Halo

I'd pass after the hesitation. Not exactly cutting my own throat anyway: partner's got a full set of cards in his bidding box.

2S Very commital, but I just prefer to play at this low level rather than speculate on a penalty.

I like to pass with a minimum in this sort of sequence. Since I accidentally passed on a good hand, I have to bid 3NT.

Another vote for 3S.

I wouldn't bid 2H and I would pull the redouble if I had, ready to apologise.

He's got spades and better clubs, I believe.

Agree with Codo.

Stayman and RKCB so 4H and 5H. I just wish I was confident about how he continues with 4 spades instead of 4 hearts. Luckily I have got two aces as well as two key cards.

3H may or may not be best, but if partner has a problem bidding his six card suit rather than playing in his three card diamond suit, the methods really are crazy.

I don't think pass over 5C is forcing. That may be because I think South's hand is quite strong for 4S. Flat board with 7C doubled -3 in the next room?

4D I'd rather give up on NT and play in a red suit. Presumably my slam interest is in the context of the weak hand I showed with 2S - so that's OK.

4C I've got a lot of clubs.

4H Get my hand across in the hope of cooperation later in the auction (vain hope maybe, but might as well try.)

I'd have a shot at 4S.

Stephen I give in. But I promise you next time you meet me and ask my name, I'll be disguised of course, and say 'dburn'. Maybe you have already modelled it - for every personating 'dburn' there is a 'halo', and the best of good fortune if that is what you conclude.

I would say that if it is IMPs, then prospects of getting it off are so poor, playing partner for QJx of hearts and singleton club looks about as good as anything. Not sure you need the ten of clubs since a diamond return looks clear. MPs may be a different story. Personally I would just try to cash two diamonds and hope that there is enough inconvenience to prevent declarer setting up clubs. I'm inclined to place declarer with singleton club because of his auction.

If your aim is to apply mathematics to Bridge for practical reasons, then approaches that cannot be applied at the table are interesting but eventually unfruitful. If this was always a 'purist' discussion independent of reality, I apologise for intruding.

Stephen I do understand. The reason in practice that it is so important to know how little the optimal strategy (play for Kxx and K10) gains is because the declarer/defender or both may adapt for minimal reasons. I gave you one reason in the spirit of the post. Consider the real world. If you had a proof or even an inference about the opponents distribution. For example, you could draw trumps and they were 23 or 32. For example, opponents lead suggested a side suit distributed 24. Your optimal line is so marginal that both experts could already be adapting their strategy if the frequency of Kxx v Kx shifted by a very small amount - if they were genuine experts. How easy do you think this game is. Not just maths and stamina.

Pass. I think partner probably is 3244 or 2344. I don't like my chances at the five level. I do have reasonable defence.

Thanks Stephen. Your exposition is very lucid. In a practical sense, though, we have to be sure people understand that if I deviate from the optimal roughly 7 times in a thousand rather than 5 or 6, you can start to exploit me by adapting your strategy. If you cannot tell the difference between 5 times in a thousand and 7 times in a thousand I start to make a small gain. The extent of my deviation from 'optimal' should in fact be the biggest number greater than 7 per thousand that I think I can get away with. This is a game of incomplete information. I suppose it is equivalent to the frequency of bidding on tram tickets before the opponents loosen up on penalty doubles etc.

Pass I am placing RHO with five spades and a source of tricks in hearts (where else?). I'm going to hope we can find a fourth trick in defence (assuming the first three stand up).

I am aware of the logic around genuine 'restricted choice', and entirely bought into it. As you know the principle doesn't contradict a priori odds - indeed how could it possibly. The position in this post simply is not restricted choice in the classic sense of your H versus HH holding example. My point is very simple. Once an optimum 'perfect knowledge of history' strategy is established for this holding, it is possible for me to choose to exploit this information, unpredictably, by adopting a different strategy that poses a different problem to you. I can't do that in a genuine restricted choice situation. Your advice in this post is excellent, but it is based on assumptions about the unrestricted behaviour of players. Sound advice generally speaking, but if a lot was at stake against a known expert, I would personally just play the a priori odds, as a rational alternative to 'he knows that I know that he knows' games'.

And yet East must as a matter of fact have K,K10 or Kx and you propose to lose to four of those 5 holdings

I think diamond to the nine is a given. Duck the club return? I've read the thread. Seems the best option, hope I would find it in real life. You have to sacrifice something, and I think it is best to sacrifrice the spade possibilities.

I think the heart discard is the key for East. The two of hearts seems to rule out South holding club KQ instead of the heart Ace and club Jack. East knows South has the eight of spades, so the Ace of hearts in South is his ninth trick and the club lead essential.