Restricted Choice in Bridge: Finesse After an Honor Drops

The principle of restricted choice says that when a defender plays an honor that could have come from two touching honors, such as the queen from Q-J, you should assume it was a singleton. So when an honor drops on your left under your ace, finesse against the other honor on the next round instead of playing for the drop. The finesse wins about 65% of the time; the drop wins about 35%.

That surprises most players the first time they see it. Before the honor fell, a singleton queen and a Q-J doubleton looked about equally likely, so a 50-50 guess seems fair. It is not, and the reason is simple: a defender with Q-J could have played either card, while a defender with a singleton queen had no choice at all. Restricted choice is card play, so it is the same in every bidding system.

This is the layout you will meet most often:

You are missing the queen, jack, nine and two. You lead a low spade to dummy's ace, and West drops the queen while East follows with the two.

Only two layouts are left that make sense:

1. West had the singleton queen, and East holds ♠J 9 2.
2. West had Q-J doubleton, and East holds ♠9 2.

(We ignore a deliberate false card from Q-x. A defender who drops the queen from Q-x under your ace usually just hands you the suit, and almost nobody does it.)

Now you lead a second spade from dummy and East plays low. If you finesse the 10, you win against the singleton queen. If you play the king, you win against Q-J doubleton. Restricted choice says finesse.

Start with the chances before any card was played. With four cards missing and 26 unknown cards split 13-13, the two exact layouts are:

The Q-J doubleton is a little more likely to start with, but with Q-J West plays the queen only half the time; the other half you see the jack. Compare the weights: 6.2 against 3.4 means the finesse wins 64.7% and the drop 35.3%, exactly 11 to 6. You can check it yourself: the singleton is C(22,12) deals, the doubleton C(22,11), and you take half of the doubleton.

The old saying "eight ever, nine never" (with nine cards, play for the drop of a missing queen) does not help here. That rule is about a suit where no honor has fallen yet. Once one of two touching honors has fallen, restricted choice is the stronger rule.

If the math feels wrong, you are in good company. The same idea is behind the famous Monty Hall puzzle. A game show host knows where the prize is, opens a door you did not pick, and always shows an empty room. Switching doors wins two times in three, because the host had a free choice of doors only when your first pick was right.

At the bridge table the defender plays the part of the host. With Q-J he had a free choice between two cards. With a singleton he was forced. Each time you see a card that a forced player had to play, the forced layout becomes more likely. That is all the name means: the defender's choice was restricted.

Count first. You have five spade tricks if trumps behave, three hearts and two diamonds: ten. The ace, king and queen of clubs are all with the defense, so three club losers are certain. One trump loser and you are down.

1. Win the ♦Q with the ♦K in hand. You keep dummy's ♦A as a spare entry.
2. Lead the ♠3 to dummy's ♠A. West drops the ♠Q.
3. Lead the ♠4 from dummy. East plays the ♠9.
4. Finesse the ♠10. It wins, and West discards.
5. Cash the ♠K to pull East's ♠J, then take your hearts and diamonds and give up three clubs.

Ten tricks. A double dummy solver confirms that 4♠ makes exactly ten, and that playing the ♠K on the second round (the drop) holds you to nine.

Notice what West's queen told you. If West had held Q-J, he could just as well have played the jack. He showed the queen, and a player with a singleton queen must show the queen. The finesse is the 65% play.

Had West started with Q-J doubleton, the finesse would lose to the jack and you would go down. That happens about one time in three. Over many deals, the player who finesses scores far more.

Restricted choice tells you where the other honor probably is. Your own cards decide whether you can do anything about it.

In the deal above, the ♠10 is in your hand and East plays before you when dummy leads. So you can finesse only East. With this layout, the useful case is an honor that drops on your left: it points to the other honor on your right, where your finesse works.

If East had dropped the queen under dummy's ace instead, the principle would say West holds ♠J x x. You have no way to finesse West, because your ♠K 10 sits in front of him and dummy has no tenace. Then you simply play the king: it wins when East began with Q-J, and nothing wins against ♠J x x on your left.

Before you lead the first round, ask yourself which honor drop would let you finesse, and plan the entries so you can lead from the right hand on the second round.

The principle works for any two equal cards: K-Q, Q-J, J-10, even 10-9 when the higher cards are gone. Two common cases:

In the second case you are missing J-10-8-2. A singleton jack or ten on your left means East holds the other honor plus the 8 and 2, and your ♣Q 9 over East picks up the whole suit. If West began with J-10 doubleton, the finesse of the 9 loses to the other honor. The odds are the same 65 to 35 in favor of the finesse.

The cards must be true equals. The queen and jack are equals. The queen and the 10 are equals only after the jack has been played; while the jack is still out, a defender with Q-10 doubleton plays the 10, so his queen was not a free choice.

The principle cuts both ways. When you defend with Q-J doubleton and declarer cashes the ace, play the queen and the jack at random, about half the time each. If you always play the queen, a declarer who knows your habit can read you: the queen means Q-J or a singleton queen, and the jack means a sure singleton.

The same goes for any equal cards. With J-10 doubleton, sometimes play the jack and sometimes the ten. You give declarer nothing he can use, and he is left with the 65% finesse, which is exactly the play that loses to your Q-J.

Restricted choice rests on one idea: a defender with two equal cards picks one at random. When that idea is false, the odds change.

• A defender who always plays the same card. If you know that East always drops the queen from Q-J, then his queen shows Q-J or a singleton queen, about 52% for the doubleton, and the drop becomes slightly better. His jack, on the other hand, shows a sure singleton.
• Vacant places. If the bidding or the play shows that West has many cards in other suits (for example, West has shown a long suit), West has fewer spaces left for this suit, and the layouts shift.
• No real choice. A defender who drops an honor because he is squeezed or has nothing else to keep was not choosing at random.
• The honors are not equals. From Q-10 doubleton, with the jack still out, a defender plays the 10, so a queen from that holding is not a free choice.

Restricted choice is one of the ideas you meet again and again in single-suit play. Practice it in BridgeDummy's suit combination puzzles, where you play one suit card by card against the best defense, and enter full deals in the free double dummy analyzer to test a line.

Related reading: the finesse, finesse or drop, common suit combinations, safety plays, avoidance play and what double dummy analysis can tell you.