[fred]

J108

K65432

You need 5 tricks. Entries are not a problem. Nor are extra undertricks.

You lead the Jack from dummy and East follows with a small card.

If you chose "sometimes", please explain how to determine when it is appropriate to play the King.

By "appropriate" I mean "the percentage play"

If you feel inclined to ruin the problem for yourself by asking a computer program, please do not ruin the problem for other people by posting what your computer program has to say.

Fred Gitelman
Bridge Base Inc.
www.bridgebase.com

[ArtK78]

I voted "sometimes."

Whenever the suit breaks 2-2, it is a pure guess whether to play the King or duck UNLESS (1) the auction has given you an indication of who holds the ace, or (2) the lie of the rest of the hand requires that you place the ace in one hand or the other. In the latter case, your play is dictated by that restriction.

If the suit is 3-1, then, once you lead the J from dummy, the only cases that matters is singleton 9 in LHO's hand. If RHO has A9x or Q9x, you must lose 2 tricks. If you were playing for a 3-1 break, you would not have led the J from dummy unless you knew that the AQ was on your right, as you would then be forced to play for a singleton 9 on your left after RHO follows small.

4-0 breaks are irrelevant, as you cannot win 5 tricks in the suit if the suit breaks 4-0.

[matmat]

hmmm... are all 2-2 really equivalent?

what's the "proper" defensive play holding Q9 with J coming off of dummy?

[kenrexford]

"A small card" seems to mean a specific card, because the 9 is not contextually small. So...

Missing: A/Q/9/x
Seen: x
Possible RHO holdings:

x
9x (your play does not matter)
Qx (guess)
Ax (guess)
AQx (riding works best)
A9x (your play does not matter)
Q9x (rising costs all three instead of two)
AQ9x (rising costs three instead of two)

On its face, it appears that you must finesse. However, you also clarified that you need five. On the last three, you lose two tricks no matter what. So, the remainder will be:

Qx
Ax
AQx

Slight lean toward riding. The only caveat seems to be when RHO cannot have four points or when an assumption must be made as to another card(s) that will reduce RHO's max to four points.

--------------------------------------------

So, what if "a small card" includes the 9?

The play of the 9 makes little sense if it accompanies the "x". So, it seems to have to be from one of the following:

9
Q9
A9
AQ9

With AQ9, a cover would be 100%. So, we eliminate that.

With the stiff 9, LHO has AQx. We have to lose two tricks. Needing five, this layout is irrelevant.

With A9, RHO will be forced to duck, making the rise better.

With Q9, RHO could duck and hope for a rise. Or, he could cover and hope for a finesse.

If the Jack is covered with the Queen, Opener losing his King to the Ace will have a "guess" at the next play. He will see a small card from LHO and will have to decide if the Queen was from Q9 or stiff Queen. That seems 50-50 to a degree, except that RHO will duck with Q9 to gain the advantage of the forced duck from A9 tight. So, Opener's odds will be to hook. But, if the odds are to hook, then RHO can also cover and win. So, RHO always wins with Q9. Because RHO always wins, we give up that scenario as a no-win for declarer.

This means that if "a small card" can include the 9, we have:

A9 (rise)
Qx (duck)
Ax (rise)
AQx (duck)

So, I think that you rise if the "small card" is the 9 but duck if it is the true small card. If RHO cannot have the Ace, mathematically or for an assumption, then you duck.

[han]

Now this post I think is funny.

It will take quite some effort to study mixed strategies, let me consider deterministic strategies first.

Strategy 1. Play low when 7 or 9 is played, play for the drop on the second round when the queen is played. Repeat the finnese when the ace is played and the 9 drops on the left.

Wins when RHO has Q7, Q9 or AQ7.

Strategy 2. Play the king when the 7 or 9 is played, finesse on the second round when the queen is played, play for the drop when the ace is played and the 9 appears on the left.

Wins when RHO has A7 A9 or Q.

Strategy 3. Play the king when the 9 is played but low when the 7 is played. When the queen is played finesse and when the ace is played play for the drop.

Wins when RHO has A9, Q7 or Q.

So these three strategies are all equal, and I don't think any other strategy is better. Maybe a mixed strategy is better but I'll look at that later.

[fred]

I seem to have blown it once again.

"Small card" means 9 or 7.

There are no inferences to be drawn from either the bidding or play. You know absolutely nothing about the EW hands.

You can assume the defenders will always do the right thing.

Fred Gitelman
Bridge Base Inc.
www.bridgebase.com

[Echognome]

Thinking through the holdings and not the percentages, we are missing A, Q, 9, and 7.

Here are the possible holdings for RHO:

---
A, Q, 9, 7
AQ, A9, A7, Q9, Q7, 97
AQ9, AQ7, A97, Q97
AQ97

We can rule out several possibilities as we need 5 tricks. We can never make with AQ sitting on the left or if there is stiff A or AQ9 or AQ97 on our right. This leaves us with:

Q, Q7, A7, Q9, A9, AQ, Q97, A97, AQ7

So now we can consider several lines:

1) Lead the 8 from dummy, cover the Q with the King, duck the A (obv) and there are two alternatives if you see the 7 or 9. Already we pick up Q and AQ.
a. Play low - to pick up Q97 in addition
b. Play the K - to pick up A7, A9, or A97 in addition

So it seems that 1b. is better and picks up Q, A7, A9, AQ, and A97

2) Lead the J (or T) from dummy. Here there are various alternatives. If righty plays the A we can play him for AQ7, AQ, A9, or A7. We don't need to worry about A97 because we will see the Q. In the latter 3 cases, we play the K next. In the first case, we will see the 9 and have a choice of finessing the Q. However, this seems inferior as there is only the one case and LHO will also play the 9 from Q9. If RHO plays the 7 or 9, then we can either run the Jack or rise with the King.
a. Run the J - picks up Q7, Q9, or AQ7
b. Rise with the K - picks up A7 or A9

So it seems that 2a. is better than 2b. The difference being that we waste an honour and lose out to A97 by leading the J initially. So let's consider 2a. as part of our overall strategy.

Now if Righty plays the Q, we could duck hoping for Q97, but covering will pick up more possible holdings (as will be shown). So we cover with the K and it will either lose to the A and then we can either:
c. Finesse the 8 on the way back if we see the 7 - to pick up Q along with Q7 and AQ7
d. Play the J if we see the 7 - to pick up Q9 along with AQ7

So it seems that 2c. is better than 2d. so we should add that to our strategy.

Finally, what if the K wins? Then we will either see lefty play the 7 or the 9. If lefty play the 9, then T and 8 are equals and it's irrelevant. If lefty plays the 7, then we lead back towards the J8 and lefty will have to either play the 9 or the A (in any case that matters) and we can always get that right. So basically LHO never gains to duck and RHO should never play the Q from AQ7. The fact that we can pick up AQ I guess is also in the mix.

Summing up the 2nd line, I'm going to go with play the J, then if RHO plays the 7 or 9, then run the J. If RHO plays the Q, then I'm going to cover. If it wins, there's nothing to think about. If it loses, then I will next lead towards dummy and finesse the 8. Finally, if RHO plays the A, I will play the K next. I believe this line picks up AQ, A9, A7, Q, and Q7.

So, if we believe the above (and I'm not sure I've done it all correct), then the difference between the lines is that the optimal line 1 picks up A97 and the optimal line 2 picks up Q7. Q7 should be more likely than A97. So I'll go with optimal line 2.

[lamford]

An interesting problem, and one that reminds me of a similar one which I looked at before. Firstly if second hand plays the seven, I think you should play low all the time, as A7 and Q7 doubleton cancel out, and rising loses to AQ7 with second hand, and on A97 you cannot succeed.

If second hand plays the nine, I think you rise all the time. The two relevant holdings are A9 or Q9 doubleton, but with the latter, second hand should cover at least some, but not all, of the time. If the queen appears you are comparing Q9 (6.78%) with Q (6.22%), but as second hand does not cover all of the time with Q9, you finesse on the way back. Basically you succeed against A9, lose to Q9 (whichever he does) and win against Q. In practice it does not matter what he does with Q9; the % success for your best line is the same.

If I am right on the above, it seems similar to K97654 opposite Q3 for four tricks, where I think the right line is running the nine, but rising if second hand plays the 8.

[Echognome]

By the way, i see the error of my ways now.

[benlessard]

To better understand this very complicated problems we have to start with the case when the Q drop is possible. When the Q drop its 3 case

CASE 1    A97 ------- Q
CASE 2    A7 ---------Q9 defense play the 9
CASE 3    A7 ---------Q9 defense play the Q

restricted choice says that finessing for the 9 is best because with Q9 defense has a choice of play. Therefore case 1 is more likely then case 3. If defense never play the Q from Q9 then when the Q drop its always a stiff and declarer will always finesse the 9. If defense always play the Q from Q9 then declarer get no edge because CASE 1 is not more likely then case 3 anymore. However here come the downside

CASE 4    Q7 ---------A9

If defense always cover with Q9 then when then 9 fall (from A9) declarer will always go up with the K. Therefore a possible defense is a mixed strategy of playing Q from Q9 to protect against case 1 but not too often since the payback is when you have A9.

So let see from declarer point of view when the 9 drop now.

CASE 2    A7 ---------Q9 defense play the 9
CASE 4    Q7 ---------A9

restricted choice says that hoping with the K best because with Q9 defense has a choice of play. Therefore case 4 is more likely then case 2. The problem is that perfect defender will also would have made the same reasonning and since case 4 is slightly more likely then case 1 (A97 ------- Q). They will never play the Q from Q9.

Another way to see it is for declarer to duplicate the holding in wich he is failing to disable deception.

A7-------------Q9

by always finessing the 9 if the Q drop and by always playing the K if the 9 appear the deceptive play of the Q from Q9 is disabled. Playing the Q or not make no difference since declarer will failed in both case. So after this duplication the problems become a simple restrictive choice of always finessing for the 9 when the Q drop.


here are some strategies to show why going up with the k and always finessing the 9 is best.



remember the important cases are

A97 -------- Q
A7 ---------Q9
Q7-----------A9




lets say we have 3000 hands
1000 each time 3 case

case 1

A97 -----------Q


case 2
A7 ------ Q9


case 3
Q7 ------A9



strategy number 1 VS defense 1.1

declarer always play the k when the 9 appear and always finesse for the 9 (and perfect computer defense know this)

defense will never cover with Q9.


A97----Q=+1000



A7 --- Q9=-1000



Q7 ---A9=+1000

so + 1000


strategy number 1 VS defense 1.2

declarer always play the k when the 9 appear and always finesse for the 9 (and perfect computer defense know this)

defense will always cover with Q9.


A97----Q=+1000



A7 --- Q9=-1000



Q7 ---A9=+1000

so + 1000




strategy number 2 vs defense 2.1

declarer play the K 50% & but always try to pick up the 9xx if the Q drop

defense will always go up with the Q with Q9

so we got

A97----Q=+1000


A7 --- Q9=-1000


Q7 ---A9= EVEN +500 -500 (since go up 50%)




strategy number 2 vs defense 2.2

declarer play the K 50% & but always try to pick up the 9xx if the Q drop

defense will never go up with the Q with Q9

so we got

A97----Q=+1000


A7 --- Q9= EVEN


Q7 ---A9= EVEN +500 -500 (since go up 50%)


strategy number 3 VS defense 3.1

declarer play the K 50% and never finesse the 9 (defense know will never cover with Q9).

A97----Q= -1000


A7 --- Q9= even since play low 50%


Q7 ---A9= EVEN since play low 50%


strategy number 3 vs defense 3.2

declarer play the K 50% and never finesse the 9 (defense will always cover with Q9).

A97----Q= -1000


A7 --- Q9= +1000


Q7 ---A9= EVEN since play low 50%



Strategy number 4 VS defense 4.1

declarer always play low and never finesse the 9 (defense will never cover with Q9)


A97----Q= -1000


A7 --- Q9= +1000


Q7 ---A9= -1000


Strategy number 4 VS defense 4.1

declarer always play low and never finesse the 9 (defense will always cover with Q9)


A97----Q= -1000


A7 --- Q9= +1000


Q7 ---A9= -1000


Strategy number 5 VS defense 5.1

declarer play high 50% and finesse the 9 50%
defense will never cover with Q from Q9)

A97----Q= even


A7 --- Q9= even


Q7 ---A9= even


Strategy number 5 VS defense 5.2

declarer play high 50% and finesse the 9 50%
defense will always cover with Q from Q9)

A97----Q= even


A7 --- Q9= even


Q7 ---A9= even

[helene_t]

I voted for "sometimes" because I think it is very close and I would in practice do what I "feel" to be right.

But assuming that opps always do the right thing, there must be an answer. Since opps have full information it cannot be a mixed strategy. Could it depend on whether it was the 9 or the 7? If RHO started with AQ9 it cannot be wrong for him to cover but if he started with AQ7 it cannot win to cover. If he can assume that we always do the right thing it may not matter for him in either case, but if he is allowed to hope for a misplay by us, the 9 should deny AQ. Also, with Q97 he should not play the 9 unless he knows we will play the K.

Good grief, after Fred explained what "appropriate" and "small card" mean, I find myself ranting about the meaning of "the right thing"

[Fluffy]

playing the king is never apropiate, you can lose 3 tricks on the suit that way, low to the king is another thing.

[jdeegan]

The correct answer is 'sometimes, it depends" because:

the relevant cases (I don't see how the nine can be a small card since it is promotable into a trick) are for RHO to hold:

a. AQ7 - playing small wins
b. A7 - playing the king wins
c. Q7 - playing small wins

With a) RHO might have played the ace, or at least exhibited a 'tell'
With c) RHO might have played the queen or at least exhibited a 'tell'

If I can rule out a) and c) because RHO always covers, OR at least gives off a 'tell', then the king is automatic.

If RHO is an automaton, then the a priori probabilities apply, and a)+c) is more likely than b. - small would be correct.
Correctly applying probabilities is a lot harder than just memorizing a few tables.

Thanks for a nice problem.

[helene_t]

Fluffy, on Oct 18 2007, 11:57 AM, said:

playing the king is never apropiate, you can lose 3 tricks on the suit that way, low to the king is another thing.

Sounds correct. Even if RHO did not have full information, I don't see how the jack could tempt him to cover when it would be wrong.

[whereagles]

Agree.. if you fetch the singleton Q, you're gonna lose 2 tricks anyway. If the K holds and the suit breaks, you seem to have made the anti-percentage play (which seems to be to run the J).

[drinbrasil]

fred, on Oct 18 2007, 12:44 AM, said:

J108

K65432

You need 5 tricks. Entries are not a problem. Nor are extra undertricks.

You lead the Jack from dummy and East follows with a small card.

making fast response as if i was in one table:
You need 5 tricks, means can loss 1 only, after player in right played x (7 or 9), we only care about:
a ) K or run J is guess
in your left with
Ax
Qx
b ) duck is better in 3 of 5 cases.
but in our right we need care about
AQ7 (look that player cant have AQ9 or we would have played the Q.)
Q9
Q7
A7
A9

so i duck...not know if right, but fast thinking at table would be my play:-)

ps: will read now all other answers to see if i am big wrong :-)

[Halo]

Sometimes.

Run the jack if x = 7, play the King if x = 9.

I think Lamford has given a relevant analysis of reasons.

[benlessard]

Im surprised most dont understand the problem at all.

1 first of all you need 4 tricks. so all case when you are sure to lose 2 trick are pointless. and extra undertricks are also pointless.

2 if RHO played the 7 you have to let run the J because of (9----AQ7)

3 playing LOW (the 8) to the king is inferior because of (9----AQ7)


So the whole point of the problem is

A7-------Q9
Q7-------A9

THESE ARE THE ONLY CASE WHERE A SMALL CARD (the 9) IS PLAYED THAT MATTERS.

i repeat when a 7 is played on 1st round you have to play the J its obvious.


Now we have a restricted choice problem because from Q9 its possible to play the Q or the 9.

http://en.wikipedia.org/wiki/Principle_of_...choice_(bridge)


Why is it possible for defense to play the Q from Q9 ?



Because of this case

CASE (1)   A97 ------- Q.

If the Q pops up on round 1 its could be from

CASE (1)   A97 ------- Q

CASE (3)    A7 ---------Q9 and RHO decided to play the Q

Again we have a restricted choice problem. Since from Q9 the defense could play the 9 or the Q. So when the Q appear on round 1 its more likely the initial position was A97 ----Q and its less likely the initial position was A7----Q9. So declarer best line is to finesse the 9 each time the Q come down on round 1. So With Q9 the defense can play the Q knowing declarer (because of restricted choice) will finesse the 9 and lose.

So what the problem of always playing Q from Q9 if it will always succeed ?.

The problem is that when you play a 9 its now way more likely to be from A9 then from Q9 (because defense will sometimes or always play Q from Q9). So declarer will go up with the king knowing that RHO holding of A9 is more likely then Q9.

So if we put the pertinent cases together we have

CASE (1)   A97 ------- Q
CASE (2)   A7 ---------Q9 defense play the 9
CASE (3)   A7 ---------Q9 defense play the Q
CASE (4)   Q7 -------- A9

All other cases are pointless.


So declarer strategy is

VS a Q on first round

putting the K and finessing the 9 on the 2nd round
putting the K and playing the T on 2nd round hoping RHO had Q9.

VS a 9

Putting the K hoping RHO had A9
Letting the J ride to win if RHO had Q9


So the 4 non-mixed strategies are

Finessing the 9 if a stiff Q appear and putting the K VS the 9 it will win in case 1+4 and will always fail in case 2 &3

Finessing the 9 if a stiff Q appear and letting the J ride VS the 9 will win in case 1+2 and will always fail in case 4

To play for D (2-2) by playing the T on 2nd round if a stiff Q appear and putting the K VS the 9 will win in case 3+4 and will always fail in case 1 & 2

To play for D (2-2) by playing the T on 2nd round if a stiff Q appear and letting the J ride VS the 9 will win in case 2+3 and will always fail in case 1 & 4

Because of restricted choice CASE 1 & 4 are more likely then case 2 & 3 therefore Finessing the 9 if a stiff Q appear and putting the K VS the 9 is better

The worst play is to not finesse the 9 is the Q drop on round 1 and to let the J ride if the 9 is played on round 1.


For the defense since Q7 -----A9 is slightly more probable then A97-----Q the mixed defense should not play the Q from Q9. But against less then excellent declarer putting the Q is probably better.

[Edmunte1]

Benlessard is right. Winning cases (RHO hand), nad their percentages:
1)AQ7 - 6.2%
2)A7 -6.8%
3)Q7 -6.8%
4)A9 -6.8%
5)Q9 and RHO decides to put 9 - 3.4%
6)Q9 and RHO decides to put Q - 3.4%
7)AQ -6-8
8)Q -6.2%

Cases 6-8-9 are irrelevant for our decision at this time. In the other cases when it's important to take the decision now:
-if 7 appears - small wins in 13%(AQ7+Q7) of cases, K wins in 6.8% (A7) of cases
-if 9 appears - small wins in 3.4% (Q9 and RHO decides to put 9 - 3.4%), K wins in 6.8% of cases.

[Codo]

If the opponents always play the smallest card, ducking is slightly better. Most cases cancle out and you win against one more holding on your right. (AQ7).

But good opps won´t be so kind.
Lets assume that they have the perfect count and know that the suit is diveded like it is. In this case, rho will take the queen in a significant numbers of cases from Q9, Q97, but the ace seldom (in reality maybe never) from Ax.
So the prohibility for a good rho to have Q9 is much lower compared to A9, which makes rising a much better choice then it appeard at first sign.


I cannot get the math correct, but I would use my nose and raise sometimes and let it run at other moments.