Compound squeeze defense

Jonathan already beat me to the correction to todays NYT column.

		♠	K Q 4 3 2
		♥	J
		♦	A Q J 9 3
		♣	A 8
♠	J 10 9 8			♠	5
♥	Q 7 4			♥	10 8 6 5
♦	10 6			♦	7 5 2
♣	Q 5 3 2			♣	J 6 4
		♠	A 7 6
		♥	A K 9
		♦	K 8 4
		♣	K 10 9 7

After a long relay auction, South placed the contract in 7♦ and got the ♦10 lead. According to the report, declarer drew trumps (West pitching a club), tested spades, then played 2 more trumps effecting a compound squeeze: West shook hearts, then ♥A, ♥K squeezed him in the blacks, and ♣A, ♠Q squeezed East in the rounds.

Of course, once a club was pitched and the spade break revealed, declarer could ruff a club, either establishing a trick or uncovering who held the sole guard and arranging either a black suit squeeze vs West or a double squeeze organized around hearts.

But, I'm more interested in figuring out whether as a pure compound squeeze how the defense should have gone. Let's imagine that the extremely revealing relay auction happened to make North declarer so that the defense knew what they were facing. Let's further imagine that the hand is in NT so we can ignore the club threat possibility and the order of discards. Picking up after testing spades and diamonds, declarer should run diamonds and come down to this ending:

		♠	Q 4 3
		♥	J
		♦
		♣	A 8
		♠
		♥	A K 9
		♦
		♣	K 10 9

West is known to have a spade guard and to have started with 7 round cards to East's 9. West must be down to only 4 round cards and so can't guard both clubs and hearts. Declarer's strategy is to cash dummy's winner(s) in the unguarded suit (perhaps pitching a spade on the 2nd heart if that suit is still guarded), then CA, SK will squeeze East. In fact, West must unguard a round suit on the 4th diamond, but the position allows cashing a 5th before attempting to read which suit he unguarded (this is not always the case in compound squeezes).

The defense's strategy is to present declarer a guess as to which suit West unguarded. To solve this fully, you need to consider all club-heart breaks, what West should do in each (possibly mixed), and the resulting guess (if any) this presents to declarer. For now I'm not going to tackle that, but I think we can get pretty close to the right answer:

West in fact pitched 2 hearts and 1 club. Unless this was hopelessly unguarding both suits (e.g. from 4=3), this is most likely from an original 3=4 in hearts=clubs (as it was in practice) or 5=2.

I think in practice declarer should assume that it is: from extreme holdings like 1=6 or 6=1, I think pitching 3 from the long suit is likely to succeed, so we can probably rule those out. From 2=5, pitching 1 heart and 2 clubs is probably best: likely to look like an original (and more likely 4=3) and lead declarer astray.

So, for starters, declarer should work out that the odds of West holding 3=4 vs 5=2 is 10:9 (if you don't know how to verify that, check out Jeff Rubens's latest book -- it's 9c3 * 7c4 : 9c5 * 7c2). You can ponder the psychology of the order of discards, but a proper expert should identify the 3 discards needed and make them in a random order. I suppose in practice most experts would try to unguard early (so pitch 2 hearts first) rather than late. Perhaps West was playing a deep game by discarding in the "easy" order (of course, in practice West had to consider many possible hands for declarer).

So, if West was playing a perfect game-theoretic game, his best bet was to discard (as he did) 2 hearts and 1 club and hope that declarer miscalculated. East could have helped this illusion by pitching clubs (as if he started with 5) instead of hearts, though theoretically his discards should be ignored (so long as he guards the suit that West unguards).

This also means that West can use some 3=4 "losers" to "protect" his 6=1 winners -- occasionally pitch 3 hearts (or clubs) from that holding so that declarer can't conclude that 3 pitches is always from an original 6. This will reduce the 10:9 odds, but not enough to change declarer's best strategy. (Of course, the HJ makes that a bad idea with this particular holding.)

So, basically I think the defense is slated to lose when West starts with 3=4 or 4=3, but succeed otherwise. By my math, that's about 64%.

In practice, I think if East had pitched a club and a heart or 2 clubs, declarer had a realistic chance to go wrong: 5=2 (West hearts=clubs) is almost as likely as the actual 3=4, but in practice there must be a 10% higher chance that East pitched clubs from an original 5 instead of an original 3, which could be enough to tip declarer to the wrong decision.