to the 1st or 6th box,
>> 40% 2nd or 5th,
>> 54% 3th or 4th.

> I would bet the most on 3th and 4th
> something less on 2nd and 5th
> and minimum on 1st and 6th

Thanks

[MathProf]

More on Variance

Let us talk first about the case where we are catching the Ace on just one hand.

You are right that in the real situation, we would want to split Aces. I think the Critical Fractions for splitting Aces against low cards (4,5,6) is near 50%, and as you note we should be betting less than that. It is more problematic against the higher up-cards. However, the frequency of these Ace splits isn't great, and so including them in the model would not shift the results a great deal.

Another thing we haven't considered is Insurance. With such a big bet out, there would be some RA reasons to insure "good hands". And take partial insurance on bad hands. But perhaps we should omit this from this study.

On Variance and Sum of Squares. Yes, the Sum of Squares is much higher with this. It isn't really clear to me which is better to use for estimating optimal bets: SS or Var. SS tends to under-estimate your bet, and be conservative. This is because the third Moment is positive, for most casino games.

For Risk of Ruin, there are results which show that using SS gives a conservative estimate of your RoR, for "reasonable" games with positive Skew. I spent a lot of time proving this in my 2000 RoR paper; part of the trick comes up with technical definition of "reasonable" games that allows a proof to go through. Probabilist was able to come with simpler criterion, using some of the results in his paper.

But I think the situation is more complex for Kelly bets, and I think it is easier to come up with counter-examples where even EV/SS could over-estimate Kelly bets.

If I get chance, I may set up some sims and or CA to deal with this situation. I agree that the discussion should probably move to another board. Before getting to this, I want to work on another issue. I was planning to post something in the thread about EoRs on your pages, and got ready to do some EoR runs. But I doing that, I noticed some "anomalies" in my software that I need to track won.

> Not a problem, but your remarks below bring
> up some interesting points, about which I
> have some questions.

> No dispute here. There are several published
> results, in more than one book, regarding
> the value of an ace for the first card of
> the hand. Wong's "Basic BJ," p.
> 112, gives 50.5%.

> Yes, agreed. But, the real-world prospect of
> having to split aces, which I don't think
> you could afford to pass up, might very well
> affect the correct wager size. You ignored
> this, but, in real-life, I don't think you
> could. If you bet a huge percentage of your
> bank, how could you pass up splitting aces
> to begin a hand with A,A instead?

> It's simple to say; it just may not be
> right! :-)

> This is a very important point, but one that
> virtually never gets made. Whenever I have
> seen a discussion of this problem,
> participants ALWAYS agree that a wager of
> substantially LESS than 50% is called for,
> because so much of the 50% expectation is
> due to the increased likelihood of the
> natural, which, in turn increases the
> variance, (or so it is always claimed).

> In fact, the average of the sum of the
> squares increases considerably, as you point
> out, but since we must subtract 0.5^2 = 0.25
> from that value, you are claiming that the
> variance of outcomes for this hand is
> actually less than that of a normal hand
> of blackjack!

> So, the question is: when calculating the
> optimal bet size, do we divide by the
> variance or do we divide by the sum of the
> squares? It is here that I think you may be
> in error.

> Recall that, when payoffs are close to even
> money, dividing by the variance is the way
> to go to get f*, or optimal bet size. But,
> as soon as the ratio of the payoff of a
> winning wager to that of a losing one begins
> to drift upward from one, dividing by the
> pure variance is no longer the correct
> methodology. Thorp made this crystal clear
> in "The Mathematics of Gambling,"
> with his roulette example. In such
> instances, we divide NOT by the variance,
> but rather by the ratio of the winning
> payoff to the losing one. This is why f* for
> positive-EV horse-racing systems based on
> betting longshots would be much smaller than
> one with an identical EV but that bet on
> favorites.

> The question is, despite the presence of the
> effect (extra EV), which is more important:
> the extra EV or the higher sum of the
> squares (ratio of a winning payoff to a
> losing one)?

> This introduces a new question: adding
> multiple hands. And, I understand the point.
> But, I don't think we've answered the
> question for the optimal bet size for ONE
> hand -- at least not yet to my satisfaction.
> And, as I've pointed out in two posts above,
> with 24 aces in a 6-deck shoe, not getting
> THIS ace in any of the other hands far from
> voids the possibility of getting one of the
> other 23!

> Agreed. I also think this discussion might
> be more suitable for the Theory & Math
> page of DD than for here.

> Not up to me. I'd imagine it was six decks,
> but I think that the hypotheticals for
> estimating in which hand the ace would fall
> are completely arbitrary. I don't think any
> player could make that kind of determination
> "on the fly," during the play of
> the next hand.

> Don

[Don Schlesinger]

> On Variance and Sum of Squares. Yes, the Sum
> of Squares is much higher with this. It
> isn't really clear to me which is better to
> use for estimating optimal bets: SS or Var.
> SS tends to under-estimate your bet, and be
> conservative. This is because the third
> Moment is positive, for most casino games.

I'm glad, in a way, to learn that it isn't clear to you. This is a tricky concept, in that, for normal BJ, we tend to use the two interchangeably, as the average squared outcome of a hand is many orders of magnitude larger than the square of the average outcome, which is, for all intents and purposes, zero.

But, as the edge gets so much higher, the above is no longer true. This fact is coupled with the idea that the edge is now coming more from a payout that is 3 to 2 and not 1 to 1, and it is for the latter kind of payout that the EV/SS or EV/Var optimal bet fraction is most effective.

As I mentioned before, when the ratio of the wining payout to the losing one starts to stray greatly from 1 to 1, the above approach gives way f* = EV/ratio, where ratio is the above-mentioned one.

So, where does this "crossover" occur? At what ratio (greater than 1 to 1) do we stop using EV/SS or EV/Var and start using EV/ratio, as per Thorp? Obviously, this must be a gradual process; there isn't a magical moment where we stop using one and start using the other.

I'd be interested to know your thoughts on this.

Don

[Sabine]

> I'm glad, in a way, to learn that it isn't
> clear to you. This is a tricky concept, in
> that, for normal BJ, we tend to use the two
> interchangeably, as the average squared
> outcome of a hand is many orders of
> magnitude larger than the square of the
> average outcome, which is, for all intents
> and purposes, zero.

> But, as the edge gets so much higher, the
> above is no longer true. This fact is
> coupled with the idea that the edge is now
> coming more from a payout that is 3 to 2 and
> not 1 to 1, and it is for the latter kind of
> payout that the EV/SS or EV/Var optimal bet
> fraction is most effective.

> As I mentioned before, when the ratio of the
> wining payout to the losing one starts to
> stray greatly from 1 to 1, the above
> approach gives way f* = EV/ratio, where
> ratio is the above-mentioned one.

> So, where does this "crossover"
> occur? At what ratio (greater than 1 to 1)
> do we stop using EV/SS or EV/Var and start
> using EV/ratio, as per Thorp? Obviously,
> this must be a gradual process; there isn't
> a magical moment where we stop using one and
> start using the other.

> I'd be interested to know your thoughts on
> this.

> Don
Don, please help me. I just read about the "global edge" when you know, a certain card is coming. Assuming the table has 15 spots(boxes)
you have a great chance receiving a ten on any spot. The ten means an advantage of +13%. If we substract the -.5% house edge for the other 14 spots, it would still result in an "overall edge" of 13%-7%= 6%. Is it true?

[Don Schlesinger]

> Don, please help me. I just read about the
> "global edge" when you know, a
> certain card is coming. Assuming the table
> has 15 spots(boxes)
> you have a great chance receiving a ten on
> any spot.

Well, you'd have to be a little more specific. "Great chance" isn't terribly helpful when trying to quantify the math. Let's use "100%" chance instead, OK? Mind you, if I deal 15 hands randomly, it would be awfully hard NOT to get a ten on any of them. For infinite deck, it would be (9/13)^15 = .004 = one chance in 250!

So, you ARE going to get a ten on one of those hands anyway!

> The ten means an advantage of
> +13%. If we subtract the -.5% house edge
> for the other 14 spots, it would still
> result in an "overall edge" of
> 13%-7%= 6%. Is it true?

No. You're forgetting to divide by 15. so, the answer would be 0.4%. But, that would assume that you know, ahead of time, that exactly one ten is coming in the next 15 hands, which may not be what you're implying.

Don