Hand of the Week, 4/18/96

This week, we're going to consider a different type of bridge problem: Given two hands, what is the optimal contract?

This hand came up in an IMP Pairs contest, where your score is determined by comparing your score with the average of all other scores (the datum) and calculating IMPS. Neither side is vulnerable.

	NORTH		SOUTH
	Kxxx	 S      void
	xx	 H	AKxx
	Tx	 D	AKQxxxx
	AKTxx	 C	Qx

			1C (strong)
	2C (G.F.)	2D
	2S		3D		Here 4D agrees diamonds, and
	4D		4H		4H, 5C, 5H, and 5S are cuebids; 
	5C		5H		5NT says "I'm still interested
	5S		5NT		in grand slam, so show me something
	6C		7D		I don't know about yet."

The obvious line in 7D is this: win the opening lead in hand, draw one round of trump (unless they led trump), and attempt to cash two top hearts, planning to ruff the third round of hearts with the ten of diamonds. If this works, then you cross back to the hand, draw trump, and pitch your remaining heart on the king of clubs.

What I want to discuss now are the following questions:

1. What is the probability of making 7D?
2. Given that probability, the vulnerability, and the form of scoring, is it worth bidding 7D?

We analyze the hand from the perspective of the heart suit.

• If H are 4-3 (62.2% chance), then grand is on unless D are 4-0 either way. Here's a cute point -- the fact that we know that H are splitting well reduces the chance that D break poorly. In the absence of other info, a suit splits 4-0 9.6% of the time; here, since H split 4-3, D will split 4-0 only 8.7% of the time. Hence from this case, we have a contribution of 62.2% * 91.3% = 56.8%.
• If H are 5-2 (30.5%), then we need to know something about how D might be splitting. I calculated the following table (in which the first number refers to the hand with 5 hearts):

  		0-4	 8.5%
  		1-3	34.1%
  		2-2	39.7%
  		3-1	15.9%
  		4-0	 1.8%
  

  (Note that the chance of a 2-2 split has decreased slightly, from 40.7%.) If D are 0-4 or 4-0, the slam will fail. If D are 3-1, the slam will succeed. If D are 1-3 or 2-2, the slam will succeed as long as the hand with long hearts holds the jack of diamonds. Thus the contribution here is 30.5% * [(34.1% / 4) + (39.7% / 2) + 15.9%] = 13.5%.
• If H are 6-1 (6.8%), we have exactly one chance, namely that the hand with long hearts also holds precisely three diamonds, which is a 10.8% chance. The contribution here is therefore 6.8% * 10.8 % = 0.7%.
• If H are 7-0, grand has no chance.

Adding all this up, we see that 7D will make 71% of the time. This answers the first question. Now we need to know: should we bid 7D, or should we settle for 6D?

Let a,b,c be the percentage of the field that are in grand, small, game respectively. The datum will be 1440a + 940b + 440c if 7D makes, and -50a + 920b + 420c if 7D goes down. (I'm making a simplifying assumption that everyone is in either 5D, 6D, or 7D, and that nobody is doubled.) To obtain an opinion as to whether or not we should be in grand, we need to estimate a,b,c. Canonical cases:

1. a=c=0, b=1. (The entire field is in small, except you.) You gain 11 IMPS if 7D makes, and lose 14 if it fails. The breakeven in this case is 56% -- i.e., if the probability of grand exceeds 56%, it is right to bid it.
2. a=b=0, c=1, (Everyone stayed out of slam but you.) You gain 14 IMPS if 7D makes, and lose 10 if it fails. However, simply bidding 6D would assure you of 11 IMPS. Hence your true "gain" for bidding 7D is 3 IMPS, and your true loss is 21 IMPS, so the breakeven is 87.5%, and hence here you shouldn't bid 7D.

I performed a spreadsheet analysis and came up with the following conclusions:

• if you think that nobody else in the field will be in grand, then you should bid 7D only if you think that two-thirds or more of the field will find the small slam.
• as long as one-third of the field is in grand, you should bid the grand.
• in between these extremes, extrapolate. (I.e., if 10% of the field is in grand, then you should be in grand even if something like 45% of the field is staying in game.)

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