Declarer play problem

[fred]

[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

[phil_20686]

I would have played the hand differently, figuring that if west has 4 hearts the spade finesse is more likely to be right, I would have aimed to pick up 4 hearts with east.

 

This gives me pretty good chances, if the spades are 4-1 the wrong way and the clubs are 2-2 and diamonds not worse than 5-3, I am surely at least 5/9 in the spade finesse, and better if the diamonds are 4-4 or 5-3 the other way, which is hardly impossible.

 

Having played the line you did, I know that diamonds were 5-3 and spades clubs were 2-2. so teh danger layouts are 2452-7132, or 5152-4432, of which the second looks distinctly more likely. For one thing, with seven spades to the K rho might have done something at some point in the auction. So looks like I am playing rho for four hearts.

[bluecalm]

So it's either:

1)xx-Jxxx-xxxxx-xx to Kxxxxxx x xxx xx or:

2)Kxxxx x xxxxx xx to xxxx Jxxx xxx xx

 

First apriori chances. There are 8*7/2 * 4 * 8*7*6/3*2*1 * 4*3/2 ways to deal 1) and:

8*7*6*5/4*3*2 * 4 * 8*7*6/*3*2*1 * 4*3/2 ways to deal 2).

Terms for hearts diamonds and clubs are the same so what is left is:

8*7/2 to 8*7*6*5/4*3*2 in spades which is:

56 to 70 so if my math is correct 2) constitutes for about 55.5% of all layouts.

 

Now is there anything in play which influence those odds ?

First, bidding is important because we need to know if club lead was obvious or if it was some random choice. From 1) W had 3 suits to lead from including 2 empty doubletons, from 2) he had 2 suits to lead from so if bidding wasn't informative club lead will more often come from 2) than from 1).

 

Now discards. From 1) discards are perfectly consistent and the only way to discard.

From 2) W sometimes would part with a spade or two before getting rid of all diamonds.

So it looks like lead increase chances for 2) (assuming not informative bidding) and discards increase chances for 1).

It's a tough call depending on how opponents lead and discard. My guess would be to play 1) against average opps if bidding indicated club lead as from two 5 carders they rarely discards exactly 2 diamonds first. Against very good opps capable of discarding in a way to suggest me wrong layout I go with a priori (+first lead information) chances so I play for 2).

Against average opps with not informative bidding I again gor with 2) as then 1st lead inference is quite strong.

 

It also seems likely that from 2) layout W would try to signal for K♠ using diamond spots so I look for that information too.

[mikeh]

I rarely see deeply enough into these hands to draw the 'correct' inferences!

 

I would play RHO for the Jxxx for two reasons.

 

One is that my gut tells me, perhaps erroneously, that finding the opps to be 2=4=5=2/7=1=3=2 is less probable a priori than their being 5=1=5=2/4=4=3=2.

 

The other is that I had an alternate line of play available. I could have pitched a heart on the 3rd diamond, which would fail only when RHO had Jxxx in hearts AND LHO held the spade K. Admittedly, if RHO has Jxxx in hearts, the spade hook is only 44.4%, but I still have to be overwhelmingly favourite to make.

 

If my opp took this line, then I have a great chance for a pickup. If he chose my line, then (subject to the impact of point 1) the choice at this juncture looks very close indeed.

[billw55]

hmm, interesting squeeze possibilities. Cash ♠A and all the minors, pitching ♠Q from hand and heart from dummy. This will squeeze either opponent holding ♠K and ♥Jxxx. Loses when LHO holds ♥Jxxx and RHO holds ♠K.

 

Alternatively, pitch the heart from hand and the ♠J from dummy. This also squeezes either opponent, but reverses the losing case. (= mikeh's line I think).

 

So maybe I can start with 4 clubs, and then try to decide which case is more likely, and take my pitches accordingly? eh I don't know.

[fred]

Sorry I did not say so explicitly, but in trying to solve this problem please don't consider other lines you might have taken, the bidding, or any inferences from the opening lead.

 

I did not intend any of these things to be relevant as far as this problem is concerned. Just try to make 7NT given the information I gave you.

 

Fred Gitelman

Bridge Base Inc.

www.bridgebase.com

[billw55]

In that case, I guess I just play RHO for four hearts. LHO has a little less room since he held five diamonds. Not much to go on but better than nothing I guess.

[gnasher]

Against perfect defenders, I can't infer anything statistical from the information that diamonds are 5-3. They should treat all the diamonds and small spades as a single suit, and so should I. At this stage all I know is that neither player had four hearts and ♠K.

 

It's a guess.

 

Edit: Or maybe it's not.

I have one other piece of information: RHO must throw ♠K some of the time when he has a singleton heart - otherwise I'd know to pick off his Jxxx when he holds that. LHO, on the other hand, will never throw ♠K holding a singleton heart. Can I use this somehow?

Edited January 31, 2012 by gnasher

[bluecalm]

Well, assuming that the opponents were told to lead club no matter their hands and that they discard perfectly it's pure guess as LHO will discard in such a way that information he gave me about 5 diamonds won't give away likelihood of him having 4 hearts.

So it looks like perfect 50-50 guess to me with your assumptions about indicated club lead.

[MrAce]

Who has ♠ K ?

 

On the 5th ♣, West can not afford to false card to pretend like he has 4♥ because dummy's ♠ J is discarded after him. However thats not the case for East, if he had stiff ♥ he cld play ♠K after we discard ♠J. He either didnt have it (which makes impossible for west to hold 4♥) or he failed to do it eventhough he is the one who has all remaining ♠. Because if there is a 4-1 ♥ split, in last 3 cards one of them is left with all ♠s and other one with all remaing ♥s.

 

Edit : Nevermind, Andy already said this.

[mikeh]

In the endgame where we are thinking about the significance of the spade K, if East has all the remaining spades....that is, West has jxx in hearts....East either holds or has already seen all the other spades, including, most obviously, the 10 and the 9. I think only a lazy expert....or maybe someone on a bad day....would hold onto the K. I mean it isn't likely that this hand was played at the speed of light.....and one's attention is generally pretty focused when defending a 7N contract where declarer hasn't claimed!

 

I don't think it even remotely necessary that E work out why this is required...it is the sort of automatic falsecard that is second-nature to most in this situation.

 

However, I confess I missed this aspect of the problem....kudos to Andy and MrAce for noting it.

 

There has to be some value to the inference that the spade K has not yet appeared, which shifts the odds in the heart suit. The a priori odds of the initial distributions appear (to me) to slightly favour playing E for the heart length....does this spade K inference negate and override that?

 

I think that depends on who you are defending against and how fast you play the cards. Against me, you'd have so much time to think about the defence that my best hope would be that, having worked out that E must play the spade K whenever he has it, E fell asleep and forgot it.

[phil_20686]

In the endgame where we are thinking about the significance of the spade K, if East has all the remaining spades....that is, West has jxx in hearts....East either holds or has already seen all the other spades, including, most obviously, the 10 and the 9. I think only a lazy expert....or maybe someone on a bad day....would hold onto the K. I mean it isn't likely that this hand was played at the speed of light.....and one's attention is generally pretty focused when defending a 7N contract where declarer hasn't claimed!

 

I don't think it even remotely necessary that E work out why this is required...it is the sort of automatic falsecard that is second-nature to most in this situation.

 

However, I confess I missed this aspect of the problem....kudos to Andy and MrAce for noting it.

 

There has to be some value to the inference that the spade K has not yet appeared, which shifts the odds in the heart suit. The a priori odds of the initial distributions appear (to me) to slightly favour playing E for the heart length....does this spade K inference negate and override that?

 

I think that depends on who you are defending against and how fast you play the cards. Against me, you'd have so much time to think about the defence that my best hope would be that, having worked out that E must play the spade K whenever he has it, E fell asleep and forgot it.

 

Yes but like all false cards you shouldnt make it all the time. If its automatic enough for us to be drawing inferences from it they should only be doing it sometimes... :)

[han]

I'm going to answer as if Fred intented this as a purely theoretical question. Not only can we not deduce anything from the lead and signals of the opponents, the opponents may also discard optimally given the way we plan to play the hand.

 

Under these assumptions, we should be careful not to fall into the trap of playing RHO for the jack of hearts because LHO has discarded 2 diamonds and ♠xx ♥Jxxx ♦xxxxx ♣xx is a much less likely hand for LHO than ♠Kxxxx ♥x ♦xxxxx ♣xx (which is true though). If we start to reason like that, LHO could discard a spade and a diamond from the second hand, and what would we do then? Or RHO could discard the spade king and we'd have a different problem. We'd have to analyse all possible discards and determine what we would do and how we would fair for all of the following west hands:

 

 

I (8/25)

 

xx

Jxxx

xxxxx

xx

 

II (4/5)

 

Kxxxx

x

xxxxx

xx

 

III (4/5)

 

xxxxx

x

xxxxx

xx

 

IV (4/5)

 

xxx

Jxxx

xxxx

xx

 

V (4/5)

 

xxxx

Jxxx

xxx

xx

 

VI (2/5)

 

xxxxxx

x

xxxx

xx

 

VII (4/5)

 

Kxxxxx

x

xxxx

xx

 

VIII (8/25)

 

xxxxxxx

x

xxx

xx

 

IX (16/175)

 

Kxxxxxx

x

xxx

xx

 

It looks like these are all possible hands, with relative probabilities in parentheses. Note that LHO cannot have Jxxx of hearts and the spade king, as he would have been squeezed. This is the reason that there are 6 hand types where RHO has Jxxx of hearts but only 3 hand types where LHO has Jxxx.

 

So this is what I think is the solution: we should play RHO for Jxxx of hearts. I suspect we should play this way regardless of what their discards are. This way we will win on hands 2, 3, 6, 7, 8 and 9, which makes for about 58% of the relavant hands.

 

(I'll likely correct my spelling and computations after posting)

[han]

The other is that I had an alternate line of play available. I could have pitched a heart on the 3rd diamond, which would fail only when RHO had Jxxx in hearts AND LHO held the spade K.

 

Not that I want to go into whether Fred's line is correct or not, but if the answer I gave in the above post is correct then Fred goes down only when LHO has Jxxx of hearts and RHO the spade K.

[TWO4BRIDGE]

RHO has no choice on his 3 Sp discards whether he is 4 4 3 2 or 7 1 3 2 .

 

So, the answer must lie in the "order" of LHO 3 discards -- which Fred said were the 2 ( good ) Diam and a small Sp.

 

If LHO were 5 1 5 2, would he discard BOTH ( good ) Diam 1st ... then another small Sp ?

 

If LHO is 2 4 5 2 ( with 2 small Sp ) I think he might just discard the 2 ( good ) Diam first .

[mikeh]

Yes but like all false cards you shouldnt make it all the time. If its automatic enough for us to be drawing inferences from it they should only be doing it sometimes... :)

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.

[MrAce]

deleted

[benlessard]

I think the quick way to see it is that if west has 4H the first D discard is insignificant but the 2nd diamond discard is significant since he cannot discard the Ks from Kx.

 

Therefore we have to assume that some of the time west has 4D and east had 3D. Wich mean that east slightly more likely to have 4H than west. (if the J of S is replaced by a low S its even)

 

But i rarely get these thing right so im probably wrong

[ron_ron]

If east started with a singleton heart and the SK, he must play the SK some fraction of the time (4/7, I believe) to balance out the times he is forced to play the SK (when he started with 4 hearts). If he plays the SK more often or less often than 4/7 of the time, declarer has a strategy that does better than break even. Declarer should therefore play east for Jxxx when no SK appears. If east never plays the SK when holding stiff H, declarer has a straight guess, so he will never do worse than break even (no matter east's strategy), and will do better than break even when east plays optimally (or in fact if east plays the SK some nonzero percentage of the time when he started with a stiff heart).

[mike777]

one email response

 

"The auction and the actual cards played are relevant here. What were LHO's clubs, what spades did LHO play? Since LHO is 2-4-5-2 if he has 4 hearts, knowing how attractive a spade versus a club lead was based on the auction and his black suit holdings is quite relevant (e.g. with xx Jxxx Txxxx Tx hard to imagine anyone leading a club, but from T9, or perhaps 98 and with Tx of spades, maybe (also did RHO get a chance to bid with his presumed Kxxxxxx of spades on that hand)?

 

 

Also, of course how good LHO is matters (the diamond discards are forced from the 2-4-5-2 hand but could be made from the 5-1-5-2 hand to trick you)."

[ron_ron]

If east started with a singleton heart and the SK, he must play the SK some fraction of the time (4/7, I believe) to balance out the times he is forced to play the SK (when he started with 4 hearts). If he plays the SK more often or less often than 4/7 of the time, declarer has a strategy that does better than break even. Declarer should therefore play east for Jxxx when no SK appears. If east never plays the SK when holding stiff H, declarer has a straight guess, so he will never do worse than break even (no matter east's strategy), and will do better than break even when east plays optimally (or in fact if east plays the SK some nonzero percentage of the time when he started with a stiff heart).

 

Actually, I think part of the above is not quite right; east can play the K anywhere from 0 to 4/7 of the time. The conclusion is still correct, though, I think.

[mike777]

Let me back up is this a math question where the issue is who has the best PHD in Math or is this a bridge question for exp bridge players?

 

 

 

Actually bridge is a timed event but in this thread no one seems to be aware there are penalties for taking all this time.

 

 

--

 

 

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.

 

In other words is this a bridge issue or a math issue where we can take hours?

[han]

My impression is that Fred intented this as a theoretical issue only. Looking at it again, it seems to me that I gave the right answer yesterday, although my numbers may have been off. If I find some time today I'll try to write a proof to show that no strategy can do better than the one I proposed (always playing RHO for the heart length no matter the discards), although I fear that such a proof will not be convincing to everybody.

[ron_ron]

Whoops. I miscounted the vacant spaces. Replace 4/7 by 3/6...

[bluecalm]

I'll try to write a proof to show that no strategy can do better than the one I proposed (always playing RHO for the heart length no matter the discards), although I fear that such a proof will not be convincing to everybody.

 

Please do.

I am still not convinced but more from not getting it than from having counter argument.