A Digression

This is a special post to discuss an issue that came up in the previous deal. Specifically, what is West's proper strategy in the following layout:

	NORTH ♠ x x
WEST ♠ Q J 10 9		EAST ♠ x x
	SOUTH ♠ A K 8 x x

That is, assuming West wants to keep declarer in the dark about how spades split, what sequence of plays should he choose in following to the first two tricks? My instinct told me that West should avoid sequences that he would be unlikely to choose from a doubleton (i.e., queen or jack followed by a non-touching card) and should choose his sequence at random from the remaining possibilities. It turns out my instinct was wrong. It is quite possible that this question doesn't interest you in the slightest. If you wish to stop reading right here, I won't be offended. In fact, I'll be none the wiser. If the question does interest you, I suspect you'll find the answer somewhat surprising.

To get a handle on how to approach this problem, let's start with a more familiar position:

	NORTH ♠ x x x x
WEST ♠ Q J		EAST ♠ x x
	SOUTH ♠ A K 10 x x

Here, as we all know, West should play the queen half the time and the jack half the time. But why, precisely, is that the right strategy? Most people, I suspect, would answer that West must randomize so that declarer cannot read him. If West tends to favor one card or the other and if declarer figures that out, declarer can gain an advantage. Actually that isn't the reason. In fact, it's not even true. Unless you play the same card almost all the time (more than 93.75% of the time to be precise), declarer can't gain an advantage.

Let's say, for example, that you play the queen from queen-jack doubleton 80% of the time and declarer knows that. How should declarer play when you drop the queen? If he finesses, he wins whenever you were dealt a stiff queen. If he plays for the drop, he wins 80% of the time that you were dealt queen-jack doubleton. Since those holdings are almost equally likely, the odds are roughly 1 to .8, or about 5 to 4, in favor of finessing. Your deviating from the correct strategy does not affect declarer's proper play.

Why, then, should you play the queen only half the time? If declarer plays correctly, it doesn't matter what you do. But what if declarer is a fool? What if declarer, perversely, decides to finesse if you play jack but to play for the drop if you play the queen? You still have an edge against such a declarer, but you don't have the edge you are entitled to. This declarer will pick up a singleton jack in your hand, and he will pick up 80% of your queen-jack doubletons - a total of 1.8 out of 3 cases (the three cases being singleton jack, singleton queen, and queen-jack doubleton). He's entitled to pick up 2 out of the 3 cases by finessing, so you do have an edge. But you would have an even greater edge if you played the queen only half the time. In that case, you would hold declarer to 1.5 wins in 3 cases. While you can hold declarer to even fewer wins than that by playing the queen even less often, such a strategy would not fully punish the opposite kind of fool: the declarer who finesses if you play the queen but plays for the drop if you play the jack. By playing the queen half the time, you hold both fools to 1.5 wins out of 3. The reason you must randomize, then, is to avoid having to guess what kind of fool declarer might be. It is to guarantee you the same edge against a declarer who misplays regardless of how he misplays.

How do we determine the optimum frequency with which to play each card? You must play each card with the same frequency with which you would play it if you actually did have a singleton. Since you hold a singleton queen and a singleton jack with equal frequency, you must choose the queen or jack from queen-jack doubleton with equal frequency. It doesn't always work out that neatly.

Let's look at the following slightly more complicated position:

	NORTH ♠ x x
WEST ♠ Q J 10		EAST ♠ 9 x x
	SOUTH ♠ A K x x x

Spades are trumps. You tap declarer. He cashes the ace and king of spades. He must now decide whether to play a third round, losing lots of tricks if trumps are four-two, or to play on side suits and allow you to ruff, losing a trick unnecessarily if trumps are three-three. As in the previous example, your correct stategy is to play your cards with the same frequency with which you would play them if you had a doubleton. There are three possible ways for you to have two honors doubleton: queen-jack, jack-ten, or queen-ten. In the first two cases, you would play up-the-line half the time and down-the-line half the time. In the third case, you would, in all likelihood, always play up the line. (To make this easier, let's assume declarer can't afford to cater to a five-one break, so there can be no gain in falsecarding with the queen from queen-ten doubleton.) If you have a doubleton, then, you will play as follows with the indicated frequencies:

Q J	1/6
Q 10	0
J Q	1/6
J 10	1/6
10 Q	1/3
10 J	1/6

So you should choose your play from queen-jack-ten with the same frequencies. I think most players intuitively know not to play queen-ten, since that effectively marks you with the jack. I suspect, however, few players are aware of the corrollary: that you should play ten-queen twice as likely as any other sequence.

Obviously, jack-ten-nine works similarly. Your plays should have the following frequencies:

J 10	1/6
J 9	0
10 J	1/6
10 9	1/6
9 J	1/3
9 10	1/6

What about queen-ten-nine? You aren't going to drop the queen at trick one or trick two, so you will play as follows:

10 9	1/2
9 10	1/2

And with queen-jack-nine? As before, let's assume there is no gain in representing a singleton. So playing an honor on the first round is pointess. In fact, it's worse than pointless, because, when you follow with the nine on the second round, you would render it unlikely that you have a doubleton. Under this assumption you would play as follows:

9 Q	1/2
9 J	1/2

Now we are prepared to solve the original problem: How should you card with queen-jack-ten-nine to offer declarer no inference as to whether we began with two, three, or four trumps? As before, the answer is: you must mimic the frequency with which you would choose each sequence if you actually began with a doubleton or trebleton.

To choose one example, how often should you play queen-jack?  You would play queen-jack from queen-jack doubleton half the time, and you would play queen-jack from from queen-jack-ten one sixth of the time. There are six relevant doubletons and four relevant trebletons - a total of ten cases. So the frequency with which you would play queen-jack from the relevant two- and three-card holdings is 1/2 times 1/10 plus 1/6 times 1/10, or 1/15 of the time. 1/15, then, is the frequency with which you should choose queen-jack from queen-jack-ten-nine. If you work it all out, you wind up with the following frequencies:

Q J	.07
Q 10	0
Q 9	0
J Q	.07
J 10	.08
J 9	0
10 Q	.13
10 J	.08
10 9	.12
9 Q	.15
9 J	.18
9 10	.12

To generalize, you should usually play either nine-jack or nine-queen. Somewhat less frequently, you should play ten-queen or the ten and nine (in either order). Less frequently still, you should play touching honors in either order.

This, of course, is of theoretical interest only. No one is ever going to bother with this strategy. But I find it interesting that my intuition was so wrong.