Declarer play problem

[han]

I haven't written it up but I've convinced myself that it is actually not as hard as I thought. In order to convince yourself that the correct solution is to play RHO for the heart jack no matter the discards, you only have to give one discard strategy for EW such that for any of the possible discards the chance that RHO has ♥Jxxx is larger than the chance that LHO has ♥Jxxx. This is not hard to do because there is a lot of room (by which I mean, the total chance that RHO has ♥Jxxx is so much larger than the total chance that LHO has Jxxx that there is a lot of freedom in how you choose the discards). I could try to write it out here but it would be long and nobody would read it, it's probably more useful if you try to work it out for yourself. Certainly more useful for me. :)

[benlessard]

If you want to prove the best math great but this is bridge not unlimited time but no one really seems to care about that.

Even if calculating this example has no real life benefit, the ideas behind the solution will be useful on others hands, of course trying to find the perfect solution or proof might have no application in real life, but in general there is always something not obvious to learn from these case and trying to find out by yourself is often a good start to understand these things IMO even if you know your tries are likely to be far from the solution.

[han]

I think the main idea that you can take out of this is that (against optimal defenders) the discards of the opponents (for example the fact that LHO pitched two diamonds) are completely irrelevant, but the fact that LHO cannot have 4 hearts and the spade king is relevant. This makes it more likely that RHO has 4 hearts.

 

For Bluecalm the following argument might be convincing. There are these possibilities:

 

A) hearts split 3-2.

 

B) LHO has the king of spades and Jxxx of hearts.

 

C) LHO has the king of spades and RHO has Jxxx of hearts.

 

D) RHO has the king of spades and Jxxx of hearts.

 

E) RHO has the king of spades and LHO has Jxxx of hearts.

 

Scenarios A and B are irrelevant since you always make it in these cases.

 

In the remaining 3 scenarios RHO has Jxxx of hearts twice (C and D) and LHO has Jxxx of hearts once (E). Since C and E are exactly equally likely, we should play RHO for the heart length.

 

To make this argument correct we have to convince ourselves that the discards are really irrelevant. See my previous post for how you can do that, it takes some work but it is not hard.

[bluecalm]

Ok, so it comes down to:

"Only honest (ie not volunteered by opponents) information is that they have 5 minor cards each; we know they both have 8 more cards. Since LHO would be squeezed if he had both Ks and Jxxx of hearts those layouts are out and all other are symmetrical leaving the ones with RHO having Ks and Jxxx unmatched, so there are less layouts with LHO having 4 hearts".

 

Now when I spelled that out it seems quiet obvious to me :)

Thanks, that was nice explanation, I wouldn't have thought of this.

[han]

It's not clear that that explanation is correct (what does it mean that the information was volunteered? If LHO has Jxxx of hearts his discards are forced), I'm convinced that the conclusion is correct.

 

Fred has posted similar threads in which the underlying question was similar: which of the plays by the opponents have impact on the odds, and which do not? Here the discards are not impacting the odds, but I think it requires some thought. Perhaps Fred has found a neat way to think about this though.

[benlessard]

Edited Sorry Han and others, Im a native french speaker and work night shift and its close to my sleeping time now.

 

IMO I don't think we should say the discards are irrelevant.

 

After cashing the fist 7 tricks we learn [hv=pc=n&w=s2h3d432c32&e=s3h2d765c54]266|100[/hv] Its symmetrical and if we make the asumption that H break 4-1 we can say for sure that E and W are equally likely to have 4H (in a no-bidding vacuum).

 

Here are the a priori west hands depending if west started with 4H or with only 1. Lets give the name "DS" for D or S idle cards.

 

with 4H

x

xxxx

xxx

xx

with 3 (DS) or with 2DS+Ks

 

with a stiff H

x

x

xxx

xx

with 6DS or with 5DS +Ks

 

on the run of the clubs west two first discards will be 2DS, he will never discard a H or the K of S. If its one diamond & one spade or 2 spades or 2 diamonds its irrelevant. He always have 2 free DS available for discarding, so these are non significant discards. However on the 5th clubs the 3rd discard by West he still cannot discard the K of S nor a heart so at this point if he got 4H & the K of S he wont have an idle DS available to discard. So in that position when he discard a DS here its a meaningful discard not an irrelevant one. So that change the odds. The fact that he knowned with 1 more free DS than EAST mean hes got one less vacant space in his hand. If we put West with 5 vacant space and 6 vacant space to east we get these H splits.

 

No-W---E--Probability--Times--Total

1--0---Jxx--12.121------1-----12.121

2--x---Jx---15.152-----2-----30.303

3--xx--J----12.121-----1-----12.121

4--J---xx---15.152-----1-----15.152

5--Jx--x----12.121-----2-----24.242

6--Jxx--0----6.061------1-----6.061

 

Bonus question what if East discard the Ks after north discard his J of S ?

[han]

Benlessard, your writing style is incomprehensible, do you honestly expect anybody to read that and understand it without spending a lot of time deciphering it?

[bluecalm]

Ok, here is my latest try:

we forced them to play 5 minor cards, one spade and one heart each. That leaves 12 cards in their hands out of which 8 are irrelevant and 4 are relevant: Jxx ♥ and K ♠

Possible layouts are:

 

a)LHO has 6 irrelevant cards; RHO has 2 irrelevant card and both Jxx♥ and K♠

b)LHO has 5 irrelevant cards and K♠ ; RHO has 3 irrelevant cards and Jxx♥

c)LHO has 3 irrelevant cards and Jxx♥ ; RHO has 5 irrelevant cards and K♠

d)LHO has 2 irrelevant card and both K♠ and Jxx♥ ; RHO has 6 irrelevant cards

 

We already know that a) and d) are not the case since we saw 3 irrelevant cards from both of them (played to clubs).

We also know that b) and c) are symmetrical and equally likely.

How does information about RHO being able to play K♠ come to play?

 

My understanding is that if they agree not to play K♠ ever then it's just a guess on actual layout but it wouldn't be a guess if layout a) occurred. RHO had to play K♠ then and we play him for Jxx of hearts too. RHO can try to some mixed strategy of throwing K♠ from layout c) but then every time he plays irrelevant cards that layout becomes less likely (because his strategy consists of throwing K♠ from it sometimes)

 

In other words:

If we always play RHO for Jxxx of hearts we succeed every time against layout a), c), d) and lose to layout b)

If we always play LHO for Jxxx of hearts we succeed against layouts b) and d).

So if we just close our eyes (despite looking for K♠) and play RHO for Jxxx ♥ we succeed more often so we should do so.

Opponent may play in such a way to give us exactly 50-50 flip on this exact hand (by refusing to play K♠ no matter what) but for using that strategy they would pay by giving us 100% in case a).

 

Are current bridge playing programs capable of recognizing such situations ?

[cherdano]

I think there is nothing to add to Han's and bluecalm's explanations, but let me try an intuitive reasoning:

 

You should always ignore information that the defenders volunteered.

 

Here we should ignore the information that LHO has 5 diamonds - he could also have thrown spades instead if he has a singleton heart. But we should also ignore the fact that we know RHO does not have ♠K ♥Jxxx - as Andy pointed out, if RHO has ♠K and ♥x he is free to choose to provide us with this information, or not to provide us with this information.

[fred]

It's not clear that that explanation is correct (what does it mean that the information was volunteered? If LHO has Jxxx of hearts his discards are forced), I'm convinced that the conclusion is correct.

 

Fred has posted similar threads in which the underlying question was similar: which of the plays by the opponents have impact on the odds, and which do not? Here the discards are not impacting the odds, but I think it requires some thought. Perhaps Fred has found a neat way to think about this though.

The way I came to the same conclusion as Han involved no real math. To me it is qualifies as "neat", but I am not sure if the mathematicians out there would consider this reasoning to be valid. Valid or not, like Han I am also confident that the conclusion is correct.

 

The problem that I posted was the same as the one I was given, but let's simplify things by changing the spades to Ax opposite xx (this changes the relative likelyhood of which defender has Jxxx of hearts, but it doesn't change the basic reasoning or the conclusion).

 

Ignoring LHO's possible discarding strategies for a minute, it is clearly more likely that RHO has the Jxxx of hearts. The simplest way to see this involves no more than realizing that LHO is known to be longer than RHO in diamonds and all other information is neutral so RHO is more likely to be the defender with longer hearts.

 

Now let's stop ignoring LHO's possible discarding strategies and ask ourselves why he would discard his diamonds. If he was 2452 then he had no choice in the matter, but if he was 5152 he gave us information that he didn't need to (because he could have discarded spades instead). Why would he do this? One possible answer is that he didn't really think about what he was doing, but it is also possible that a strong LHO might discard this way intentionally from 5152 trying to make it look like he had 2452 instead.

 

Several posters basically said the same thing in their analysis - well done by them.

 

But let's go a step further. If LHO is a super-strong player (and gives declarer credit for being able to get this far), he might switch horses and discard 3 spades from 5152. Now declarer is left with a complete guess, but declarer might then ask himself "if my super-strong LHO is actually 5152, why didn't he discard his diamonds in order to make it look like he is 2452? Maybe he is really 4432 and had no choice about his discards". Of course that would leave RHO with 5152 and then the same the same question could be asked about him (if he is also super-strong).

 

This sort of spy versus spy game can go on forever - it is one of those "he knows that I know that he knows that I know..." type of situations. The only reasonable way for declarer to attempt to win this game is not to play it. He should not get involved in trying to figure out how many rounds of bluff and double bluff a super-strong LHO might have considered and just use the actual information that he has.

 

That means going back to square one and playing RHO for Jxxx of hearts.

 

Fred Gitelman

Bridge Base Inc.

www.bridgebase.com

[gnasher]

I'm going to stick my neck out and say that Fred's reasoning is invalid.

 

If this is a theoretical problem, LHO will treat all his diamonds and small spades as equivalent, and discard them randomly. Throwing ♦6, ♦8 and ♦10 is exactly equivalent to throwing ♠4, ♠9 and ♦7. LHO has three suits:

- Small pointed cards

- Hearts

- King of spades

The particular three small pointed cards that LHO chooses to throw have no bearing on his length in the other two suits.

 

If this is a real-world problem, we should try to outthink them, obviously.

 

 

[Edited because I very briefly posted some complete nonsense. I hope no one saw it.]

Edited February 1, 2012 by gnasher

[han]

If this is a real-world problem, we should try to outthink them, obviously.

 

I absolutely agree with this. Especially if I think that I'm better at this than my opponents I should be willing to play this spy vs spy game against them.

 

Your idea that the remaining low spades and diamonds should be considered as equivalent is a very nice way of thinking about the problem and makes the solution obvious. I regret I didn't think about it earlier.

[JLOGIC]

Very smart people in this thread. Great read, enjoyed it.

[bluecalm]

Your idea that the remaining low spades and diamonds should be considered as equivalent is a very nice way of thinking about the problem and makes the solution obvious. I regret I didn't think about it earlier.

 

Well I did think about it in my latest try post thanks exactly to your previous posts and before I read them I didn't see the problem at all :)

 

I think Fred's reasoning is invalid too. This:

 

The simplest way to see this involves no more than realizing that LHO is known to be longer than RHO in diamonds and all other information is neutral so RHO is more likely to be the defender with longer hearts.

 

Is incorrect. If we are that naive LHO will just discard diamonds from 2-4-5-2 but spades from 5-1-5-2 and we will be outplayed badly. Or in similar hands where diamonds where 4-4 only defender with Jxxx discards a diamond thus fooling us if use that information.

I think that if spades are xx to Ax it's pure 50-50 guess.

 

You should always ignore information that the defenders volunteered.

 

I tried this but while it seems to be good general rule I think there will often be problem interpreting what information was volunteered and which wasn't.

[fred]

If we are that naive LHO will just discard diamonds from 2-4-5-2 but spades from 5-1-5-2 and we will be outplayed badly. Or in similar hands where diamonds where 4-4 only defender with Jxxx discards a diamond thus fooling us if use that information.

I think that if spades are xx to Ax it's pure 50-50 guess.

I am now (mostly) convinced that I had lost my mind (hopefully only on a temporary basis) in my earlier thinking about this problem. Thanks to those who helped to point this out with the above quote by Bluecalm really getting the job done for me.

 

Fred Gitelman

Bridge Base Inc.

www.bridgebase.com

[benlessard]

It a simpler but equivalent solution to what i was thinking before. Before the run of the clubs West has 7 know cards and 6 vacant spaces & so do East. This IMO is the important starting point. We know that West has at least 3 idles cards (he cannot have Ks and the 3 remaining hearts) and that west has at least 2 idle cards (for him the K of S is an idle card since he can discard it easily after dummy discard his J of S). So the H distribution should be ---5 vacants spaces for West and 6 vacants spaces for East. West is going the have Jxx in hearts slightly more than 6% and East has twice the odds (more than 12%) to have the 3 remaining H.

[Cascade]

Isn't the easy way to see this that the problem is symmetrical in the defenders.

 

If either defender started with four hearts to the jack then he had no choice in his discards (only the order).

 

In addition if either defender held four hearts and the spade king then we would have seen either a heart discard or the spade king.

 

There is no information that suggests one rather than the other opponent had a choice in his plays.

[HighLow21]

No matter what you assume about discarding strategies, it is more likely that RHO has the long hearts than LHO, simply because RHO has more vacant spaces to hold JXX♥ after trick 7. I personally think that RHO is far more likely to have it than LHO, but it's still always bigger no matter how you think through the math.

 

Here's my explanation.

 

After trick 7 (1♥ 1♠ 2♣ and 3♦ have been played) there are 12 unseen cards: 2♦ 3♥ and 7♠.

The defenders randomly choose 6 of them; there are 12!/(6!x6!) = 924 unique ways to distribute them.

 

Of these, LHO will have no diamonds 210 times, 2 diamonds 210 times, and 1 diamond 504 times. (These are 6/12 x 5/11 x 924, 6/12 x 5/11 x 924, and 6/12 x 6/11 x 924 x 2, respectively.) By the end of trick 9, we're restricted to the 210 cases with LHO having started life with 5 diamonds.

 

Of these:

- LHO will have the JXX♥ without the K♠ 5 times. This is about 2% of the time.

- LHO will have the JXX♥ with the K♠ 2 times. (By the time the J♠ is pitched from dummy, these 2 cases can be eliminated, because LHO is squeezed.)

- LHO will have 2 hearts 63 times and 1 heart 105 times. This is 80% of the time.

- RHO will have the JHH♥ 35 times. This is about 16.7% of the time. Of these, LHO will usually have the K♠, but certainly not always. (I still haven't decided how to treat the inference that RHO does or does not pitch K♠ after this trick ends. I tend to think it's only marginally relevant.)

 

NOW: An intelligent defender who fully understands the situation after trick 7 will reason as follows: "The only relevant cards are the diamonds, the K♠, and the 3 ♥. I will ONLY pitch diamonds on the clubs if I hold 3 hearts or no hearts; otherwise I will pitch spades."

 

(Note that in this reasoning diamonds ARE relevant because of restricted choice, in a sense: each defender must have at least 2 spades, but neither defender must have another diamond.)

 

So back to the 924 cases after Trick 7 has ended.

1. 7 times, West will have all 5 remaining ♥/♦. Of these, 2 can be eliminated because 2 of them will include K♠.

2. 7 times, East will have all 5 remaining ♥/♦.

3. 42 times, West will have 3 ♥ and 1 ♦.

4. 42 times, East will have 3 ♥ and 1 ♦.

5. 35 times, West will have 3 ♥ and no ♦.

6. 35 times, East will have 3 ♥ and no ♦.

 

Under this strategy, West will only pitch two diamonds on the clubs and FAIL to pitch the K♠ while holding the ♥ stack 5 times; he will do so 35 times when East has the ♥ stack. Different reasoning, exact same result.

 

There are other lines of reasoning that get you different probabilities. For example, you could start at the beginning of trick 10 and say there are 4 relevant cards (K♠ and the 3 hearts); there are 70 ways to arrange them, and after the J♠ is discarded, 4 times West has all the hearts without the K♠ and 5 times, East does. In this case, the relative probability of East having the ♥ stack is lower, but it's still higher than the probability of West holding them.

[phil_20686]

While that statement is, I think, fair in general, there are types of falsecards that one should use all the time....so-called mandator falsecards.

 

For example, here is a well known one:

 

Dummy has AQ10x and declarer is known to hold 4 in the suit.....you hold J9xx in front of dummy.....declarer cashes the A and partner follows low. If you play low, declarer leads to his K and hooks your J. So you have to play the 9...now declarer can pick up Jxxx (assuming he has the 8) by cashing the Q....and you have created a stopper....this only works when you falsecard the 9....you have to do it all the time....you never gain when you don't.

 

I think this is in that category. I think there is no edge to ever not playing the K...the more frequently you falsecard, the closer to making declarer's choice a guess.

 

I got started on an attempt to show this mathematically and realized that I was far too ignorant of the math to even attempt. I suspect that I could have done this readily enough 40 years ago, but my engineering degree has sat unused for some 40 years, and my math skills, never great, have languished along with it.

 

We basically agree. But I should point out that in mandatory false cards represent situations where without the false card declarer has no losing options. In this case he has a losing option, he can finesse the other way. Attempting to look like I was squeezed is flat out deception and as such one should try to avoid doing it so often that declarer can draw any inference from your failure to attempt the deception.

[billw55]

Maybe slightly off topic, but I wonder what the actual best line of play is against a club lead. Is the line given in the OP best, or one of the squeezes, or something else?

[HighLow21]

I was wondering the same thing. Wouldn't it make sense to play the major suit aces first (discarding a blocking middle ♥ on the A♥), run the clubs, and then and only then play off the 3 top diamonds? The Q♠ can be discarded on the 3rd ♦, and the early ♣ plays force the defenders into discards when much less is known about the unseen hands. Expert defenders may not have any trouble, or they might; anyone below expert level probably will give something away. The Vienna coup play enforces an autosqueeze on anyone holding 4♥ and K♠, and communications still exist to play either opponent for 4♥ if they have them. I think this line makes it much more likely that declarer will get either (1) sufficient information to play hearts with certainty, or at least (2) enough information to make a strong inference one way or the other.

 

I realize the post was about which defender is more likely to hold the 4♥ strictly given the actual line of play, but it still is an interesting question to me. Does anyone see an improvement on the line I described?

[benlessard]

Another way to check your solution is to find at what point do you know East or West is favorite to be longer in H and why does it change at that point. When declarer cashed 3D+1H+1S+2C the position at this point is totally symmetric (so both defender have the same expected H lenght). After the 3rd club (defense 1st discard) are the defense significant discard (IMO no) ? Does it change something that west discarded a D when he could have discarded a S (no again) ? What about the 2nd discard ? (imo its still even no matter what the discard are since both players have idle cards) What if on the 3rd discard (on the last club) you see west a computer player discarding the K of S or you see a human player conceding or see the player go into thinking and discard a H ? How do you think the H split now ? IMO its clear that the odds are shifting exactly when the 5th club is cashed and west make is 3rd discard.

[benlessard]

In addition if either defender held four hearts and the spade king then we would have seen either a heart discard or the spade king.

The problem is that east can discard the K of S after the dummy. So he can discard the K of S from spade lenght randomly or to fool you into thinking he is long in H.

[HighLow21]

The 2 diamonds are the only significant discards on correct defense as my previous post explains. They reveal the number of spaces that are available for the only 4 relevant cards remaining (K♠ and Jxx♥) and the probability of them lying in one particular hand. I'm becoming more and more convinced that West's pitching them at the first opportunity was either a horrible blunder or a genius bit of reverse psychology if he were indeed 2542 without K♠.

 

Of course, a heart discard or the K♠ discard is also relevant, but I'm assuming we're not playing against relatives here.

[gszes]

[hv=pc=n&s=saqhak98dk2caj432&n=sj2hqt76daq3ckq65]133|200[/hv]

You get a club lead against 7NT.

 

Rightly or wrongly, you play a second round of clubs (both follow) and then 3 rounds of diamonds (both follow) discarding the Queen of spades from your hand. Then you cash a top heart from your hand (both follow small) and the Ace of spades (both follow small) before finishing the clubs.

 

On the run of the clubs, LHO discards 2 diamonds and a small spade while RHO discards 3 small spades. You discard dummy's Jack of spades on your fifth club.

 

Which opponent do you play for Jxxx of hearts and why?

 

Fred Gitelman

Bridge Base Inc.

www.bridgebase.com

 

deleted garbage due to misread of trick 10