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I was hoping it was a joke; thank you for confirming. The odd fact is that despite growing up in Palo Alto and now 25 years in the East Bay I don't much know my way around San Francisco. I did attend a conference at the Cow Palace in the 70's and I have noticed that Daly City is not a place that people strive to drop into their conversation.

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What's the motive for lessening the chance of the human defending? Weak players supposedly don't like to defend, so perhaps it's pandering to them. Alternatively, players good enough to enjoy the challenge maybe don't want the infuriating experience of defending with an inexplicable robot partner. Am I close?

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If Ed Davis overtakes the ♦K at trick 1 I have to reconsider. My calculations were predicated on the assumption that overtaking can't be right. Consequently I took a point of view that led to computing conditional probabilities (per edit just added above), but forgot to include a correct description of the numbers I presented. So my numbers are right but wrongly described. (A pendant might call that a distinction without a difference but I avoid being so judgmental except toward others.)

Ed, calculating all possibilities from trick 1 as you did is a lot more complicated that my calculations which assumed the first 2.5 tricks given (as ♦K, ♠A, ♠Q & the ♠4,5,7 have now appeared). My guess is that you are right but confirmation would be more work than I'm up for.

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If we're thinking about risking a lot to make 7 because it's matchpoints we also have to consider matchpoint field odds. It's convenient on paper to pretend that 100% of everyone bids this easy 6♠, but in reality out of a large field some pair somewhere missed slam altogether (making the conservative line for 6♠ more attractive) and also some (better bidders than us?) found 6NT, which argues for us to swing for the 13-trick fence to catch up. These are judgments more than calculations.

That said, for trick probabilities I find [Edit - conditional on seeing RHO follow small twice, LHO once, in trumps.]

Risky line: If we overtake then finesse ♥, we take 13, 12, or 11 tricks with relative chances 6 : 12 : 5, i.e. 26% : 52% : 22% ♠J drops 12/23 cases, & ♥ finesse wins 6 cases. ♠ are 1=3 11/13 case, of which ♥ finesse still 6 cases (slight favorite because of the trump asymmetry).

Safe line: If we don't overtake then play ♥x if a trump is out or ♥A, ♥x if safe to do so, and assuming RHO hops ♥K - Make 7 when trumps split & ♥K singleton = 1% Make 6 if not making 7 and RHO has either ♥K or at least two hearts = 91% Down 1 otherwise = 8%

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Note that the ♥ finesse when you really need it in the overtaking line isn't 50%, it's 6:5 because it's through the short trump hand. This increases the benefit of this risky line.

In the “safe” line, if trumps split might as well cash the side suit Ace first - why surrender the chance for 7? This gives about a 1% chance for 7, somewhat negating the previous point.

The calculations in the overtaking case are all conditional on the information that RHO is known to have exactly 2 small spades. If the calculations for the safe line are based instead on original dealing probabilities then the two sets of numbers would not quite be comparable. Anyway, I get a bit different numbers for singleton kings for example.

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“I considered ZERO option, but I did not find a possible distribution for that…”

Since all plans beginning with 1+ trumps are covered by options other then “play differently”, the only effects of including “no diamonds” would be to allow those who consider attacking diamonds to save needless mental energy deciding if one round matters, and to save non-bridge energy to work out that checking “play differently” and “diamonds” might fit the bill.

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“Lemma” comes, according to the noble Internet, from Ancient Greek. λῆμμα (lêmma) meant “premise, assumption”, but doesn't any more. It's a junior theorem - a claim that needs proof (unlike an axiom or premise) but is ancillary, a stepping stone to the main theorem being proven. In this way it differs from “lemming” which comes from the Norwegian word “lemming” meaning lemming.

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East's club switch was predicated on ♣KJxx and a misguess by declarer being a possibility, so that the defense could score ♣Q, ♣A, ♣ruff. That's a reasonable idea. Couldn't a West looking at ♣Ax sensibly hope East's low club is from ♣Qxx (or ♣Kxx if declarer played the ♣Q) with a similar defense in mind?

For West to duck declarer's ♣K (or ♣Q) could be moronic or genius, but it's not everyday proper defense.

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Kit's analysis pertained to a different situation. Here the defenders cannot see the honors and even if deep analysis shows that a West with ♣Ax may as well play for the actual layout the pro's remark sounds like a good tip in real life.

Kit's analysis rested on a certain claim, a lemma to the effect that declarer may as well decide on a strategy under the assumption that the defenders will know what it is. Even if that is a valid simplification for the case Kit was discussing I don't regard a showing as “conclusive” when it rests on non-obvious lemma whose proof is omitted. And at best surely the lemma is only valid in limited circumstances. Perhaps it applies when the defenders will know the layout.

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Craig's numbers reinforce the related theory that given a weak minor 5-5 for responder to 2NT and assuming a system that allows stopping in the better fit between 4♣ or 4♦, choosing the minor part-score is way better than 2NT. (I came to believe this years before the availability of double-dummy software by computer-dealing a set of qualifying hands and evaluating them by eye. More specifically the bidding method I was evaluating allowed stopping on an 8-card minor fit but ended in 5m if opener has a 4-card minor.)

So notwithstanding Marty's valid cautions, I subscribe to Craig's general conclusion.

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Charles Brenner

Ed, calculating all possibilities from trick 1 as you did is a lot more complicated that my calculations which assumed the first 2.5 tricks given (as ♦K, ♠A, ♠Q & the ♠4,5,7 have now appeared). My guess is that you are right but confirmation would be more work than I'm up for.

Charles Brenner

That said, for trick probabilities I find [Edit -

conditionalon seeing RHO follow small twice, LHO once, in trumps.]Risky line: If we overtake then finesse ♥, we take

13, 12, or 11 tricks with relative chances 6 : 12 : 5, i.e.

26% : 52% : 22%

♠J drops 12/23 cases, & ♥ finesse wins 6 cases.

♠ are 1=3 11/13 case, of which ♥ finesse still 6 cases (slight favorite because of the trump asymmetry).

Safe line: If we don't overtake then play ♥x if a trump is out or

♥A, ♥x if safe to do so, and assuming RHO hops ♥K -Make 7 when trumps split & ♥K singleton = 1%

Make 6 if not making 7 and RHO has either ♥K or at least two hearts = 91%

Down 1 otherwise = 8%

Summary:

Matchpoint edge – probability of r tricks for risky line while s tricks for safe:

22.3%=0.6+21.7 that the safe line takes more tricks.

That edge favoring “risky” looks to me good enough to overwhelm a realistically small number of pairs who stopped short of slam.

Charles Brenner

In the “safe” line, if trumps split might as well cash the side suit Ace first - why surrender the chance for 7? This gives about a 1% chance for 7, somewhat negating the previous point.

The calculations in the overtaking case are all conditional on the information that RHO is known to have exactly 2 small spades. If the calculations for the safe line are based instead on original dealing probabilities then the two sets of numbers would not quite be comparable. Anyway, I get a bit different numbers for singleton kings for example.

Charles Brenner

Since all plans beginning with 1+ trumps are covered by options other then “play differently”, the only effects of including “no diamonds” would be to allow those who consider attacking diamonds to save needless mental energy deciding if one round matters, and to save non-bridge energy to work out that checking “play differently”

and“diamonds” might fit the bill.Charles Brenner

Charles Brenner

For West to duck declarer's ♣K (or ♣Q) could be moronic or genius, but it's not everyday proper defense.

Charles Brenner

Kit's analysis rested on a certain claim, a lemma to the effect that declarer may as well decide on a strategy under the assumption that the defenders will know what it is. Even if that is a valid simplification for the case Kit was discussing I don't regard a showing as “conclusive” when it rests on non-obvious lemma whose proof is omitted. And at best surely the lemma is only valid in limited circumstances. Perhaps it applies when the defenders will know the layout.

Charles Brenner

(Raw loser count, without A/Q adjustment as Marty)

For what it's worth

Charles Brenner

So notwithstanding Marty's valid cautions, I subscribe to Craig's general conclusion.

Charles Brenner

Charles Brenner

Remembering numbers from one moment to the next may be important too but apparently I'm not qualified to say.