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Misconceptions on automatic false cards
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An "automatic falsecard", is defined as a card which must be played in order to create some ambiguity for an opponent. Some automatic falsecards genuinely do create a guess, but in many of the common positions, it's a guess the opponent will get right provided he trusts you to have thought a little. In that case, why should we bother with them? This article will consider one well-known position in some detail.

To begin with, consider an easy hand:

North
A8xx
Qxx
KQxx
AK
South
KQ7x
AK
AJxx
xxx

7 by South - West leads J. Left to yourself, you'd doubtless have reached 7 or 7NT, but partner took control after a Stayman & Blackwood sequence. How do you play?

Is there any chance if spades are 4-1? If West has a stiff J/10/9, you could pick up the suit if you lay down the ace first. But you won't, in practice - if East splits on the second round, I think you'll win the match whatever you do. But there is another play worth considering. Cross to A at trick 2 and lead a low spade from table. East MAY insert the 9 from J9xx. Especially if he reads and trusts this article.

Why should he play the 9? Consider this well-known layout, which is often misanalysed:

West
x
J109x
xxx
xxxxx
North
A8xx
Qxx
AJxx
AK
East
J9xx
xxxx
xx
xxx
South
KQ10x
AK
KQxx
xxx
D

South declares 7 on J lead. A typical summary would be: "If declarer leads K from hand at trick 2, East must play the 9 to give declarer a guess on the next round. If however declarer leads a small from table, East dare not play the 9, lest West hold a singleton 10." There are 3 errors in this summary.

(1) If declarer leads K from hand the automatic false card of the 9 will not help. Declarer should still finesse through East.

(2) If declarer leads from table East should sometimes play the 9, as we calculate below.

(3) In practice on the hand on page 1 declarer rarely thinks of crossing to dummy to give East a problem with J9xx. So the chance of partner having a stiff 10 is less than it appears.

So let us consider East's and South's strategies in some detail. As usual, the best strategies involve some unpredictability. A "strategy" must be of the form "In a well-defined scenario I will play card X with a fixed probability P."

This is a clean problem - none of the 3-2 breaks matter. Declarer is known to hold 4 cards including the KQ. The only divisions of interest to declarer are

(a) W: x          E: J9xx

(b) W: Jxxx    E: 9.

East also has to consider the possibility of

(c) W: 10         E: J9xx.

From declarer's point of view (a) is 3 times as likely as (b) while (c) is impossible. From East's perspective, (a) is twice as likely as (c) while (b) cannot happen.

Suppose declarer leads K from hand and East follows with the 9. There are 3 times as many divisions of type (a) than type (b). So unless declarer takes the view that in this fairly well-known position you would play the never-costing 9 from J9xx less than 1/3 of the time, he will always finesse through you. The only reason to play the 9 from J9xx is so that declarer can't infer that case (b) exists when the 9 appears. So the automatic false-card doesn't gain directly, but it protects you on a hand you don't actually hold!

The only thing you can do to improve your chances with J9xx is to exude an aura of innocence and hope that declarer insults you. Of course, the fact that he led the K from hand in the first place, may suggest that he doesn't think much about these things.

The interesting position is when declarer crosses to table and leads a low trump at trick 3. Now East has a critical decision. Suppose we, as East, play the 9 with probability p and a small card with probability (1-p). Here p could be 0 or 1 or any value between.

It's now declarer's turn. The only case requiring a decision is when the 9 appears from East. After winning with the Q, declarer can choose to play to the Ace with probability q, or lead the K with probability (1-q).  East and South naturally aim to choose p and q to maximise their expected gains.

Consider South first. The chance of us (as East) holding J9xx is 3 times the chance of a stiff 9. South wins (1) if we don't play the 9 from J9xx, (2) if we do but he finesses through us, or (3) if we actually hold stiff 9 and he finesses through partner. Thus his expected gain is proportional to

3 * (1-p) + 3 * p * q + (1-q) = 3 + (1-q)(1-3p).

Declarer chooses his q to maximise this outcome. For example, if he knows p=1 he would choose q=1 with a gain of 3, but if p=0 he chooses q=0 with a gain of 4. This is why we must sometimes play the 9 from J9xx.

Now East must also decide on his p. If we play small we lose unless partner has a stiff 10 when we win. As 10xx are missing, twice as often partner has an x rather than a 10. In those cases, we lose always if we play small and q times if we play the 9. So East's expected gain is proportional to

1 * (1-p) + 2 * p * (1-q) = 1 + p(1-2q)

and East chooses p to maximise this. Clearly if q=1,  (i.e. declarer will always finesse through East), he has nothing to gain by playing the 9 and so he chooses p=0. But if q=0, he should choose p=1.

East and South choose their p's and q's simultaneously to maximise these different functions. As so often seems to be the case, partner and declarer really are playing different games!

The best strategy for both sides has q=1/2 and p=1/3. (This is called a "Nash equilibrium point", for those who care.) If either side varies from these values the opponent could do better. Essentially, East chooses p as small as he can without making it advantageous for declarer to play him for a stiff when he does play the 9. South chooses q at the value which means East loses equally on the stiff 10 and the stiff x positions.

 So East's best strategy is to play the 9 one time in 3, and then declarer should toss a coin.

Now we come to a slightly surprising point. With best strategy for both sides, declarer crossing to dummy at trick 2 has an expected gain of 3, exactly the same as if he leads the K from hand or just claims with "I make if Spades are 3-2, the Jack falls or if East has 4 to the Jack." If we hold KQ10x, we do not gain ON THIS HAND by making the correct play of crossing to dummy at trick 2 and leading a trump down. Indeed, there is a tiny chance of an adverse ruff if are 8-0. So against optimal defence, the best play has a slightly worse chance of success on this hand than the 2nd best!!

Nevertheless, the correct play gains in the VERY long run. If we play enough we will reach this contract many times against the same opponent. If we never cross to table, East will observe our strategy and never play the 9 from J9xx, and we will go down whenever we don't hold the 10. With the correct strategy, on the occasions where declarer actually holds KQ7x. East should play 9 from J9xx a third of the time, crashing his partner's 10, and earning partner's anger, teammates' scorn, the kibbitzers' derision but the mathematician's approval.

Which brings us back to the first hand. Crossing to table to lead towards KQ7x is the correct play, but is found much less often than with KQ10x, even though that it is on the former occasions when it "should" gain!

But those are the theoretical odds. In practice humans try to maximise the pleasure they get out of life, which isn't always the IMP expectation on playing a particular board many times. (Though surely all here would agree that the two are close.) Some people would find the pain of crashing the 9 and 10 more than twice the pleasure of enticing declarer to misguess with J9xx. And while by choosing p=0 they lose more often, they pay out when they hold a stiff 9, when partner may not blame them for following suit! If it seems fanciful to blame partner's play on another board for what happens on this one, consider the much more important case of players who are well known to give true count almost always - their behaviour on other hands certainly can cause disaster on a particular board.

For the record, I recall holding this combination in trumps only once, and not in a grand slam. On that occasion I played the 9 and Declarer misguessed and then accused me of misdefending. I have held it in non-trumps once, but then playing on the side suits revealed the layout before the critical decision. I have observed many declarers "misplay" KQ7x opposite A8xx, but of course usually it doesn't matter. Yet crossing to table and and leading a low one increases your overall expectation by about 1%!

Finally, of course in practice we don't really know what p or q the opponent has selected, and in any one instance we still have to play a definite card. You can do better than the theoretical odds if the opponent assumes you are using an inferior strategy and you outguess him. To this end, it would clearly be a bad idea to write an article on the holding and reveal your thinking in public.

Perhaps the real moral is that when partner follows with a singleton and declarer guesses correctly, it can still be partner's fault for how he played last week on a similar hand. I look forward to telling him - he's a reasonable guy, I'm sure he'll take it with good grace.

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