Strategy for next card when an opponent plays after you.

That's true on one level, and was my starting point for getting into this debate - if you average the effect of removal of each of the 13 cards on any given quantity relating to your opponent's hand (e.g., EV, probability of winning, probability of reaching a particular total), you must necessarily get the same average value for the 12 remaining cards as you started with for the original 13. So, given that you don't know what card you will draw if you hit, hitting changes nothing for your opponent.

However, what impact might hitting have on the distribution of win/lose vs lose/win outcomes between the two players?

Suppose 5 out of 13 remaining cards would be of benefit to either player (the same 5). On 5 occasions out of 13, hitting will simultaneously improve things for you and make things worse for your opponent (reducing their probability of also drawing a beneficial card from 5/13 to 4/12). Of course, the flip side of that is that on 8 occasions things will get worse for your hand and better for your opponent (5/12 chance of drawing a favourable card), balancing out all the quantities I mentioned above.


So if the aim is to maximize the chance of a swing, it looks like hitting can help achieve this. The opponent's overall probability of losing the hand is unchanged, but when he does lose it is slightly correlated with you winning your hand.

And conversely, you have not only reduced your chances of winning your own hand (if the hit was in defiance of basic strategy), but when you do lose the hand, that is similarly correlated with the opponent winning.

 

I have one little thing to add to this.

Suppose you have 12 and dealer has 4. You have a virtual minimum bet and a person behind you has 10 or 11 with a big bet.

Hitting/standing 12 vrs 4 is a borderline anyway. If you hit and bust you have taken a 10 away from the player behind you. One less 10 for him to get. You might take away a 10 and really aggravate the individual, if the hand does not work out to their advantage. I know at least one person who really gets upset at people doing this. This might prompt the individual to perhaps make an error later, because of irritation. The situation will not happen often, but it is something to think about.

 

Sorry, I hadn't quite taken this in before I made my previous post. I'm a bit confused about what this example shows. It looks like you might be addressing the point I was trying to clarify- the potential impact of the common card on the swing probability - and saying that it will always average out to zero, just like all the other quantities I referred to. Is that right?

 

As I stated in the beginning of my previous post we all, so far, are mixing a whole bunch of approaches and ideas. Perhaps we should find and define what we are looking for and then possibly try to quantify it.
Whatever is written afterwards, kudos to Chairman for looking at a novel way to approach the tournament plays.

Colin, I am sorry as my example with both players having 16 could have been misleading as a proof that all hits don’t change the chances of swings, it was just one of possible examples where the swing rate doesn’t change.
We could come up we lots of other, very clear and (even more so) not so clear, examples where if we are looking for a swing, taking a card, which is against basic strategy rules, increases the chances of the swing.

But....
One of the ideas was to find out how much taking away one of the cards changes the chances of the swing.
Yes, the hit card will belong to one of the groups that improve (by various degrees) our hand and increases our chances for a swing, or don’t change our chances of winning/pushing, or diminishes prospect of our push/win. At the same time that “removed’ card may improve/not change/worsen the final outcome of our opponent.
Removing one card has minimal effect for most playing conditions and may become somewhat important if you get to play two rounds from a single deck to six players, or so.

We could come up with bunch of tables for different remaining decks (number of cards) and possible denomination distribution, that will show how taking away one card changes “the swing” chances based on both players cards and dealer’s upcard. Full Monty would have hard 4 to 20 and soft hands for both players, and 10 dealer’s upcards. That’s 27x27x 10 and would create 7290 positions for each table.
They can definitely be shorten by using subgroups that will have likely outcome: like soft hands lesser than 16, or both players same hands, or groups of stiff hands with difference by x points – still it would be tables with hundreds positions.
Then, each position should have a value for changing specific “swing” – W-L, W-P, P-L.
Then, it has to be applied to very specific tournament situation where different swings values will vary for reasons of hands remaining to be played, bankrolls, bets limits, betting position, opponent skills, etc.

I wish that I missed something and the discussion can be directed to even minimally practical or theoretical aspect of the game with a little less dispersion.
Though my conclusion is: yes it happens, minuscule effect with thousands possibilities, and “it all goes back to tournament strategies” – it was fun to try to embrace the subject.

S. Yama

 

For myself, I really just wanted to get to the bottom of a single, fundamental yes/no question - is it mathematically possible for the 'common card', which player A can either take or give to his neighbour, player B, to have an impact on the combined results of the two players (specifically the chance of a swing), such that there is a difference compared to the case when they play the same two hands (in the same way), but with one or more additional players seated between them. That, as I understand it, is Chairman's thesis in a nutshell.

I've been taking it for granted that if such an effect were to exist it would be too tiny to be of any practical use. But before worrying about quantifying something, the question of whether or not the laws of nature allow for that thing to even exist has got to be the first hurdle!

I started out as a naysayer, became convinced, and now I am unsure again! My head is spinning.

Indeed. The problem is that we routinely go against basic strategy in the hope of a swing, regardless of where the opponent is seated, e.g., by hitting to a specific total. It's normally only the impact on our own hand that concerns us, and changes the chance of getting the swing. So any analysis of the impact of the common card would have to subtract that out.

Just to clarify - when you say 'it happens', are you referring to Chairman's thesis, as I summarized it above?

 

I am not sure I understand what are the exact questions that we are trying to find answers to.

Colin wrote: “fundamental yes/no question - is it mathematically possible for the 'common card', which player A can either take or give to his neighbour, player B, to have an impact on the combined results of the two players (specifically the chance of a swing), such that there is a difference compared to the case when they play the same two hands (in the same way), but with one or more additional players seated between them.”
If the play is identical for both cases, where a player(s), acting in between player A and player B, take or don’t take card(s) -the combined result will be the same as assessed before other players acted.
If we compare plays where player A takes or doesn’t take a card then these are different plays.
As Colin stated previously, taking card by player A does not change combined chances for player B acting after him, considering the composition of the remaining cards before the player’s A card is seen.

However, it is possible for player A to take or not take a card, that is against basic strategy plays, that it will increase chances of gaining or full swinging player B.
It is usually at a cost of increasing the gap (and losing conventional bj EV), but in specific situations it may be the proper play considering the “count” or a specific deck composition.
For example, with a small negative count hitting 12 vs. dealer’s 6 is against basic strategy, but right play bearing in mind the deck composition, and it is the only way to gain/swing an opponent who has a stiff and will not hit his/her hand.
A different example would be if player A has TT (no splitting possible), player B has 10, dealer has one card Ten, and the remaining cards are three Tens and one Ace.
Standing causes three of four hands pushed and one loss to the dealer (EV -.25%), and three same results and one increased gap to player B.
Hitting player’s A 20 wins once and loses three times (EV -.50%), but it would have one gain, one negative swing, one increased gap, and one same result vs. player B.

Part of the discussion was removing one card (by taking a hit). Any of the ten denominations removed has an effect and always have to be considered for all three changed outcomes: players A and B and the dealer. The only exceptions may be a Ten value for players with soft pat hands where all other negative and positive hands outcomes are balanced (though hitting Ten leaves the soft pat hand at the same value it reduces chances for Ten to bust stiff hands and making pat hands for totals 7 to 11).
Also, counting number of denomination that will improve/not improve both players’ hands (and let’s not forget the dealer’s) may be misleading as each removed card influences the outcomes by very different values.

S. Yama

 

My question was, has anyone looked into the possibility of increasing the likelihood of a swing by using this next card relationship. So far it seems the answer is no. The implied question if the answer is yes would be what did they find.

The hands I found interesting were you have soft 18vA and hard 14 vA for your opponent. The other was You have soft 18vT and your opponent has hard 16vT but any hand set could be interesting (The latter is less complicated). The premise was you are needing a swing in the next couple of hands to be in position to play the final hand and be relevant. Now to get a swing you need to beat your opponent and beat/push the dealer OR beat the dealer and have your opponent lose/push with the dealer. Relative EV's do not tell that story. Most of the EV difference can be a large difference in a rare set of circumstances. Or they may be a small difference that cover a large amount of what is possible. If you hit your soft 18 and end up busting you can't beat your opponent but if your opponent doesn't beat the dealer you have the next round to try to get a swing. If you make a better hand it increases your chances of beating the dealer. Another concern may be a loss for both of you may cripple you if your stack is low enough. So if you bust and your opponent loses it may be the same as him winning and you losing in terms of your chances. You are out of it if you lose. But that is a special case.

Soft 18vT there is little difference for you no matter how you play it. There is even less difference for your opponents 16vT.

1 of 13 next card causes the same result for both of you. No change except for lose/lose in the special case mentioned above. If you hit you would have 20.
8 of the 12 remaining ranks half would hurt your hand and the other half would have no effect on your total but they all bust your opponent. Leaving the fate of the swing at 47.33% that the dealer busts or gets 17 for full swing or 18 for a half swing. The 4 ranks that hurt you have you standing on 17 or hitting 14, 15 or 16.
1 of the 12 remaining ranks gives your opponent 17 which makes the possibility of a half swing 24.07%. If you took the card you would have 19 opponent hitting 16vT.
1 of the 12 remaining rank gives your opponent 19 which gives him a half swing chance of 24.44%. If you took the card you would have 21 opponent hitting 16vT.
1 of the 12 remaining ranks gives your opponent 20 which gives him a half swing 49.12% of the time and a full swing 12.21% of the time. If you took the card you would have 12vT opponent hitting 16vT.
1 of the 12 remaining ranks gives your opponent 21 which gives him a half swing 15.95% of the time and a full swing 49.14% of the time. If you took the card you have 13vT.

The numbers presented assumes only cards on the table as known cards out of play in a 6D S17 game.

I never worked the rest of the problem because I got too busy. Maybe I will pick it up sometime but it won't be for a while. The chances of getting a both a partial or full swing are defined above if you stand. I just need the chances if you hit and that was a little more complicated but could be done. I got part of the way into it and decided I didn't have the time to finish. Maybe I will look at it again later and finish when I get more time. Of course a real answer to all possibilities would be worked out on a computer sim. Or a program using composition dependent analysis as was my approach.

 

I can generate such strategies and, since I use simulation to do it, the resulting strategy automatically accounts for all possible factors that exist. I can't tell you what percentage of the swings generated are attributable to each possible factor, however, or even what all of the possible factors are.

 

You can insert some number of additional players between the two of them and repeat the simulation. Any difference in the results will be due to the next card relationship when the players are adjacent, and nothing else. Though the statistical significance of any slight difference (for a given sample size) would have to be calculated.

Yama, I'm afraid it's still not entirely clear to me whether you believe the above test would (or rather could) reveal a difference or not, though it sounds like you don't. This is a really hard subject to speak unambiguously about.

 

That's similar to my thought process, except that I am still not entirely sure that the first reaction was not correct. Could it be that we are making a mistake by focusing on the impact of a single card, rather than accounting for the totality of all the things that can happen to the set of three hands that are involved (two players, plus the dealer)?

Or have you actually managed to run through a complete example in this fashion, in order to compare the swing probabilities in the two cases?

 

Just to have fun with the numbers before Chairman narrowed down the situation to hand of soft 18 I looked into another one.
Let’s say that all we care is a full swing. We have hard 17 our opponent (P2) acting after us has 16, but he makes a bonding hit signal to his 16, and the dealer shows 7.
[A good player trying to avoid full swing should stand on 16 if his opponent’s total is 17 (and 18), and hit to 17 with the opponent’s totals of 19 to 21, when dealer shows 7.]
If we stand on 17 we reach our goal when the dealer busts and P2 busts.
Depending on how many cards are left to be dealt from, dealer busts 25.99% when a full deck of 52 cards is left and 26.23% from the infinite deck.
Our chances are 15.99% with single deck and 16.23 % with infinite deck.

Let’s see what happens if we decide to hit (and hopefully take away one of the 4 cards that improve our hand and at the same time reduce chances of P1 improving his hand).
This becomes somewhat tricky as removing one card does not create intuitive results for the dealer. For example removing 9, which would make dealer upcard 7 a “bustable” 16 reduces total dealer’s busts from 25.99% to 25.14%, and removing a 4 (which with 7 makes a nice 11) increases busts to 26.4%
If we hit our 17 (and P2 hits his 16) effects of removing one card change to a small degree everything else.
Taking into account all possibilities for removing only one card, we get a full swing 10.93% of the times if the play was from a full single deck, but it drops down to 10.63% if it was dealt from full 100 decks.

S. Yama

 

Colin, I think you hit the nail right on the head.
Removing a single card from the remaining deck(s) changes to some degree all outcomes for hands to be played. On super rare occasion the composition may be balanced so there is no change, and that would be rather with limited number of cards left. With more decks remaining there will be a change but with a smaller effect, and it could move in either direction.
I think the effect is not observable by looking for the numbers of denominations that improve/worsen particular player(s) hand as they change dealer’s outcome too, and the hands that need to be hit again can’t easily be calculated.
As to three specific hand of soft 18 vs. P2 16 and d’s A, I am sure the right play is to stay if we are looking for a gain (W/P and P/L) and even a full swing with dealer hitting soft 17. The reason is that P2 trying to avoid those swings should stay on 12 (and stay on 14 vs. d’s hs17), so letting him hit 16 only worsen his chances.
If we are looking for a full swing (s17) P2 should hit to 17 making it a closer case in terms of what is a better decision hit or not to hit.

Gronbog, if you get a chance to do the sim do it for multiple decks, single deck, or even more so, particular smaller subsets may create results that change the outcomes one or another way for other reason that were originally intended (looking for cards that could be removed, that seemed to benefit/worsen players hands.
Thanks,

S. Yama

 

I've got a few things on my plate over the next few days but, after that, I'll try to design an experiment in an attempt to demonstrate whether the effect exists or not.

 

Yep, I'm pretty sure we can all agree on that, which allows us to eliminate any notion of there being a "common" card that player 1 is considering giving to, or withholding from player 2 and also therefore any need for player 2 to be the next player to act when thinking about the original question. Such considerations would only come into play if we knew the exact composition of the remaining cards and were somehow able to compute the effect of removal of one of each of the remaining ranks on player 2's hand. Its unlikely that any human player could achieve this.

I think that there is still a question to be considered here, however, and with the simplifications above, perhaps the original question can be restated along the following lines:

We know that there are good reasons to either take another card or to stand when trying to generate a swing. The reasons are usually based on the current state of our hand vs that of our opponent. First of all, for a 1/2 swing, we need our opponent to bust while we push, or our hand total needs to be at least 17 and higher than that of our opponent. For a full swing, we need our opponent to bust while we win, or we need at least 18 and a difference of at least 1 between the hands. Once a swing is possible, the question becomes one of the balance between improving things by taking more cards vs the risk (i.e. the possibility of busting a hard hand or losing the gap for soft hands). There is a table for this in Wong's book. The case where our opponent has already acted is not in question. When our opponent acts after us, the right answer depends on what our opponent will do in response. I don't know whether Wong's table is based on our opponent playing basic strategy or optimal strategy, although I suspect the former given the computer resources that he would have had at the time.

Given all of that, here's how I think Chairman's original question could be rephrased: If we know that our opponent will take a card if we stand, would that change the recommendations in Wong's table? The question is not one of the opponent acting next or taking "the" card that we leave, but rather one of knowing that he will hit no matter what we do. I think that the answer is probably "yes". For one thing, if we know that our opponent will hit his hard 16 vs a dealer 6, then we are less likely to need to hit to 17 or 18. I should be able to show this formally by generating the proper strategies for each of these 3 cases:

1. our opponent will play basic strategy
2. our opponent will play optimal strategy
3. our opponent will always hit with 16 or less (hard or soft)

Hopefully one of the first two will match Wong's table and the 3rd will be different from both of them. Now, whether the results will be practical remains to be seen. e.g. relying on our opponent to be dumb enough to hit a stiff vs a 6 after we stand stiff is unlikely to be of any practical value!

 

It turns out that I am mistaken and that there is no table for generating a swing in Wong's book. The table I was thinking about is a table for correlating -- "Win Both Ways". Maybe I saw one somewhere else. Has anyone else seen one?

In any case, I will generate the three tables that I proposed above and we can see how they differ.

 

I stand by my statement that taking or not a card(s) does influence the results between the two players and the outcome of their hands (w/p/l).
It has really a minimal effect but from a mathematical point of view it is there, I believe.
The reason for it is that removing a card(s) reduces the number of cards left creating a “smaller deck” and changes the statistical outcomes if more than one card is to be played.
Without knowing what exact card is (will be) removed we assume that the chances for the next card dealt to player (or dealer) is X number of particular denomination left minus one, and it applies to all denominations, so the chance is the same. The chances, in our examples from previous posts, for the opponent hand remain the same if he hits only once, but if he hits more than once they will change.
The same applies to dealer’s outcomes since in some instances (even if nobody takes a card before the dealer hand is played) dealer takes more than one card.
Changing dealer’s outcome is enough to affect ours and the opponent’s hand and their correlation.

Let’s look at dealer’s outcomes when we have left: 8 decks, single deck, and subset of 13 cards with balanced denominations. Dealer’s upcard is one of the remaining cards (I had those numbers handy). For no special reasons I just picked dealer’s upcard Ten.

Final results for 8 decks in % and difference to 1 deck and 13 cards:
17 .... 18 .... 19 .... 20 ... 21(and bj) bust
11.18 11.16 11.18 34.06 11.185 21.24 ----8 decks
+0.26 +0.13 +0.28 -1.17 +0.30 +0.19 ---1 deck
-0.05 -0.67 +0.06 -8.10 +0.09 +8.67 --13 cards

It can be looked at as removing a set of large numbers of unknown cards, but each single removed card contributes to the change.
The above numbers are somewhat skewed; taking away a Ten from the deck(s) affects it more then it would show with full decks (balanced denominations) remaining, but Norm's calculator didn't work when I was writing it, but changes are there.

S. Yama

S. Yama

 

I liken it to playing 3rd base in a ENHC game. You are considering hitting a 17 against an ace. If the next card is a T you lose to a bust (you get the T) or a BJ (dealer gets the T) no matter what you do. In effect your decision is based on a deck with no T's in it because you lose no matter what if the next card is a T. It doesn't matter what you decide. Now you are considering 9 ranks instead of 13. Obviously 4/9 (44.4%) is significantly more than 4/13 (30.8%). If you weren't playing immediately before the dealer you would be worried the next card is a T. That T does not give the dealer BJ but it will still bust you. The dealer likelihood of getting BJ is not affected by you taking an unknown card but for the player playing before the dealer there is also the 7-9 that will cause you to lose either way. If you reduce these ranks from the ranks that leave a win possible if they are the next card you are down to 6 ranks that it matters whether or not you hit. So 4/6 ranks give you a chance to win hit or stand instead of 13 ranks for the next card if you were not playing before the dealer and stood. Now let's look at the analysis of the off the top 2 deck (including dealt cards) S17 ENHC using a CDCA:

If you are not playing directly before the dealer T,7vA:
EV standing is -63.46%, EV hitting -69.27%. STAND.
Dealer outcome percentages:
17: 0.1321%. PUSH
18: 0.1247% LOSE
19: 0.1327% LOSE
20: 0.1335% LOSE
21: 0.0535% LOSE
BJ: 0.3069% LOSE
BUST: 0.1167% WIN

IF you are at third base base the EV's remain the same but a new relationship can be formed for the next card:
T: LOSE 100% either stand or hit
9: LOSE 100% either stand or hit
8: LOSE 100% either stand or hit
7: LOSE 100% either stand or hit
So 7 ranks you lose whether you hit or stand if you play immediately before the dealer. Since the EV of the hand is unchanged whether you hit or stand you should stand but if you needed a swing against the dealer and lose 100% of the time no matter what on 7 ranks of next cards the ranks that it matters whether you hit or not will be loaded toward a correction to keep the average. The standing chances are listed above as dealer outcome percentages if you stand. Now if you hit there is no affect if the next card is any of the 7 ranks listed above. It is a lose in the above cases and in the cases of next card 2-6. But if you also eliminate the 7 ranks that you lose either way from the above standing percentages he remainder will also inflate the winning probabilities to compensate to maintain the average EV.
6: Dealer has soft 17
HIT: LOSE 100%
STAND: PUSH 100%
5: Dealer has soft 16
HIT: LOSE 100%
STAND:
Ace counted as 11:
17: 0.1573% PUSH
18: 0.1089% LOSE
19: 0.1086% LOSE
20: 0.1027% LOSE
21: 0.0939% LOSE
Ace counted as a 1:
0.4286% Dealer is hitting a hard stiff.

I will stop here. This is the part that is important. The rest of the next cards A-4 increase your chances if you hit but as I said in the previous paragraph both hit and stand results are skewed to maintain the original EVs when eliminating the 7 ranks that have no effect. The EV's being the same mean over many hands the average result will reflect the EV. We are interested in the result of this hand. The deciding factor will be how the next card 6 affects your prospects in the tournament. If that push if you stand means the same thing as a loss if you hit to your chances of meeting your goal there is a big gain to hitting. It may not be enough to change the decision but these kinds of affects are what I am talking about and we are down to 6 ranks at this point. We are worried about how your chances to meet your tournament goal are affected not the relatively unchanged EV's. I guess this may not be worth the trouble to figure out. Plus the problem becomes fuzzier when you include the dealer and opponents hands which is necessary for a swing in the tournament.

A short recap. 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. In our example you can see that the ranks are reduced to 5 that make a difference once the 6 is added due to the push being the same as a lose. There are lots of push is the same as a loss in a normal tournament problem but I guess there is already a chart for that.

I appreciate all the input. But at this point some kind of sim results is the only thing that can say anything with certainty.