A Suit Combination

by Phillip Martin, New York City

A deal in the May, 1988, Swiss Match involved the following suit combination:

Against three notrump, West led the heart five. East won and returned the jack, which might have eliminated a guess for declarer. The author ponders whether this is the right return. He concludes, incorrectly in my opinion, that it is.

The author advances two arguments for why South is likely to guess right: (1) East might cash ace-king if he has it, in hope of a doubleton queen. (2) East is a priori likelier to hold honor-jack-fourth than ace-king-fourth, by restricted choice.

Assuming fourth best leads, argument (1) doesn't hold. Although the text does not say so, I assume declarer played the heart deuce at trick one. (If he played the seven, it is automatic for East to return the jack; declarer cannot have queen-nine-seven.) West was marked with a four-card suit, so East would always return a low card from ace-king-fourth.

As for argument (2), it is true that ace-king-fourth is only half as likely as honor-jack-fourth, but it is honor-jack-empty-fourth we are concerned with. East will always lead the jack from honor-jack-nine-fourth. The true frequencies of the relevant cases are as follows:

Case 1 (relative frequency 6)

Case 2 (relative frequency 5)

Case 3 (relative frequency 4)

Case 4 (relative frequency 2)

I'll skip the computations of the relative frequencies. You are welcome to verify them if you feel so inclined.

We can now construct a table to compare the various possible strategies. (A fourth strategy for declarer--low from Qxx, high from Q9x--is obviously ridiculous, so we won't bother with it.)

This table shows how many times out of seventeen declarer will take a trick with each of the strategies. For example, if the defender will lead low from honor-jack-fourth and declarer will always duck (strategy A), declarer will be right in Case 1 (6 times) and Case 3 (4 times), a total of 10.

Let's consider the problem first from declarer's perspective. If we don't wish to guess how East will defend, the best strategy is (B): rise with queen-third, duck with queen- nine-third. No matter which strategy East uses, we will be right nine times out of seventeen.

Theoretically, then, (B) is fine. As a practical matter, though, I think (C), hopping with or without the nine, is better. By playing (C), we are in essence laying even money that our opponent is more likely to lead the jack than low from honor-jack-fourth. In my experience, most players are quite disinclined to lead low in this situation. So anyone who plays (B) is a sound theorist but a poor bookmaker.

What should you do as defender? If declarer plays strategy (B), it doesn't matter what you do; you will lose nine times out of seventeen. So you might as well assume that declarer does not play (B). Which way, then, is he more likely to deviate? If you think he is more likely to play low without the nine than he is to hop with the nine (as the Swiss Match author apparently thinks), then you should lead the jack. If you are right, this holds declarer to four wins instead of the nine he is entitled to. If you are playing against someone who thinks as I do, who might hop with the nine, you should lead low, holding him to seven wins.

If you don't care to guess how declarer will play, you can still enjoy an edge by playing a mixed strategy: lead low 70% of the time, lead the jack 30%. If declarer plays strategy (A), 70% of the time he will score ten wins in seventeen; 30% of the time, four wins. This gives him a net expectation of 8.2 wins. If he plays (C), he will score 70% of seven plus 30% of eleven: again, 8.2 wins.

This isn't as good as guessing right, but it's better than what you are theoretically entitled to.

Reprinted by permission of The Bridge World.

Afterthoughts

Many players don't seem to understand what mixed strategies are all about. For example, take this well-known position:

I've heard people say, "I know that when declarer cashes the ace or king, East is theoretically supposed to play the queen half the time and the jack half the time. But in practice, what difference does it make? That is, unless you play against the same declarer over and over and he keeps records."

This comment betrays a misconception that the reason you play a mixed strategy is to keep your opponents guessing. In fact, you play a mixed strategy to keep yourself from guessing about your opponents' quirks. In this example, for your choice to matter, we must assume it actually effects declarer's play. In other words, strange as it may seem, one card is going to entice declarer to finesse; the other is going to entice him to play for the drop.

Which card is which? Who knows? All we can say for sure is, if you play one card consistently, it is quite possible that, against this particular declarer, you will never take a trick with queen-jack-doubleton. If you randomize your play, however, you are guaranteed to take a trick half the time.