Free Weekly Instant Tournament - July 5 - Board 8

Board 8
Neither side vulnerable

♠ A 4  ♥ A 10 6  ♦ A J 7 4 2  ♣ Q 10 7

LHO opens with one spade. There are two passes to me. 

I double, and partner bids one notrump. This shows five to ten HCP according to the tooltip. Eight to eleven makes more sense. You don't want to have to jump to two notrump with eleven opposite a balancing double. But I have no say in the robots' methods.

Given the bid shows five to ten, we could have 25 HCP combined, in which case we belong in game. But it's unlikely partner has a complete maximum, so it isn't worth getting to the two-level hoping he does. Besides, we've wrong-sided it. Opener might be hard-pressed to avoid giving away a trick on the opening lead. Responder will have an easier time.

I pass, and RHO leads the eight of spades.

	NORTH Phillip ♠ A 4 ♥ A 10 6 ♦ A J 7 4 2 ♣ Q 10 7
	SOUTH Robot ♠ Q 10 9 3 ♥ 8 4 2 ♦ K 8 6 ♣ A 4 2

West	North	East	South
Robot	Phillip	Robot	Robot
		1 ♠	Pass
Pass	Double	Pass	1 NT
(All pass)

It looks normal to duck this trick to set up my spade queen. But if diamonds run, I can always lead toward the queen later. If they don't run, I'd like to concede a diamond trick before the defense establishes the heart suit.

I play the spade ace--six--three. I lead the deuce of diamonds from dummy--ten--king--three. Now the eight of diamonds. West plays the five.

Let's assume for the time being that East has a doubleton diamond. If he has three, it makes no difference what I do. We can consider his holding a singleton later. What is my correct play if East has two diamonds? A priori, the odds are three to two in favor of finessing. But there are special considerations here that may change that.

Some, noting East's play of the ten, might reason this way: East has Q10 or 109. By restricted choice, Q10 is twice as likely, so it's two to one to go up. That's faulty reasoning, however. This is not a restricted choice situation. If we know nothing about the opponent's high cards, the right play is to finesse. (More on this in the post mortem.)

But we do know something about the opponent's high cards, since East opened the bidding. On average, the 16 missing high card points will be distributed 13-3. We already know four of East's high card points. Of the remaining 12, East has, on balance, nine to West's three. So East is roughly three to one to have the diamond queen. Roughly because of granularity. High card points come in clumps; they aren't distributed one at a time. Still, three-to-one is a fair approximation. So, even with the three-two split, East is a heavy favorite to hold the diamond queen. That means the percentage play is the ace.

What about the case where East has a singleton? Then my best play is to float the eight. That play is out of the question, however. It loses to any doubleton in the East hand. So my choice is between the jack and the ace. My choice is irrelevant in the diamond suit itself. I lose one trick either way. But the ace does make my life more difficult, since I have no convenient way to get to my hand to play another diamond. This makes playing the jack somewhat more attractive. But the queen is such a heavy favorite to be on my right, that I don't think it tilts the odds.

I rise with the diamond ace; East plays the nine. I continue with a diamond, and East wins the queen. It made no difference what I did in the diamond suit. West discards the five of hearts. That's probably from a five-card suit, so West is probably 2-5-2-4.

I expect a heart shift, but East cashes the spade king. West follows with the seven. Now East shifts to the three of hearts. West plays the jack. Surely West's opening lead would have been a heart if he held king-queen-jack. So East must have led low from an honor. If I'm right that West has five hearts, it was a doubleton honor. That gives West king-jack or, less likely, queen-jack fifth of hearts. Either way, East must have the club king.

It must be right to duck this trick to tighten up the position for a squeeze against East. Suppose I duck and West continues hearts. I can win in dummy and cash diamonds, coming down to this position:

	NORTH Phillip ♠ -- ♥ 10 ♦ -- ♣ Q 10 7
	SOUTH Robot ♠ Q 10 ♥ -- ♦ -- ♣ A 4

East can't afford to come down to a singleton in either black suit. And if he comes down to king doubleton of clubs and jack doubleton of spades, I can play ace and a club and force him to lead into my spade tenace.

I duck the heart jack, arriving at this position:

	NORTH Phillip ♠ -- ♥ A 10 ♦ J 7 ♣ Q 10 7
	SOUTH Robot ♠ Q 10 ♥ 8 4 ♦ -- ♣ A 4 2

West valiantly tries to get his partner off the endplay by switching to the eight of clubs. I play the ten from dummy, and East plays the jack. Did this club switch save the day? No. If I duck, the position just converts to a simple squeeze rather than a squeeze endplay. If East returns a heart, I win and cash a diamond, reaching this position:

	NORTH Phillip ♠ -- ♥ 10 ♦ 7 ♣ Q 7
	SOUTH Robot ♠ Q 10 ♥ -- ♦ -- ♣ A 4

Now the last diamond squeezes East. He must stiff one black honor or the other.

I duck the club jack. East sees the squeeze coming, so he exits with the club king to save time. Making three.

	NORTH Phillip ♠ A 4 ♥ A 10 6 ♦ A J 7 4 2 ♣ Q 10 7
WEST Robot ♠ 8 7 ♥ K J 9 7 5 ♦ 5 3 ♣ 9 8 6 3		EAST Robot ♠ K J 6 5 2 ♥ Q 3 ♦ Q 10 9 ♣ K J 5
	SOUTH Robot ♠ Q 10 9 3 ♥ 8 4 2 ♦ K 8 6 ♣ A 4 2

We would have reached this game had I been dealer. I would open with one notrump, partner would invite, and, with three aces, two tens, and a five-card suit, I would accept despite my 15-count. Whether I would make it or not without the blueprint the opponents' auction gave me is open to question.

But, as is often the case in matchpoints, you don't need to bid the game. You just need to make it. Plus 150 is worth 86%.

Back to the restricted choice problem. Why is it right to finesse if we know nothing about the opponents' high cards?

The argument for refusing the finesse is as follows: East can have Q10 or 109. If he has Q10, his choice is restricted. If he has 109, he might have played either card. So 109 is half as likely as Q10.

I know from experience that some would reason this way. And the argument is right so far as it goes: 109 indeed counts as only half a case. But the fact is, Q10 isn't a full case either. If East has 109, then West has Q53, and his play is restricted. He would always play the five and the three. But if East has Q10, then West has 953 and could choose to play any of three pairs: 95, 93, or 53. So, by restricted choice, East's Q10 counts as only a third of a case.

This means the odds in favor of the finesse are 1/2 to 1/3, or 3 to 2, the same as the a priori odds. East's play of the ten did nothing to change the odds.

This is all very complicated, but it needn't be. The way to avoid these complications is to recognize this isn't a restricted choice situation in the first place and just stick with the a priori odds. After all, if East had played the five of diamonds on the first diamond trick, no one would even think about restricted choice. Why should the ten be any different?

Some seem to think it should be different because the ten is an honor. But that's not the criterion. Here's the rule:  Apply restricted choice only when the cards involved are critical. What makes a card critical? A card is critical if the other defender would never play that card voluntarily.

To illustrate, suppose dummy's diamonds were headed by A10 rather than AJ. You cash the king, and East drops the queen. When you play a second diamond, West would never play the jack voluntarily, so the jack is a critical card and restricted choice applies. East is twice as likely to have a stiff queen rather than queen-jack doubleton.

But in the case under discussion, when you play a second diamond, there is no reason West can't play the nine if he has it. The nine is not a critical card, so restricted choice does not apply. Or, more accurately, it does apply, but it applies to both opponents, so it cancels out. That means you can just ignore it and stick with the a priori odds.

Be sure to play in this week's Free Weekly Instant Tournament on BBO, so we can start comparing in my next blog post.