Strategy for next card when an opponent plays after you.

[Chairman]

I am not finding a push is the same a s loss. charts. I find push the same a s a win but not loss. Do they already exist? If so, where can I find them?

 

[gronbog]

There is a thread here from a while back in which I presented my optimal strategy for this, for the initial hand, and then discussed several simplifications for dealing with split hands. S. Yama then came up with a strategy for handling split hands which achieved almost all of the gain from the complete optimal strategy. There are simulation results there as well as links to my strategy tables.

https://www.blackjacktournaments.com/threads/strategies-for-must-win-one-bet.8201/

 

[London Colin]

Chairman,

Rather than comparing hit and stand scenarios for 3rd base, compare hitting from 3rd base with hitting from any other seat. The push/win/lose probabilities are identical, just as is the case for standing from any seat. So the difference between standing and hiting is the same, whether or not you are at 3rd base.

I put together a program to fully enumerate all the probabilities in your example, using CA. I run through the hitting and standing probs firstly for any player, and then again for a notional third base. In truth, just writing out the code makes it clear they must give the same answers, without the need to run it, since all that is being changed is that some of the calculations for standing are effectively being moved from within the CA library code, to a top-level loop that calls into the CA ...

for card = 1 to 10
  stand -> give the card to the dealer and run the CA for T,7 stands vs. A+card
  hit   -> give the card to the player and run the CA for T,7+card stands vs. A

instead of

stand-> run the CA for T,7 stands vs. A
for card = 1 to 10
  hit ->  give the card to the player and run the CA for T,7+card stands vs. A

For what it's worth, here are my results -

(Sorry, I can't find a font that makes the columns line up like they are supposed to.)
And the post turns out to be longer than the forum software will accept. I'll have to break it up...

 

[London Colin]

Breaking up turns out to be hard to do! I've uploaded it as a file instead.

 

[S. Yama]

Yep, Ken you are right.
It is obvious that removing one card, that remains unknown, does not change the statistical chances for the final hand of either the players or the dealer.
I haphazardly concluded that since the dealers’ outcomes are different for different numbers of decks used it have to come after more than one card is removed.
The differences in tables are caused by excluding the dealers’ upcard, and they are more pronounced with smaller number of cards left.
If there would be any “effect” it would have to be present with removing a whole deck, or any number of cards, even one.
It maybe somewhat deceiving when we deal with a smaller group of cards. One can bring an argument that for example having less small cards left reduces chances of making a pat hand if a stiff hand is hit, because a particular small card denomination can be completely exhausted with one or two hits. The similar thought can be created for any denominations of cards used. But the total effect of chances for a final hand remains the same.
S. Yama

 

[Chairman]

Thanks for the CDCA results. I knew the EV's would be the same or subtly different. The point I was making with that analogy of 3rd base and dealer is in this quote.

All the EV when ranks are eliminated is skewed but the average remains in the remaining ranks. The part that will really change the chances of meeting your tournament goal is when different outcomes (win/loss/push) are equivalent. That will change your chances of meeting that goal from what the EV predicts significantly. EV gained/lost by a push over a loss/win is not a gain in your tournament chances. That will have your tournament chances be different from EV. The question is in what situation does these changes things change your likelihood of meeting your goal enough to change the decision that a goal of maximizing EV would have you make.

If I didn't state it well getting a swing is not just the win and loss if a partial swing is enough. so a push and an opponent loss OR a win and an opponent push gets you a partial swing.

Going to the simpler problem of the dealer and third base you summarized as:

STAND:
push 13.206%; win 11.669%; lose 75.126%; EV -63.457%
HIT:
push: 3.400%; win 13.667%; lose 82.933%; EV -69.266%
Maximizing the EV of this hand says to STAND by almost 6% margin but if your goal can only be met if you win you have to HIT by about a 2% margin.

Now if a push means the same thing as a win for your tournament chances (push and opponent loss) in this dealer analogy the results would change to:
STAND:
win (push + win above) 24.874%; lose 75.126%
HIT:
win (push + win above) 17.067%; lose 82.933%
Now you would STAND by almost an 8% margin.

So as you can see the chances of meeting a tournament goal in a next card situation simplified to be between a ENHC dealer and you is not strictly tied to EV. In a must win case it was opposite the EV maximizing decision and in a must not lose it was the same as the EV maximizing decision. Now the actual tournament situation is complicated further by having 3 hands determine 2 results that combine to determine success or failure. If you need a swing or partial swing on the 2nd to last hand to be in position to have a chance to meet your current tournament goal the strategy will not necessarily be the same as if you need a full swing.

It seems the answer to my initial question is NO, nobody has looked into it. That really doesn't allow for the implied follow up question.

 

[London Colin]

Thanks for the CDCA results. I knew the EV's would be the same or subtly different. The point I was making with that analogy of 3rd base and dealer is in this quote.

I understand (I think) the point you were making. I was attempting to refute it. I probably shouldn't have included the EV figures in my results, as they are irrelevant.

Ignore EV, and ignore the Hit versus Stand comparison for 3rd base. Instead, compare the win/push/lose probabilities for carrying out either action, hit or stand, at 3rd base versus the general case. There is absolutely no difference, and when you think about it, it is obvious that there can't be.

So -

Now if a push means the same thing as a win for your tournament chances (push and opponent loss) in this dealer analogy the results would change to:
STAND:
win (push + win above) 24.874%; lose 75.126%
HIT:
win (push + win above) 17.067%; lose 82.933%
Now you would STAND by almost an 8% margin.

You get the same 8% margin whatever seat you are in. The 'next card' is a total red herring.

As you say, the tournament scenario with two players and the dealer is more difficult to analyse, but this ENHC/3rd-base analogy doesn't offer any support for your thesis; if anything, the opposite. [But, then again, I'm still not entirely certain that your thesis has really been disproved in any of the preceding posts.]

 

[Chairman]

It is more of a question. I think you all may be right about the red herring but it overlaps some of the outcome possibilities. The point in the analogy was to show things change concerning your decision not that the change reverses is in that case. The trouble is it gets so complicated with 3 hands that you aren't going to longhand the answer to any next card tournament matchup. When you include partial swings with the 3 hand tournament format things get really screwy. I don't see anyone interested enough to give it a serious look and I don't have the resources or time. I can use one set of logic that is based in EV maximizing that tells me rather quickly nothing is there but then when I shift to maximizing the chance of getting the swing and the affect on that of a win/push or a push/loss as success I can see how there might be something here. One issue that discourages investigation is the limited number of interesting matchups. It is probably not worth pursuing further because of that. I was just looking for an extra edge late in the tournament.

 

[London Colin]

It is more of a question. I think you all may be right about the red herring but it overlaps some of the outcome possibilities. The point in the analogy was to show things change concerning your decision not that the change reverses is in that case.

The forum title is 'Blackjack Tournament Strategy'. By definition, everyone contributing already knows that strategy decisions change, based on your goal.

The focus really has to be firmly kept on the search for a mechanism by which the 'next card' relationship can have an additional impact. Otherwise, there is just confusion all around.

The trouble is it gets so complicated with 3 hands that you aren't going to longhand the answer to any next card tournament matchup. When you include partial swings with the 3 hand tournament format things get really screwy. I don't see anyone interested enough to give it a serious look and I don't have the resources or time. I can use one set of logic that is based in EV maximizing that tells me rather quickly nothing is there but then when I shift to maximizing the chance of getting the swing and the affect on that of a win/push or a push/loss as success I can see how there might be something here. One issue that discourages investigation is the limited number of interesting matchups. It is probably not worth pursuing further because of that. I was just looking for an extra edge late in the tournament.

This whole thread has been a serious look. There is serious doubt that the effect you are seeking to measure can exist. Ken has declared that it cannot. It ought to be possible to come up with a proof, one way or the other, that all of us are capable of understanding. Only then, if the proof were to be for, rather than against your basic premise, would it be time to break out the computers and start trying to measure stuff.

 

[London Colin]

Pardon my brief lapse into silliness above. There is no effect.

If player 1 stands, player 2's probabilities are unchanged whether he gets the "next" card, or a randomly chosen card from the remaining deck.
This is all that needs to be realized to show there can be no effect.

While I'm at it, please forgive my unfortunate use of "effect" as a verb in my earlier post.

I think it is the scenario in which player 1 hits that has to be focused on.

If player 1 hits, then it can be argued that each possible card he may draw simultaneously affects both his own hand and his opponent's.

For each possible card -

1. Player 1 gets a new total, obviously.
2. Player 2 gets to draw a random card from the remaining deck, compared to drawing the specific card that went to player 1.
3. There is one less card in the deck when/if the dealer draws.

So, if we compare that to the same actions when the two players are not seated next to each other -

1. Player 1 still gets the same new total.
2. Player 2 still draws a random card from the remaining deck, but that would now have been the case even if player 1 had stood (assuming there is at least one hit before player 2 acts).
3. There are 2 or more less cards in the deck when/if the dealer draws. (again, assuming at least one additional hit before player 2 acts.)

If we can show that both of the above are entirely equivalent, then we have proved conclusively that there can be no effect of the type Chairman is looking for - any and all strategy decisions will depend purely on the content of the hands involved, not where they are located.

 

[Chairman]

Everyones logic to approaching this problem is different than the one that makes me believe there might be something here. The logic I am thinking about is that you can either take the card or waive it off. If you take the card you will have a specific hand or set of possible hands if you take more than one card and the opponent hits without those cards in the set of possible cards. If you waive off the card your opponent gets it and he has a specific outcome. These two outcomes form a set of pairs of outcomes that depend on your decision that do not exist if the opponent gets a different card should you waive it off. The net affect of this may be zero but it will redistribute the outcome if you hit or stand to a subset of what they were. This affect is only realized if you stand. If you it any other cards drawn in between might as well be dealt face down. What matters is the information you have when you make the decision. What card you draw and what card your opponent draws and the probabilities for the various dealer hand resolution that remain. So we are talking about what gain you might have in expecting a swing (either partial or full) by violating BS and stand rather than hit.

For this to be the case you need a hand that you would normally hit that if you stood the opponent would still be hitting his hand. That means either it is BS for the opponent to hit or he is trying to mimic another player that is more in contention in terms of results. If you stand on a stiff he can just stand on a stiff if he is trying to maintain his lead. This is why I liked A,7vA and 16vA as your hand and your opponents. There is not much of a loss in EV if you stand but your opponent should definitely hit just once even if he uses tournament strategy for same result equals a success as long as you have a pat hand of 18 or more. I think the key to this is in the pushes giving you a partial swing.

Anyway I am not sure you would do a fair job of analyzing this unless your approach looked directly at the relationship between the pair sets of outcomes that depend on your decision.

For each possible card -

1. Player 1 gets a new total, obviously.
2. Player 2 gets to draw a random card from the remaining deck, compared to drawing the specific card that went to player 1.
3. There is one less card in the deck when/if the dealer draws.

I am with this so far and is exactly what I would think would need to be done but after that you see what the possible outcomes are if you stand that correspond to each next card analyzed for hitting. These are option pairs if the next card is either taken by you or waived off and taken by your opponent. Compare these and see if you have a greater chance of any swing by waiving the card off given the distribution of dealer outcomes for the remaining cards. My suspicion is that the added chances of half swings from not busting will increase the odds of of getting your goal. Your EV doesn't increase by much by hitting but you eliminate the chance of busting by standing which increases the likelihood of a half swing from a push/loss. EV ignores pushes but in a tournament they are relevant since you don't push with your opponent you push with the dealer.

In our situation of A,7vA and T,6 vA our dream hit ranks of 3 and 2 which almost certainly give us our goal become failures to meet out goal since our opponent now beats us or ties us. Either outcome makes a swing impossible. That is a big change in the wrong way in terms of chances of getting a swing. There may be enough to overcome (after all you aren't going to bust if you stand) this in the rest of the corresponding outcomes but maybe A,87vA and 14vA is a better one to look at.

 

[London Colin]

Everyones logic to approaching this problem is different than the one that makes me believe there might be something here. The logic I am thinking about is that you can either take the card or waive it off.

I don't think there is any uncertainty about what it is you believe. The point is there is a chance your belief is based on a fallacy, which people have been trying to get to the bottom of.

If you take the card you will have a specific hand or set of possible hands if you take more than one card and the opponent hits without those cards in the set of possible cards. If you waive off the card your opponent gets it and he has a specific outcome. These two outcomes form a set of pairs of outcomes that depend on your decision that do not exist if the opponent gets a different card should you waive it off. The net affect of this may be zero but it will redistribute the outcome if you hit or stand to a subset of what they were. This affect is only realized if you stand.

Whether you are next to your opponent or at the other end of the table, the effects of hitting and standing on the set of pairs of outcomes will be identical. This is what the ENHC results were intended to illustrate. It can be counterintuitive, but it is a fundamental law (I think) -

• If you stand on a total, say 18, and your opponent draws the card that you waived, then if the opponent has, say, a 10% chance of scoring 19, they will have that exact same chance even if they draw a different, random card (e.g., after some number of other players have hit).
• Conversely, if hitting gives you, say, a 10% chance of scoring 20, and gives your opponent next to you a 5% chance of scoring 21, they will have the exact same 5% chance of scoring 21, whether they draw the card that immediately follows the one you waived, or a later one.
• The common card simply has no bearing on the combined results of the two players.

I said earlier that this sort of reasoning can be seductive, and I got slightly seduced once again, the complexity of two players versus the dealer making me think this might be a special case. But now I'm highly doubtful; I just can't quite find a way to formulate a rigorous (and comprehensible) proof that the common card makes no difference in this case, just as it makes no difference as a general rule. But I have had an idea for such a proof...

These are option pairs if the next card is either taken by you or waived off and taken by your opponent. Compare these and see if you have a greater chance of any swing by waiving the card off given the distribution of dealer outcomes for the remaining cards. My suspicion is that the added chances of half swings from not busting will increase the odds of of getting your goal.

As I've already said, there is no point in comparing the benefits of hitting versus waiving off (the common card) in this way. Nothing about the results is going to tell you whether the common card played any role in producing the differences. What you need to do is compare hitting with hitting, and standing with standing (for the common-card versus non-common card scenarios).

 

[London Colin]

It occurred to me that there is nothing special about blackjack. It ought to be possible to gain insights and derive proofs by looking at much simpler (albeit contrived) games. We just need to have the same fundamentals -

1. Players draw cards (or whatever) from a finite stock.
2. They act in turn.
3. The cards aren't replaced before the next player acts.

So with that in mind, the simplest game you can have is drawing for the high card. Imagine a two-player game (no dealer, for now at least) -

• Each player is dealt a card.
• Acting in turn, they have the option to either stand or draw a replacement card.
• High card wins.

Let's say the deck consist of just 3 individual cards (one of each of the ranks - 1, 2, and 3); that is, the remainder after the players cards have been dealt is just these three cards.

Suppose player A was dealt a 2, and somehow knows for certain that player B will draw. Ignoring any ideas of good/bad strategy, we can just enumerate all the possibilities for how the game will play out...

Firstly, just consider hitting versus standing as completely separate questions-


If A stands on his 2, B has an equal chance of drawing any of the three ranks, and there is a 1/3 chance of a win/lose/tie -

A     B   A's result
--------------------
2     1    win
2     2    tie
2     3    lose

If A hits his 2, the chance of a tie is removed. It is now a 1/2 chance of win/lose. (So hitting is good if a definitive result is needed.)

A     B   A's result
--------------------
1     2    lose
1     3    lose
2     1    win
2     3    lose
3     1    win
3     2    win

Now think about it in terms of the three possibilities for the next card -

Next Card    A stands  A's Result    A hits    B hits     A's Results
-----------------------------------------------------------------------
1             2 v 1        win         1     2 or 3         lose,lose
2             2 v 2        tie         2     1 or 3         win,lose
3             2 v 3        lose        3     1 or 2         win,win

As you might expect, the results are the same.


And we can also go through the possibilities if a card is burned before player B acts -

Next Card    A stands  Burn       B hits     A's Results
--------------------------------------------------------
1             2        1            2 or 3         tie,lose
2             2        2            1 or 3         win,lose
3             2        3            1 or 2         win,tie

Next Card    A hits   Burn       B hits     A's Results
--------------------------------------------------------
1             1        2            3         lose
1             1        3            2         lose
2             2        1            3         lose
2             2        3            1         win
3             3        1            2         win
3             3        2            1         win

Again the results are no different - If we don't know the next card, then standing gives equal chances of win/tie/lose, and hitting gives equal chances of win/lose. And this is true regardless of whether the card we waive off goes to the opponent. (Specifically, the probability of each matchup of player A and player B scores is the same, whether or not there is a common card.)

As far as I can see, the only thing missing, preventing this game from being an exact analogue of tournament blackjack, is an additional dealer card, which the players would be attempting to beat. Then we could count up swing/no-swing results in the tables.

I know I've been contradicting myself at various points in this thread, but right now I'm struggling to think of a reason why the results for that game would be any different.

 

[tirle_bj]

I would like to consider a very simple situation, to demonstrate the ups and downs of Stand vs Hit.
Let's say we're playing the last hand, where only one player advances. We are BR1 and there is only one runner up (BR2), who is playing after us.
We took Low and hope for the best. Our hand is irrelevant (we have a hard stiff hand),
but BR2 has hard 12 vs Dealer's 10 (not a BJ). Also lets imagine that there are only 4 cards left (including Dealer's hole Card):
three Ts and one 9. Here we come to the main Topic: if we Hit (we can do it max once) then we will get 9 with 1/4 and T with 3/4. So BR2 will advance with p=0 in 1/4 of a time and with p= 1/3
3/4 of the time.
The probability for BR2 to advance is p=3/4 x 1/3 = 1/4
Now if we Stand then BR2 advances with the same p=1/4 (only if he gets a 9)
Of course, NO surprises. Mathematically no difference. BUT, here's what we see:
In the first case with small probability (1/4) we completely eliminate BR2's chases to advance, but instead when we get a T (3/4) we increase BR2's probability to advance from 1/4 to 1/3. So with 1/4 (smaller probability) we decrease BR2's probability to advance by 1/4 (25%), and with 3/4 (higher probability) increase BR2's probability to advance by 1/12 (~8.33%).
In other words, more often we increase BR2's chances by little, and less often we decrease BR2's chances by much.
The choice is yours, mathematically no difference, psychologically there is a difference.
If one out of 4 people is sentenced to death by getting one marked ticket out of 4 - then it is up to us to draw first, second, third or forth. Probability is the same, but someone before us can get a marked ticket...