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A Beautiful Hand

This deal came up six or seven years ago and remains my favorite bridge problem. 

North
Qx
xx
KQxxxx
xxx
South
Axx
AKx
Axx
10xxx
W
N
E
S
1
2
P
3NT
X
P
P
?

Imps, KO, regional final, your LHO is David Grainger, then a younger Canadian bridge pro. Your side is vulnerable and the auction proceeds, (1C)-2D-(P)-? Vulnerable, 2D promised 2/3 6th. 1C was 3+. You bid 3N, Grainger doubles, back to you.

This is a classic game theory problem with a minimax solution and a Nash equilibrium. Give West the AKQJ of clubs, but does he have 4, 5 or 6? Assume they occur with the probability of 4=45%, 5=34%, 6=21%.  And what should he do with each? Similarly, declarer can stay, run or RDB. Both sides have a clear mixed strategy.

One should be able to set up the classic matrix and solve for best strategy on the imp payouts of each action. My current recollection of game theory is such that I am unable to work out the answer.  The problem is more complicated than the simple game theory problem because you have to factor in the likelihood of West's club holdings.

We were playing on a client sponsored team as was Grainger. I was dummy. Annie ran to 4D, Grainger had 4 only clubs (and later said, "I didn't open 1C to defend 3N") and we lost by less than the swing on this board.

Our captain was Chris Compton who was unhappy with the result and observed, "They always have four." Only a guy who plays 45 weeks a year, year after year after year, has seen this type of problem enough to have a feel for what opponents do at the table.

But I am posting it here to see if anyone can show how to arrive at the optimal mixed strategy for both West and South.

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