Monday, 23 September 2013


How do you play A752 opposite J943 for one loser?

The technician will advise you to take an intrafinesse by leading towards dummy's jack.  If LHO has K10 he will win the trick with the king, and you can subsequently run the jack, pinning his ten.  Similarly if he has Q10.  Or if he has 108, you can cover his eight (or ten) and subsequently pin his ten (or eight).  (There is no winning line if he has 106, this is different from a combination where declarer or dummy has the eight.)

However, against expert opponents life is harder.  They may play the ten from K10 or Q10 or 108 or 1086. and they may play the eight from 108 or 1086.  So what is the correct combination of plays?

I'll attempt a game-theoretic analysis.   Assuming that declarer starts with a low card towards the jack, there are five, equally likely, holdings for LHO where a successful play is available but might not be found - the four I listed above, plus KQ.  Let's suppose that the defender follows a mixed strategy: he plays the king from K10 with probability p, similarly from Q10; he plays the ten from 108 with probability q; and the ten from 1086 with probability r (and otherwise the eight: it's a mistake to play the six, which gives declarer no losing option).  And he plays the king from KQ with probability half (other approaches may be as good, but symmetry considerations say they can't be better).

To start with, let's assume that the values p, q, r are all known to declarer, who follows a pure strategy - this assumption is unjustified, but it's a useful starting point.

a) LHO plays the king or queen.  (a0) declarer tries to pin the ten: for every five times there's a critical layout this succeeds 2p times. (a1) declarer tries to drop the other top honour: this succeeds 1 time.

b) LHO plays the ten, and the jack loses to the queen or king.  (b0) declarer tries to pin the eight: this succeeds q times.  (b1) declarer tries to drop the other honour: this succeeds 2(1-p) + r times.

c) LHO plays the eight, and the nine loses to the queen or king.  (c0) declarer tries to pin the ten: this succeeds (1-q) times.  (c1) declarer tries to drop the other honour: this succeeds (1-r) times.

So the combined successes per five critical layouts are:
a0b0c0: 1 + 2p
a0b0c1: 1 + 2p + q - r
a0b1c0: 3 - q + r
a0b1c1: 3
a1b0c0: 2
a1b0c1: 2 + q - r
a1b1c0: 4 - 2p - q + r
a1b1c1: 4 - 2p

We can simplify this somewhat by putting = q - r, getting:
a0b0c0: 1 + 2p
a0b0c1: 1 + 2p + s
a0b1c0: 3 - s
a0b1c1: 3
a1b0c0: 2
a1b0c1: 2 + s
a1b1c0: 4 - 2p - s
a1b1c1: 4 - 2p

We note that by making the appropriate choice from each pair, declarer can make the value of s irrelevant, and that it can't help the defender to choose a value other than 0, but may hurt him.

Putting s = 0, we see that a0b1c1 succeeds 3 times,  but a1b1c1 will be better if p < 1/2 .   So overall a sound strategy for the defender is to rise with K10 or Q10 at least half the time, and to choose between the ten and the eight on the same basis with either 108 or 1086.   If the defender follows this strategy, a sound strategy for declarer is to play for the pin if LHO plays the king or queen, but for the drop if LHO plays the ten or eight.

This morning I was watching the final round of the World Championship round robin stage on BBO, when the following deal came up:

Board 4         ♠ K 10 9 5
Game All         4 2                
Dealer West      Q 9 8 7
                ♣ Q 8 6
♠ Q J 4                         ♠ A 8 2
 K J 8 3                        A Q 10 6
 6 3                            A 10
♣ J 9 4 3                       ♣ A 7 5 2
                ♠ 7 6 3
                 9 7 5
                 K J 5 4 2
                ♣ K 10

Robson          Fritsche        Forrester       Rohowsky       
West            North           East            South
Pass            Pass            1♣              Pass
1 (hearts)     Pass            4              All pass
Rohowsky for Germany led a diamond to the queen and ace. Forrester drew trumps and exited with a diamond, won by South who played a spade. Three rounds of spades put North on lead to concede a ruff and discard, which did the defence no harm. Declarer now needed to play clubs for one loser, so he led towards dummy, South playing the ten, and North beating dummy's jack with the queen. North returned a low club. The Vugraph commentators thought declarer would have to get it right, apparently not seeing the possibility of playing South for 108.  But Forrester got it wrong.

Tony Forrester and Andy Robson - it's an old and successful partnership, but the two were selected primarily to play with David Gold and Alexander Allfrey respectively - have been England's leading pair in the Butler rankings, so Tony won't mind my saying that I think he was wrong in theory as well as in practice on this one - strategies including b0 are never best.

One interesting aspect of this is that the same hand was played 22 times in the Bermuda Bowl, 22 times in the Venice Cup (for women), and 22 times in the d'Orsi Trophy (for over-60s).  In the Bermuda Bowl, 4 was played 21 times: four declarers went off; it was played all 22 times in the Venice Cup where five declarers went off; in the Seniors only 18 pairs reached 4 and six of them went off.  Perhaps it matters which hand is dummy: in the Bermuda Bowl all twelve West declarers, with A752 in dummy, made the contract, but only four out of nine Easts.  In the Venice Cup, fourteen out of seventeen West declarers made the contract, and three out of five Easts.  But in the d'Orsi Trophy four out of eight Wests succeeded, eight out of ten Easts.  Overall, West was 30/37 (81%), East 15/24 (63%).

BBO's vugraph archive (in the future you may need this link) records the play at fourteen tables, including the one above: alternatively one can find a record of the play at the same matches by following links from the results tables I linked to above (apparently play is recorded in seven of the 33 matches in each round).

This lets us see how two other declarers went off.  Multiple world-champion Lorenzo Lauria played the hand as East on a trump lead,  He drew trumps in three rounds and played ace and another diamond.   North overtook his partner's jack to switch to the ten of spades, won in dummy.  Declarer now unexpectedly played ace and another club and went off.  However, this surprising line is not much worse than the recommended intrafinesse strategy - it makes against KQ doubleton in either hand and K or Q singleton with North.  And perhaps there's some chance of success if North has king doubleton.

Furuta Kazuo played the hand as East for Japan on a spade lead to the nine and ace.  He drew trumps, knocked out the king of spades, won the diamond exit, and led a club towards dummy.  South rose with the king, cashed the king of diamonds, and exited with a spade.  Declarer now played for the drop in clubs.  This might be right in theory if South would play the ten from K10 sufficiently often, but having gone off I wouldn't want to advance that argument too loudly in the post mortem.

We have the play in 4 at nine other tables (both pairs in the Seniors match on record went off in 3NT).  Interestingly, whereas all three East declarers above went off, the nine declarers who played the hand as West all made it.  Usually the play involved leading a club off dummy in the middle of the play, won by South's king, and later leading the jack to pin the ten.  One notable exception was in Netherlands v Monaco: Verhees ducked the diamond lead, won the continuation, and immediately led a club off dummy.  Geir Helgemo, arguably the world's best player of the cards, won with the king and continued with the ten of clubs.  Two other Souths also continued clubs when in with the king, but much later in the play.  Paul Thurston, South for Canada, put the ten in when a club was played off dummy at trick ten, but Kevin Bathurst for USA1 dropped his king.  And the play went wrong when France declared against England in the Venice Cup: North led a spade to declarer's queen, declarer drew trumps and played ace and a diamond, won by South who played a second spade to dummy's ace.  Declarer now cashed dummy's ace of clubs, apparently a horrible line since unlike Lauria she didn't have the entries to take advantage of a singleton honour with North, but in practice an easy way to succeed because South dropped the king.

Most of the declarers fiddled about in diamonds and spades, hoping to learn something, but it seems to me that all they achieved was making it easier for the defence to read the position - a couple induced North to switch to a club, but that didn't help.  I like Verhees' line.

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