The Majority Rule

by Phillip Martin, Brooklyn, NY

Often, knowledge of how one suit breaks will change your odds in the play of another. For example:

You cash the ace and lead toward the dummy; West follows low. Since East has twelve unknown cards to his partner's eleven, he is a slight favorite to hold the queen. So, the percentage play is to go up. But if you know that East started with, say, five clubs to his partner's three, the percentage play is to finesse. It is now West who has more unknown cards.

That combination is easy, since there is no lie of the cards where both plays succeed. If West has the queen, you must finesse; if East has it, you must go up. All you have to do is decide who is more likely to hold the queen, then play him for it. Unfortunately, not all combinations are so easy.

Here, it would scarcely be right to finesse West for the jack simply because he is more likely to hold it. Since most of the time the jack will drop anyway, the finesse must be a heavy favorite before it becomes right. Just how heavy? That's what I had to decide--with 26 imps at stake--in a Reisinger (the Eastern States Knockout) match.

I know five diamonds looks like a grand-slam try in light of my subsequent six-spade bid, but it wasn't. It was an attempt to transfer responsibility to partner for getting to a bad six. Unfortunately, John refused the transfer, and I had to bid it myself.

West led a third-best deuce of diamonds to East's ten-spot; I ruffed. Five minutes later, I led a spade to dummy's nine and East's king. East led a second diamond. I ruffed with the ace and led a spade--jack, queen, low. On the third round of spades, East followed, I pitched a club, and West pitched his remaining diamond. Both opponents followed to the ace and king of clubs. It was time for another intermission.

Since East was known to hold either 3-1-6-3 or 3-2-6-2, a first-round heart finesse against West was attractive. But was it really the percentage play? In general, the way to solve these problems is to count the combinations where one play succeeds over the other. If we consider only the heart suit, we find that the finesse is necessary in four cases--when East has a singleton two, three, four, or five; the drop is necessary in five--when he has jack-two, jack-three, jack-four, jack-five, or a singleton jack. So, it would appear that the drop is a five-to-four favorite. Unfortunately, this analysis ignores a critical factor: Any assumption we make about the heart suit affects the lie of the club suit as well.

If we credit East with a singleton heart, we are giving him three of the seven missing clubs. Since there are 35 ways to be dealt three cards of seven, a particular singleton heart occurs not once but 35 times. Similarly, if we credit East with a doubleton heart, we are giving him two of the seven clubs. There are only 21 ways to be dealt two out of seven cards; each doubleton occurs 21 times.

The finesse, then, caters to 140 hands (four singletons times 35) while the drop caters to 119 (four doubletons time 21 plus 35 singleton jacks). Accordingly, I led the eight of hearts and ducked in dummy. It turned out not to matter; East had two small hearts.

At the other table, our opponents stopped sensibly at four spades and made only four. When declarer led a spade to the nine, Kit Woolsey ducked. Fearing a bad trump break, declarer now abandoned trumps, eventually conceding three ruffs to the defense.

A few months later, I encountered the same theme. I picked up,

♠ A K ♥ Q J 5 ♦ A K 10 9 7 4 ♣ 10 4

I opened a big club; a relay auction revealed that partner held five spades, four hearts, and two doubletons with six Schenken points (A+3, K=2, Q=1). No one ever said relay was good for choosing the right game. But at least partner had bid notrump first, so I tried three notrump.

RHO led the nine of clubs, showing zero or two higher honors. LHO won the ace, dropping partner's queen, and returned a low club to partner's king. Seeing that we weren't down off the top, I turned my attention to mental counterpoint exercises.

Soon, I became aware of a break in tempo. Dummy had been reduced to,

♠ -- ♥ Q J 5 ♦ A K 10 9 7 ♣ --

Partner and RHO had low diamonds on the table. Apparently, partner had cashed the ace and king of spades, then had led a diamond to his queen. In order for this huddle to make any sense, RHO must have shown out on the second round of spades. (She could not have shown out on the first round, or partner would not have bothered to cash the king.) Since I was sure partner would ask later, I abandoned my counterpoint and began working out the percentage play.

Assuming, as it appeared, that RHO began with five clubs, the finesse would be necessary when she started with four diamonds to the jack (four combinations), thus three of the six outstanding hearts (20 combinations)--a total of 80 combinations. The finesse would be wrong when she started with three small diamonds (four combinations), thus four out of six hearts (16 combinations)--a total of 60. It would also be wrong when--At this point, partner called for the ten of diamonds. LHO produced the jack, and we went down three.

"Was that the percentage play?" partner asked.

"I don't know yet," I said. "Give me another minute."

As partner scored the ticket, I continued the computation. The finesse would also be wrong when RHO held two small diamonds (six combinations), thus five hearts (six combinations), bringing the total combinations where the finesse was wrong up to 96.

"No," I said. Partner shrugged.

On the subway back to Brooklyn Heights, I thought about these hands again. Perhaps there was an easier way to choose between the finesse and the drop in these situations. After all, I play slowly enough without having to reconstruct Pascal's triangle in my head all the time. It occurred to me that on the first deal, where the finesse was right, the putative four-one heart break would still leave room in West's hand for a majority of the clubs; on the second deal, where the finesse was wrong, a four-one diamond break would leave the unknown suit, hearts, evenly divided.

Perhaps this was the dividing line: You should finesse if, and only if, a four-one break would leave the long hand with a majority of cards in the unknown suit. If so, if I had actually found a way to play these hands in under ten minutes, my partners and teammates would be forever grateful. So I pulled out my pen and convention card and began calculating.

About the time the train pulled into Coney Island [Our author was now some 45 minutes beyond his destination.--Ed.] I was convinced I had hit upon something. This was indeed the dividing line, provided we make two assumptions: (A) We assume that you can take some measures to limit the hands you pay off to. Either you can get enough of a count to rule out jack-third offside (as I had done in the first deal) or you can cash one honor and rule out a singleton jack offside (as partner had done in the second deal). (B) We assume that the opponents hold from five to eight cards in the unknown suit. Assumption (B) may seem more tenuous than (A), but in practical play the auction is likely to give you a count on a suit in which the opponents hold an extreme number of cards. So I shall take both assumptions for granted and propose the following general rule to join ranks with "eight ever, nine never":

When you have an eight-card fit missing the jack, if a partial count of the hand (e.g., a count in two other suits) suggests playing one opponent, say West, for jack-fourth, consider the implications of a four-one break on the lie of the unknown suit. When the four-one break would still leave West with a majority of cards in the unknown suit, finesse.

If not, play for the drop.

While the rule deals specifically with suits missing the jack, there are obvious parallels. Take, for example, this deal, given to me by Dave Berkowitz from a Vanderbilt match:

After an auction that begins two hearts (weak) by South, pass, three hearts by North, you manage to find your way to six clubs from the East seat. South leads a heart, which you ruff, and both opponents follow to one round of trumps. Since you must decide between trying to drop the spade queen in three rounds and finessing against RHO to pick up queen-fourth, you are in essentially the same position as on the previous examples. In effect, your ability to ruff the third round of spades has demoted the opponents' queen to the status of a jack. How, then, should you play?

When both opponents follow to the second round of trumps, then RHO will be three-two in the rounded suits: a four-one spade break would leave diamonds four-four. So, by the majority rule, you should play for the drop. When trumps are three-one and RHO has the long trump, the drop is obviously the better line. And when LHO has the long trump, you have no choice but to play for the drop; you can no longer pick up queen-fourth. Since you intend to play for the drop however trumps split, your best play is to cash the ace and king of spades now for the extra chance that LHO began with two black singletons. As it happens, LHO ruffs the second spade, and trumps were two-two; but at least you have the satisfaction of knowing that you played the hand correctly.

Again, you are East. RHO (North) opens two hearts (weak); and you finish in a spade contract. LHO leads the king of hearts. RHO overtakes, cashes the queen (to which his partner follows), and leads a third round. You ruff high as LHO discards. You cash a high trump, then lead the nine; LHO follows low. Should you finesse?

This time, the opponent's ten has been promoted to the status of a jack. So the majority rule would apply except that we have a count in only one suit. What about treating the minors as one large suit? One could say that, since a four-one trump break would leave South with seven minor-suit cards to his partner's six, the majority rule says you should finesse. But the majority rule breaks down when the unknown suit is extremely large or extremely small. In fact, if you work it out the long way, as I did in the first two examples, you will discover that the finesse is wrong. But it would be right had hearts broken seven-one instead of six-two.

Since deciding how to play trumps after a preempt is a recurring problem, I worked out an extension to the majority rule to deal with hands where you have a count in only one suit:

When you have an eight-card fit missing the jack and a count in only one suit, finesse the short hand for the jack if the known suit has broken seven-one, eight-three, or worse. With a less extreme break (including six-zero), play for the drop.

I must be getting faster. It took me only 17 minutes to figure that out.

Reprinted by permission of The Bridge World.

Afterthoughts

As a matter of historical interest, the six spade contract in the first example occurred against Larry Edwards in the same match as John's famous psychic Lightner double. John was also declarer in the three notrump example where he finessed (incorrectly) in diamonds. When this article appeared, John chastised me for referring to him simply as "my partner" in that example.

"But you misplayed it," I protested.

"If I'd done something good, you would have identified me. It's dishonest reporting not to identify me when I do something bad," he said. That's vintage John.

As long as we're on this honesty kick, I suppose I should also mention that I didn't actually ride the subway all the way to Coney Island. I went only one stop past Brooklyn Heights before I woke up.