[jvage]

The discussion about Fred's problem made me think about this classical suit-combination:

QJ97

A865

You need 4 tricks, entries no problem, optimal defense (?) and no outside info.

You start by leading the Q from Dummy, how do you play if;

a: RHO covers with the K and LHO follows small when you win the A and play another one.

b: The Q holds, both opponents following small.

John

[whereagles]

Book play is finesse the 10, I believe...

[BillHiggin]

There are 16 possible holdings where the K is onside (not all conform to all the conditions):
RHO holds
Stiff K
KT
Kx (*3)
KTx (*3)
Kxx (*3)
KTxx (*3)
Kxxx
KTxxx

The last two do not conform to LHO following suit with a small card so can be eliminated. The first two are not possible for case b. The possibilities are not quite equal, the 4-1 splits each occur 293,930 times to 352,716 for each 3-2 split. You are dead to RHO holding KTx or KTxx (cannot win).

With no considerations for RHO strategy (covers or not randomly):
a: Take the back finesse - wins when RHO held K, Kx or Kxx - loses only to KT - not even close (about 10:1)

b: Choices are to drop the K (from Kx) or smother the ten (RHO having Kxx). 50-50.

So, what is RHO's optimum strategy? Do not cover except with KT doubleton or stiff K!
This changes case a to always play for the ten to drop (slight favorite for 3-2 over the 4-1 split). If no cover, you are still on a 50-50 guess.

edited to fix d*$# smilie glitch

[ArtK78]

BillHiggin, on Oct 26 2007, 09:03 AM, said:

So, what is RHO's optimum strategy? Do not cover except with KT doubleton or stiff K!
This changes case a to always play for the ten to drop (slight favorite for 3-2 over the 4-1 split). If no cover, you are still on a 50-50 guess.

If RHO's optimum strategy is to never cover except with KT doubleton or singleton K, and that makes playing for KT doubleton declarer's optimum strategy whenever the Q is covered, then the first statement is not true. While I am no game theory maven, I know that RHO's optimum strategy would be to cover with Kx some of the time so that you cannot rely on the cover being from KT or singleton K.

[BillHiggin]

ArtK78, on Oct 26 2007, 09:25 AM, said:

BillHiggin, on Oct 26 2007, 09:03 AM, said:

So, what is RHO's optimum strategy? Do not cover except with KT doubleton or stiff K!
This changes case a to always play for the ten to drop (slight favorite for 3-2 over the 4-1 split). If no cover, you are still on a 50-50 guess.

If RHO's optimum strategy is to never cover except with KT doubleton or singleton K, and that makes playing for KT doubleton declarer's optimum strategy whenever the Q is covered, then the first statement is not true. While I am no game theory maven, I know that RHO's optimum strategy would be to cover with Kx some of the time so that you cannot rely on the cover being from KT or singleton K.

Should declarer switch strategy becuase he thinks RHO will only cover with stiff K or doubleton KT? Only if 100% sure of that strategy. The back finesse is only a 6:7 underdog in that case, and is nearly a 10:1 favorite with random covering. If declarer has any doubts at all, he will back finesse.

So what percentage of opportunities should defender cover with when he has a realistic option? None! Every time he covers with Kx or Kxx, he is commiting suicide.

Declarer's strategy switch is questionable (and in a practical sense, unwise).
Defender's strategy is clear. Cover only when you must (things change if the 9 is not visible).

[TimG]

whereagles, on Oct 26 2007, 08:46 AM, said:

Book play is finesse the 10, I believe...

Little nit. Don't you finesse against the Ten? Or, finesse the Nine?

[FrancesHinden]

ArtK78, on Oct 26 2007, 03:25 PM, said:

BillHiggin, on Oct 26 2007, 09:03 AM, said:

So, what is RHO's optimum strategy? Do not cover except with KT doubleton or stiff K!
This changes case a to always play for the ten to drop (slight favorite for 3-2 over the 4-1 split). If no cover, you are still on a 50-50 guess.

If RHO's optimum strategy is to never cover except with KT doubleton or singleton K, and that makes playing for KT doubleton declarer's optimum strategy whenever the Q is covered, then the first statement is not true. While I am no game theory maven, I know that RHO's optimum strategy would be to cover with Kx some of the time so that you cannot rely on the cover being from KT or singleton K.

No, a strategy to cover none of the time with Kx is perfectly good, as also is one to cover some percentage of the time. You can see this with a slightly easier example, in which you can ignore 4-1 breaks.

Take

QJ9

opposite

A87xx.

You lead the queen, RHO covers with the King and you win with the ace. The only holdings that matter now are Kx and K10 doubleton on your right.

RHO will always cover with K10
Suppose RHO covers with Kx y% of the time.

Then finessing on the way back is right as long as

y% * chance of Kx > chance of K10

As Kx is 3 times as likely as K10, RHO just needs to keep y to 1/3 or lower. That is, if he covers more than 1/3 of the time you have an edge; but anything between never and 1/3 is fine.

Of the 3-2 holdings, there are 7 relevant ones - RHO having Kx (3), K10 (1), Kxx (3).

- If he only EVER covers with K10 doubleton (y = 0) you will get the suit right every time he covers (1/7) but you are a complete guess as to whether he has Kx or Kxx when he doesn't cover (50% of getting it right 6/7 of the time) giving you a total chance of 4/7 of bringing the suit in whenever it can be.

- If he covers with Kx 1/3 of the time, you will lose into K10 doubleton whenever he is dealt it but you have an edge when he doesn't cover, as he is now more likely to have Kxx rather than Kx - you will lead the Jack next, and still bring the suit in 4/7 of the time (all the Kxx's and the 1/3 of the Kx's when he covered).

Your chance of bringing in the suit is exactly the same whether he covers 0%, 10%... or 33% - it's just that the holdings you win on are different.

However, if he covers (say) all the time you are better off as now you get the suit right everytime he has either Kx or Kxx, or 6/7 of the time, losing only to K10 doubleton.

The original 4-4 example is more complicated as you have to include the singleton K holdings. I think he should cover anything up to about 11% of the time, but that was a fairly quick calculation.

[pclayton]

When RHO covers, RHO will always cover with stiff K or K-T doubleton. K-T-x doesn't matter. What if RHO has K-x? More on that later.

When RHO doesn't cover, we have to decide if RHO has K-x or LHO has T-x.

That's the basics. Since this is drifting into conditional probabilities, I will leave this for the math people to sort out.

[whereagles]

TimG, on Oct 26 2007, 03:09 PM, said:

whereagles, on Oct 26 2007, 08:46 AM, said:

Book play is finesse the 10, I believe...

Little nit. Don't you finesse against the Ten? Or, finesse the Nine?

Well, where I live "finesse against the ten" literally translates to "finesse the ten"

[Halo]

1 Finesse against the ten

2 Play the Jack.

[BillHiggin]

Back to "optimum"

For case b, declarer strategy mix does not change his success rate, it is 50-50. If he choses to always try to pin the 10, he will pick off Kxx on his right regardless of defensive strategy. For case a, declarer can choose to always try to drop the 10 and his odds will be 1.41:1.35 favorable regardless of defensive strategy (since declarer is conceding to Kx, no mix can help the defense). Declarer wins for Kxx or KT on right, loses to stiff K or Kx on right (the fractional advantage comes because stiff K is slightly rarer than doubleton KT).

If declarer suspects that the defense is covering from Kx some portion (say z%) of the time, it might be advantageous to switch to back finesse for case a. There will never be an advantage to switching strategy for case b. If z=0, his odds will drop to 1.35:1.41 (he is now winning against stiff K and losing to KT). For non zero z, we will switch to long numbers (more digits = author is smarter ).
The odds for back finesse-pin become:
(1,352,078 + z*1,058,148) : (1,410,864 - z*1,058,148)
If z is greater than 5 out of 90 opportunities (just over half of edit:Helene's Frances' guess), declarer will be doing better than sticking with drop-pin.

The best the defense can possibly hope for is to actually cover less often than 5 of 90 and hope declarer believes they are covering more often. The original post specified "optimal". That would be cover only with stiff K or doubleton KT and settle for being a slight dog to optimum declarer strategy on the cases that matter.

In practical situations, nobody will have perfect information. Then declarer should take the slight hit against perfect defense by switching to back finesse-pin and gain when the defense covers a bit too often from Kx.

Always try to pin the 10 when the first finesse wins. At least the defense cannot attempt to get inside your head.

[dburn]

jvage, on Oct 26 2007, 07:49 AM, said:

QJ97

A865

You need 4 tricks, entries no problem, optimal defense (?) and no outside info.

You start by leading the Q from Dummy, how do you play if;

a: RHO covers with the K and LHO follows small when you win the A and play another one.

b: The Q holds, both opponents following small.

John

I had read this question and answered "play the jack on the second round" before realising that there was a part b. A moment's reflection convinced me that this did not matter.

[Halo]

This is very much a bidding forum. So I will ask dburn - why play the Jack on the way back round when the Queen is covered? I can see no reason to do that.

[Stephen Tu]

Quote

So I will ask dburn - why play the Jack on the way back round when the Queen is covered? I can see no reason to do that.

Read Frances' post carefully. The idea is you are supposed to pick up Kxx onside & KT doubleton onside. When the Q is covered, RHO has either Kx or Kt doubleton. But a good defender is supposed to pursue a strategy of not covering with Kx, at least not more than 1/3 of the time, since if he ducks you are likely to get it wrong anyway. So if he covers, it is more likely that he started with KT than that he started with Kx.

If you are playing a poor defender who covers the first honor with Kx way too often, then you can exploit him by picking up both Kxx & the Kx where he covers, which would be greater than the frequency of Kxx + KT, if he covers more than 1/3 of the time.

[Halo]

Does this even work a priori Stephen - against anyone whose eyes are half closed and they cannot see whether they have Kx or K10?

[Stephen Tu]

Quote

  Does this even work a priori Stephen - against anyone whose eyes are half closed and they cannot see whether they have Kx or K10?

That's not a well specified question. Is this a good player with a vision problem who can't distinguish between spot cards but can see the K (so I guess would play low), or a bad player with a vision problem who covers all the time? Or someone who can't see either card and thus plays totally randomly? Or someone who knows enough to cover from KT, but plays totally randomly from Kx not knowing best strategy?

If (n% cover from Kx > 33.33%), then hook on the way back, otherwise play J. Decide what you think n is based on your knowledge of the defender & play accordingly.

[dburn]

Halo, on Oct 27 2007, 02:43 PM, said:

I will ask dburn - why play the Jack on the way back round when the Queen is covered? I can see no reason to do that.

It's an example of the General Principle of Restricted Choice: assume that someone did something because he had to, not because he chose to. RHO must cover with K10; he may choose to cover with Kx, but he may not - indeed, he should not (or at least, not often enough to unbalance John Forbes Nash).

We adopt the shorthand notation:

pin-finesse for this strategy: if the queen is not covered, try to pin the ten; if the queen is covered, finesse against the ten

drop-finesse for this strategy: if the queen is not covered, try to drop the king; if the queen is covered, finesse against the ten.

drop-drop for this strategy: if the queen is not covered, try to drop the king; if the queen is covered, try to drop the ten.

and now be very quiet, for you are about to hear a

pin-drop for this strategy: if the queen is not covered, try to pin the ten; if the queen is covered, try to drop the ten.

An optimal defender should follow this strategy: never ("What - never?" "Well, hardly ever") cover except with K10 or K. Trust me on this for the moment - I will clarify later, when you can give one cheer more for the hardy captain of the Pinafore.

Now, in the relevant cases:

pin-finesse succeeds against an East holding of Kxx (three 3-2 breaks) and singleton K (one 4-1 break).

drop-finesse succeeds against an East holding of Kx (three 3-2 breaks) and singleton K (one 4-1 break).

drop-drop succeeds against an East holding of Kx (three 3-2 breaks) and K10 (one 3-2 break).

pin-drop succeeds against an East holding of Kxx (three 3-2 breaks) and K10 (one 3-2 break).

Since a 3-2 break is more likely than a 4-1 break, it is clear that drop-drop and pin-drop are better than the other two strategies, and that they are equivalent for present purposes. At the table, you should follow drop-drop because this minimises undertricks against K10xx in either hand, but here we are not concerned with such trivia - we are used to being in grand slams with this trump suit by now.

I hope that Halo has been following this, because I can sense this question lurking in the back or even the front of his mind: "if declarer is going to follow X-drop, I can beat him by covering from Kx (or K), so isn't that optimal defence? What does dburn mean by saying that an optimal defender should cover only from K10 or K?"

Welcome back to my table, Halo my sub-optimal friend. Was it only last week that you covered from Kx? Did you beat me then? I guess you did. And yesterday - well, I had my suspicions, but I lost a coin-flip. Today, though, you have no chance at all unless you happen to have been dealt K10, because against you I have switched to pin-finesse, so I'm going to beat you now and for evermore when you have Kx. And Kxx too (six of the 3-2 breaks in total) - oh, and singleton king. Though "bother it" you may occasionally say, never - never - use a big big D.

[Halo]

OK

If you know the player (and therefore his strategy) you are clearly correct.

[BillHiggin]

Halo, on Oct 28 2007, 05:10 AM, said:

OK

If you know the player (and therefore his strategy) you are clearly correct.

If you do not know the player then:

1) You can use pin-drop which always wins against KT or Kxxx and always loses to K and Kx without regard to defensive strategy.

2) Or you can assume RHO covers too often (since the actual break even point is 5.5% rather than 33%, this is almost always a correct assumption) and use pin-finesse. Then you still win against Kxx every time. But now you win against Kx when they cover and stiff K, always losing to KT and losing to Kx when they do not cover. Since stiff K compared to KT is almost but not quite even (similar to quart:liter), it does not take many mistaken covers from Kx to put declarer ahead of 1). This declarer strategy is very slighly worse than pin-drop against defenders who never cover with Kx, so declarer does not risk much by making this assumption.

The mistake some declarers will make is to try drop-drop or drop-finesse. Trying to drop the K when there is no cover loses to Kxx (where covering cannot work for the defense) and only picks up Kx when defender has not covered. Essentially declarer is conceding a sure win situation (Kxx) for an equal probability maybe win situation (Kx no cover). The best declarer can do is break even when the defense never covers from Kx with this (non) option.