The Fallacy of Restricted Choice

The Principle of Restricted Choice in contract bridge is a howler (or possibly a hoax) attributed to Alan Truscott and advocated by the great Terence Reese. I can only think that it is Reese's reputation which has led to the fallacy being accepted and mindlessly trotted out in publications by other expert players.

The suggestion that the choice of a player with touching honours could have any bearing on the outcome of a finesse is manifestly absurd. The result of a finesse is determined when the cards are dealt: nothing can change it.

The very term Principle of Restricted Choice is just woffle. Bridge players should be guided by reason, not by principles. Advocates woffle on, claiming that the Monte Hall problem supports their case. There is no principle involved. It is a fact that that the presenter's choice was restricted if the contestant guessed wrong. The contestant knows the probability that he has guessed right is only one third and so the probability that the presenter had restricted choice is two thirds. When the presenter had restricted he wins by switching and that is to two thirds of the time. So the contestant switches; not on principle, by reason.

Evidence that advocates have abandoned reason is afforded by the following extraordinary quote from no less than Zia Mahmood:-

"West is slightly more likely to have been dealt an original holding of queen-jack doubleton in spades than singleton queen. Does this mean that you should go up with the king? No, for you should assume that West played the queen of spades because he had the singleton queen and thus no choice, rather than because he had a choice and chose to exercise it by playing the queen from queen-jack doubleton. You should therefore run the 10 of spades on the second round, which will work roughly twice as often as playing the king."

How crazy is that?

One of several justifications for the Principle of Restricted Choice is in the book “How the Experts Do It” by Terence Reese and David Bird.

They present this example:-

S A J 10 7 3

S 9 6 4 2

You finesse the J, losing to the queen (or king). When you lead a second round of spades from hand, West follows with a low card. Do you finesse again or play for the drop?Most players know that a second finesse is the better play. Some might be surprised to hear that the odds in favour of the finesse are almost 2 to 1. Why is this so? Well, playing for the drop on the second finesse gains in only one situation – when East has K Q doubleton. It loses in two situations – when East started with a singleton K or singleton Q. Since these three situations are roughly as likely as each other, it follows that you will win two times out of three by taking a second finesse.

Do you follow? Not really, but if the great Terence Reese says so... .It is incredible that so many have been taken in for so long! Let me spell it out.

When East wins the first finesse with the Q, South leads a spade and West follows low, only the location of the K is unknown. Therefore only two situations are possible, viz:-

A,J,10,7,3                           A,J,10,7,3

8,5                 K,Q    &       K,8,5               Q

9,6.4,2                                 9,6,4,2

So the finesse wins once and loses once. As a matter of fact, because East has one more unknown card than West, it is marginally more likely that the Q is with East. So to play for the drop is the percentage play!

Similarly, when East wins the first finesse with the K, the possible situations are:-

A,J,10,7,3                           A,J,10,7,3

8,5               K,Q      &      Q,8,5               K

9,6.4,2                               9,6,4,2

So, as before, the finesse is slightly odds against. It is clear that the three situations are NOT, "roughly as likely as each other," as Reese and Bird assert. KQ is always possible for East but one of the singletons can be ruled out (the card that won the first finesse). In other words KQ is twice as likely as the singleton K (and twice as likely as the singleton Q).

Putting it another way, the three situations for East (KQ, K and Q) are NOT roughly as likely as each other because K and Q are never East's options.

Andrew Robson's website presents this hand in support of the fallacy.

S A 10 5 3 2

H A Q 3

D A Q 3

C K Q

S Q 9 8                   S J

H 10 9 8 6               H 5 4 2

D 8 4 2                    D 10 9 6 5

C 9 7 5                    C 10 8 4 3 2

S K 7 6 4

H K J 7

D K J 7

C A J 6

West leads the 10 of hearts, declarer wins with the jack and cashes the king of spades. West follows with the 8 and East the jack.South leads the 4 of spades and West follows with the 9.        The only outstanding spade is the queen. Is West or East more likely to have the king?

Any rational person knows that East is the slight favourite because East has eleven unknown cards and West has only ten. Rational people play for the drop.

However, Andrew Robson asserts “ declarer used PRC to deduce that West was twice as likely to hold the adjacent card ”.

So, Robson believes that playing for the drop is 2 to 1 against. I have a very generous offer for him or any other believer in the Principle of Restricted Choice!

Assuming that North and East played low on the first round of hearts, then at the point of decision (when declarer has to play from dummy on the 9 of spades) the 21 outstanding East-West cards are:- S Q

H 9.8.6.5 4

D 10 9 8 6 5 4 2

C 10 9 8 7 5 4 3 2

If we take these take these 21 cards, shuffle them and deal them to East and West (starting with East because needs 11 of them) and leave the other 31 cards as they are in the diagram above we have a deal consistent with the facts.

If you believe that playing for the drop is 2 to 1 against surely you will lay me 6 to 4 against if I bet on the drop.

If you will then I propose that we shuffle and deal, I bet £20 on the drop at 6 to 4 against. Then we shuffle, deal and if the finesse wins I lose £20 or if it fails I win £30. We do this again and again until one of us admits that he or she is wrong.