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Jonathan M.,
That’s certainly a plausible pattern of learning. There are thousands of pages on learning processes which might get you to equilibrium, without any universal answers.
The second paragraph of Charles Brenner’s comment is a good view on equilibrium. Rephrasing, equilibrium isn’t necessarily what you *should* do; if you have good reason to think your opponent is away from equilibrium in a certain direction then don’t randomize, go ahead and play the best reply. The equilibrium is a useful baseline of what would happen if good players faced this situation a lot, and of course it defends, on average, the tricks you’re entitled to.
Jan. 12
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South actually has *four* strategies when holding KQTx:
A – Start with K from hand, play East for stiff when you see 9.
B – Start with K from hand, play East for J9xx when you see 9.
C – Cross in clubs, small spade from dummy, play East for stiff when you see 9.
D – Cross in clubs, small spade from dummy, play East for J9xx when you see 9.

As you correctly analyzed, South never uses A in any equilibrium. The falsecard is free, so East does it at least often enough to make A worse than B.

When we ignore 8-0 clubs, B and D are completely equivalent. They perform equally well against any defense. Hence the equilibrium you found, where South plays C and D 50-50, is actually part of a continuum of equilibria where South plays C 50% and B/D in any combination totaling 50%. East’s equilibrium strategy is unique (the one you found).

When we *account* for 8-0 clubs, however unlikely, B *dominates* D. That is, it sometimes does better and never worse. Accordingly, South should *never* play D. The now-unique equilibrium is for South to play (50% B, 50% C), and East to play the 9 a *tiny* bit less than 1/3 of the time on a small lead from dummy. Why is that the equilibrium for East? Because it gives South a tiny reward when his club lives: he will pick up a tiny bit more than 3 cases, namely stiff 9 and a bit better than 2 of 3 J9xx cases. This tiny reward compensates for the risk of a ruff. Meanwhile, East knows that when South crosses to dummy, he’ll have the T half the time and will always fall for a falsecard if holding the T. This makes East indifferent to falsecarding.
Jan. 11
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Regarding your paragraph about the 8-0 side suit, page 5: In equilibrium, no one *ever* makes a play which is *inferior* given their current information. The resolution of that particular paradox is that we have simply calculated the equilibrium wrong, when we take the 8-0 into account. The equilibrium is then quite a bit different from the one you gave. It turns out that when we ignore 8-0 splits as in your analysis, the game actually has a continuum of equilibria of which you gave just one; factoring in the 8-0 actually reduces the set of equilibria to a singleton which is *not* close to the one you documented. I’ll write up the details as soon as I can.

In the game-theory literature, we say the correspondence from game to equilibrium is discontinuous; this phenomenon happens to be a focus of my research.
Jan. 11
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Thanks, Kit! I had settled on a line very close to this, also starting with K,A of hearts – then I had the 3-2 case correct, but on the 4-1 case I had resigned myself to going down against 2=4=5=2 because I was taking the second spade ruff after 1 round of clubs, losing one of my club tricks. Your version is a clear improvement.

I find it sort of amazing that a hand with 0 non-trivial card combinations can require such care. My guess is that well less than half of declarers in a Blue Ribbon final would find Kit’s exact plan.
Jan. 11
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See Kit's line, which makes at least 5 and sometimes 6 when trumps 3-2, and still almost certain to make against 4-1.
Jan. 11
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OK, I think it’s not a percentage issue because you can combine essentially all your chances. Heart to Q at trick two. (1) If it wins, K, run T. Cold unless someone has Q9xxx, even then you have good chances in hearts. (2) If heart loses, you get to check for Jx or KJ and then try for 5 diamond tricks when that fails.
Trying diamonds first would lose the chance of Jx with righty when the diamond hook fails.
Jan. 10
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Good article!

Has anyone commented on the play of the very first hand? Looks to me like a close percentage-play issue whether to go for hearts or diamonds first. If you do tackle diamonds for five tricks, either before or after a failed heart play, low to the T gives you the best shot.
Jan. 10
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Yes, that's all true. Just woke up and came straight here wanting to delete that comment but wasn't soon enough :-o.
Dec. 21, 2016
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Ducking Qx doesn’t seem that obscure. If declarer had AJTxx this would be a very well-known position where you need to duck. Ducking Jx is more obscure.
Dec. 20, 2016
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Kit’s play gains for sure in 2 cases (LHO’s Ax) and loses for sure in 1 less likely case (LHO’s stiff A), call it a net gain of 1.2 cases against the worst opponents, who never find the smooth duck. It puts you in jeopardy in 4 cases, LHO’s quack-x. So if the opponents find the smooth duck 30% of the time it’s a wash. The claim that if they give it away 80% of the time, you gain, is correct. Your success rate would be 1.2% higher than with the technical line. Against the worst opponents your success rate is 4% higher than with the technical line.
Dec. 20, 2016
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Rephrasing Peg’s comment, “evidence insufficient for a criminal conviction” is worlds away from “no evidence.” To which, amen. It was Edgar Kaplan’s great innovation to adjust results due to minor ethics violations (using inadvertent UI) without requiring the impossible standard of conclusive proof. When it comes to score adjustments, as opposed to suspensions, Kaplan's principle should apply to major ethics violations (intentional UI.)
Dec. 16, 2016
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ACBL should be giving much more information about what happened and why each penalty is the appropriate one for each violation. Major goals of these penalties include deterrence and development of community standards about what punishment is proportional to each crime. Each case should help establish clear precedent for the future. This document as it stands does very little to help handle future cases in a consistent manner. There are probably good reasons some were penalized more than others, and they should be shared.
Dec. 14, 2016
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Yes, on balance this looks like a (very small) improvement.
Dec. 7, 2016
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Danny,

I consider that a genuine cost…you have added some variance to the draw, for everyone including totally innocent teams. I'm saying it's a cost I would be willing to bear. Rub of the green … “Too bad we got Helgemo-Helness first round, they are great players but had no seeding points because they got stuck with non-bridge-playing teammates for five years.” I see avoidance of certain teammates when there are credible whispers about them as a positive side effect.
Dec. 7, 2016
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Very close whether you should start with K or A. If you start with K and righty shows out, your options in are not as good as in Kit’s line (when you play A and lefty shows out), BUT the spade hook is 57% instead of being 43%. The net difference is within 1-2%, and that's *conditional* on having a trump loser. So start trumps however you like. I agree with Kit on starting trumps immediately.
Dec. 6, 2016
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The travesty that is the WBF rankings page was pointed out months ago on BW, and still remains a travesty: http://www.worldbridge.org/open-classification.aspx . In other news, Bernie Madoff is one of the nation's top investors…
Dec. 6, 2016
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Seeding points are one of the major perks of winning a title. If the title is voided, no seeding points. I, for one, am more than willing to see top teams play each other early as a price for upholding that principle. This isn’t a “punishment,” really…choosing the right teammates is a major part of bridge success, and if you choose so badly that your teammates aren’t even playing bridge, well, you won’t win (and keep) any titles and your ranking will suffer. Yes, even if the choice was just “unlucky” in the sense that you couldn’t know.
Dec. 6, 2016
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Agreed, and I would then bid 6 over 6. Can't risk playing 6 opposite 6=2=3=2 or 5=2=4=2. So unfortunately clubs just not in the picture, I'll settle for an intelligent choice of 6 or 6.
Dec. 1, 2016
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David,

I guess you’re joking? People who devote copious time and energy to a game want the game to be correctly scored and officiated, or why waste their time? Also, the NAP is played more for qualifications than for masterpoints.
Sept. 26, 2016
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Not being able to find out about the J, West should seriously consider settling for 6NT in a club game. I would expect about 80% MPs for that. But perhaps the various chances make 7NT a big enough favorite. Your 9 gives you 75% opposite Q432, close to 90% opposite QT32 and of course 100% if you catch the J. Could even catch the Q, another few percent.
Sept. 1, 2016
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