[fuburules3]

Perhaps I'm seeing complexity where there is none, but anyway . . .

Suppose I have
AJ9 ♠ in dummy opposite

xxx

in a no trump contract with plenty of entries. It is well-known that the best way to play this for two tricks is low to the nine and then low to the jack. I am interested in how I should change my initial plan based on what card LHO plays. A secondary question is what is the optimal strategy for LHO.

Assume for simplicity that LHO has exactly 3 spades and that he has a holding where my play matters (either K10x,Q10x,KQx). There are 8 (equal probability ways) to be dealt (K/Q)10x and 4 (equal probability ways) to be dealt KQx.

If he will always play an honor from KQx and never from (K/Q)10x then it is clear that if I see an honor on the first spade play, I should win the A and lead towards the jack and if I don't see an honor I should finesse the 9. On the other hand, if LHO will never play an honor from KQx and always from (K/Q)10x then I should adopt the opposite strategy. So how I play this combination when LHO plays an honor should definitely depend on how I think LHO will play his cards.

Assume LHO adopts a strategy where he will play an honor from KQx A% of the time and an honor from K10x B%. So (still assuming LHO has a holding that matters) I will see an honor

(A/3+2B/3)%

and when I see an honor there is

(A/3)/(A/3+2B/3)

chance it will be from KQx and

(2B/3)/(A/3+2B/3)

change it will be from (K/Q)10x. This seems to suggest that if I see an honor, I should play LHO for KQx if 2A>B and for (K/Q)10x otherwise.

This analysis does not even get into what I should do if I see a spot card or what the optimal strategy for LHO is. It would not be hard to do this, but these sort of calculations are not reasonable to do at the table and in the real world things are much more complicated.

My question I guess is if I faced this (or a similar) situation at the table, how can I decide what to do in a reasonable amount of time? My best guess is that when LHO adopts his optimal strategy in a situation like this, then my strategy will frequently be to just go with the a priori odds (and if he deviates from his optimal strategy I can gain an edge if I notice).

[CSGibson]

My best guess is that you will not play that combination against an opponent often enough to notice his deviation, and that even if you do so, an opponent will notice that you've played differently (guessing right) and change his strategy mix as a result.

[Statto]

If LHO produces the King or Queen, unless I've missed something, I don't see what you could gain by not covering with the Ace. The decision will come on the next round, whether to play 9 or J. SuitPlay says the 9, though my instinct would be the J, expecting it to be much more normal to have played the honour from KQx than K10x/Q10x, but maybe that's not the case. It probably depends quite a bit on the hand in question. Your post illustrates restricted choice in a more complex setting

[cherdano]

The optimal strategy is to play LHO for HTx; here "optimal" means that it cannot be exploited.
Against weak opponents you should of course play them for KQx if they play an honor on the first rounds. Against good opponents who think you are weak you might want to play them for KQx if they play low on the first round.

[Phil]

I find that the falsecard of HTx occurs more in books than it does at the table.

[mikeh]

I'm pretty sure that this combination was the subject of one of the occasional articles in the BW talking about optimal strategies, taking into account the 'correct' defensive and offensive plays, with the usual result being that it pays defender to adopt a mixed strategy.

Of course, a mixed strategy, to be of maximal effect, requires that one plays against declarer enough that these situations arise so often that declarer will recognize your tendencies.

I agree with Phil in that, in real life, only a handful of defenders will insert the H from H10x.

If a meckwell rose with the Q, for example, I'd probably play him for Q10x since I'd assume that he would think that there was at least a chance that I didn't know this tactic

But if one of the club regulars did it, I'd assume either a stiff or KQ tight

[lalldonn]

At the table, it's easier to play the Q from QTx than the K from KTx since in the first case he is surely finessing anyway but in the second he is likely not playing the 9, such as holding Qxxx. So it's not true restricted choice when they play Q or K IMO.

[Phil]

With KTx, the King could definitely be right. Declarer is leading the Q with Qxx, although probably not Qxxx.

With QTx, even Meckwell would probably play low and then save the creative play for the next round and drop the Q to encourage a finesse when declarer holds K8xx. This is also interesting because shouldn't the defender be playing the Q from some Qx combos?

KQx is certainly the most interesting holding from the defense. Low seems obvious. I had this identical situation a few years ago, and a really good defender played small without a hitch. Of course I 'misguessed' and the guy had QTx and just sort of gave me a funny look. Level two is to play an honor, and we get into the roshambo game.

None of this discussion is about when the defense has KTxx or QTxx. In either of these cases, playing the honor is outright silly. Unless you had a good count on the defender's shape and could reliably determine how many cards they had in the suit, trying to develop a mixed strategy when a defender holds precisely KQx, KTx or QTx seems like a difficult proposition because most of the time their play is forced.

So if LHO plays the K, I'm more likely to play him for KTx instead of KQx. When LHO plays small, it still looks right to play the 9 which caters to QTxx(xxx) or KTxx(xxx). With QTx I think the defender is seldom rising, and KTx he often is, but not always. The J only works with KQx(xxxx) onside.

Some of this feels like gibberish to me. Maybe this combo is more interesting than I thought

[lalldonn]

The Q from Qx wouldn't be so bright if declarer had a two-way finesse! I agree this combo is interesting despite appearing simple, and there are few guarantees as to what the best play ever is unless you know declarer's exact holding.

[aguahombre]

cherdano, on 2012-April-18, 21:41, said:

The optimal strategy is to play LHO for HTx; here "optimal" means that it cannot be exploited.
Against weak opponents you should of course play them for KQx if they play an honor on the first rounds. Against good opponents who think you are weak you might want to play them for KQx if they play low on the first round.

This leads to the optimum strategy for a good player, who knows you know he is a good player. You will assume he would play an honor if holding the ten; so, his strategy should be to play an honor with both KQX and HTX. You know he would only play low on the first lead with none or both of the honors and playing the Jack cannot lose, so he must play an honor.

However, if lefty is an unknown, Declarer just sticks with the original plan on the second round. Lefty might play the other honor...perforce or by mistake...if he has it.

Phil, on 2012-April-19, 09:40, said:

I find that the falsecard of HTx occurs more in books than it does at the table.

This observation is valid because, at the table we encounter good players less frequently. I don't find that to be the case at higher levels.

[gnasher]

Phil, on 2012-April-19, 11:11, said:

With KTx, the King could definitely be right. Declarer is leading the Q with Qxx, although probably not Qxxx.

With Qxx opposite AJ9, there are several reasons for starting with a low one:
- It loses only to singleton ten offside, but gains against singleton king onside, so the two plays are equally good in theory.
- You may play against LHO again tomorrow, and next time you might have xxx opposite AJ9
- If he doesn't play the king, you might infer that he doesn't have K10x, so you might finesse the jack and then look elsewhere for another trick, or try to endplay him.

[Phil]

gnasher, on 2012-April-19, 15:38, said:

With Qxx opposite AJ9, there are several reasons for starting with a low one

Low to the 9, and low to the Jack usually doesn't work so hot when RHO has KT. Low to Jack is inferior to playing for Kx onside as well.

Is that what you are saying?

[Fluffy]

Please han don´t tell us the same story again (ok, its a good one, else I wouldnñt remember)

[gnasher]

Phil, on 2012-April-19, 16:34, said:

Low to the 9, and low to the Jack usually doesn't work so hot when RHO has KT. Low to Jack is inferior to playing for Kx onside as well.

Is that what you are saying?

I was assuming, for no particular reason, that we knew we needed three tricks, so yes I was suggesting low to the nine.

[nigel_k]

I think there are two kinds of opponents:

1) the kind against whom you take your best chance and don't try to get into mind games
2) the kind who never play high from HTx

So decide which kind of opponent LHO is and play accordingly. No need to get into the percentage stuff.