What would you do with a 3-card limit raise in hearts on this auction? Redouble seems pretty normal. And who do you think has the thirteenth heart if not LHO? Did partner double light in high cards with a doubleton, or did RHO fail to rebid his 6-card suit?

I think the likeliest hand for LHO is a 3-card limit raise in hearts.

One is frequently willing to defend 1NT doubled at IMPs with only moderate expectation of beating it, since -180 is not a disaster. 1NT redoubled, however, is a different matter. The risk/reward ratio has changed, and I don't thinking defending has a positive expectation.

It is for this reason that I believe, in the auction 1NT-P-P-X, you should play that opener redoubles automaticallly. If you don't redouble, LHO might have a marginal decision, decide to gamble a pass, and get lucky. It's much harder for him to take that gamble if you redouble. Even knowing the redouble doesn't mean anything and that responder might well run if he passes, he may simply be unwilling to risk a pass and will bid something, letting you off the hook.

This isn't a restricted choice problem, but it is a real-life example of a principle of trading strategy, so perhaps you can use it.

When I was small (not sure what age, but I wasn't in school yet), my uncle took a coin out of his pocket and showed it to me. “If you can tell me what this is, I'll give it to you,” he said.

I didn't know. I knew it was either a half dollar or a quarter, but it was of the Standing Liberty type and I had trouble telling them apart unless I saw them side by side. What I did know was that, if I guessed “quarter” and it turned out to be a half dollar, I would more upset than if I guessed “half dollar” and it turned out to be a quarter. So I said, “It's a half dollar.” My uncle's jaw dropped, and he handed it over. It wasn't until years later that I understood I was instinctively following the rule, “when faced with a random choice, choose the option where the cost of being wrong is smaller.”

Yes. If your possible holdings are not equally likely, it might be necessary to adopt a mixed strategy. For example, if for some reason or other you are 40% to have AKxx and 30% each to have AJxx or KJxx, then, if declarer knows you will always play the king from AKxx, he has an edge. To eliminate that edge, you must play the ace at least 1/4 of the time. That way, when you play the king, you are no more likely to have AKxx than to have KJxx.

It doesn't matter what declarer thinks. That's the point. He can't do better than 2 wins out of 3. That's true whether you are in “statistics class” or at the table.

You may be correct that, if you always play a certain a card, a good declarer will pick up on this fact over time. Even so, your conclusion that you must randomize does not follow.

Let's take the case of holding AKxx in front of declarer's Q10x after partner leads the deuce. You have two possible pure strategies: Always play the ace, and always play the king. You also have a mixed strategy: randomize between your two pure strategies according to some percentage. Declarer has four possible pure strategies: (A) Always play the 10; (B) always play the queen; (C) play the 10 only if you win with the king; (D) play the 10 only if you win with the ace. He wishes to choose the strategy that maximizes his success when you hold AKxx, AJxx, or KJxx–three equally likely holdings. Strategy (A) will pick up two of these three holdings whatever strategy you deploy. Suppose declarer suspects you will always play the king from ace-king. In that case, he may attempt to do better by adopting strategy (D) instead. If he does–and if he is right in his suspicion–he will succeed when you hold AKxx or AJxx but will fail when you hold KJxx. So he still wins in only 2 out of 3 cases (and does worse if his suspicion is wrong). Even if declarer knows for certain what strategy you will adopt, he cannot gain an edge. He can never do better than two wins out of three.

Thus, as far as declarer's play is concerned, it makes absolutely no difference what strategy you adopt as defender. Given that fact, if one of your strategies makes life easier for partner, you might as well choose it. You lose nothing and gain in those cases where you help partner in the later defense. Thus you should always win with the king holding ace-king. Similarly, you should always play the 10 in the example above.

You think when partner has a slam invitation with only two aces he's supposed to bid Blackwood?

Partner is correct. It makes no difference to declarer whether you play the 10 or the 9. If partner ducks, he should finesse either way. This is true unless declarer is 100% certain you will always play the 10 from J10 and knows (or must assume) that trumps are not 4-1. (And even then, his play is a 50-50 guess, so he might as well finesse in case he's wrong.) Accordingly, you should always play the 10 from J10 so partner knows what to do.

A similar situation arises when partner leads a 4th-best deuce against notrump, you hold AKxx, and dummy holds a small doubleton. If declarer holds Q10x, he should finesse the 10 on the way back no matter which honor you win with. But if you falsecard with the ace, thinking you must randomize, partner may misjudge the defense if declarer happens to have QJx and wins with the queen at trick two. Thus you should always win with the king to help partner out. Again, this is wrong only if declarer is 100% sure you would never falsecard. And he can't be 100% sure. He can't even be sure I would never falsecard on the basis of this post, because it's possible I'm not serious.

The context is, you hold ♠984 ♥86 ♦Q96 ♣J9642, neither vul at IMPs, and partner opens 1♣ in first seat.

I didn't say “all other plays.”

Nicely done. Though it's the percentage play regardless of what you think of your position with the field. Other plays for 11 tricks risk holding yourself to 9.

I didn't think I had ever before seen a squeeze operate without an entry in either threat suit, but Jeff Aker reminded me I had: the last deal in “The Bet,” The Bridge World, Jan '78.

None of the above. Double is takeout with a doubleton heart, and East should balance with 2♠.

I agree with 4♥ but not with that last reason. If I think the field is doing something wrong, why should I go along with them? If I don't exploit my superior judgment, I'm giving up my edge.

Other

I must discard the heart nine. I need the ten, and the all the spots except the nine are known to be hearts. The nine of hearts, however, might be a diamond, so I must play it to avoid revoking.

Unfortunately the auction took an unexpected turn when partner chose one of my suits. If only the auction had gone as planned, I would know what to do.

The initial pass was normal. When you subsequently bid diamonds, partner will know you have a good hand, since you wouldn't fail to bid the cheapest strain with a bad hand. The fact that the auction took an unexpected turn is unlucky.

As for what to do now, note that partner doesn't even have to have four diamonds. He thought we were in a scramble, so 1D with 4-4-3-2, intending to redouble if they double, was his correct bid. In other words, I'm pretty much in the same position as if RHO had bid 4S on the previous round. I would pass then, so I'm passing now.

I once saw Marshall Miles make a grand with AKQ10xxx of trumps opposite two small with Jxxx offside. He led low from dummy, and when his RHO showed out, he played the 10 in tempo. It helped that his left-hand opponent had the nickname of “Rocket.”