Deception is the act or state of deceiving or the
state of being deceived. In the game of bridge, an opponent may falsecard in order
to deceive the declarer into altering his opinion about the lay of the cards. This
deceptive play is used by all bridge players, and Mr. Albert Dormer and Mr.
Terence Reese analyzed the situation and postulated their results in the Mathematics
of Deception. They applied the rule of multiplication of probabilities when the declarer
has to decide whether a played card is intended to deceive.
Simplified Rule of Probability
The probability that a suspected card is true
is the probability that the players holds a distribution that leaves him no choice
but to play that card. The probability that it is false is the probability that he
has a distribution from which the deceptive play would be the better choice, multiplied
by the probability play that he would in fact decide to falsecard. The following
example is the illustration Mr. Albert Dormer and Mr. Terence Reese used for their
probability formula.
After winning the opening lead, South plays the King. West plays low, and East plays the 9. It has been determined mathematically that the 9 is a singleton by 2.8%. On the other hand, East may hold J-9-x-x, and the probability of this holding is approximately 8.4%. Therefore, the probability that East would play the 9 from J-9-x-x is greater than 50%, that the distribution would be more likely than the singleton 9. Both Mr. Albert Dormer and Mr. Terence Reese assume it to be self-evident that the play of the 9 from a holding of J-9-x-x is obligatory, in order to present South with a choice of plays on the second round.
In the case that South decides to share this view, then South must play the Ace the next time. Assuming that the only deception imaginable is the play of the 9 from J-9-x-x, or that East holds either J-9-x-x or the singleton 9 when he plays the 9, South then give himself a better chance. South enters the dummy and leads low toward the Q-10. If East does have only the 9, South plays the Queen, and finesses West for the Jack.
In order to mislead South, East is obligated to play the 9 not only from a holding of J-9-x-x, but also from a holding such as 9-x-x or even 9-x. In this case, South will then have to guess whether to finesse the 10 or play the Queen. Since 9-x-x and 9-x each have a mathematical probability of about 10.2%, South would establish a better situation by playing the Ace from the dummy on the second round. That is, unless South estimates or guesses only a very small probability of the 9 being played from a doubleton or tripleton.
In the postulation of Mr. Albert Dormer and Mr. Terence Reese, it is assumed that East will always play the 9 from J-9-x-x. Then the possible convictions of South become:
1. |
Play low to the Ace in the dummy, in order to be able to finesse against East if West discards. |
2. |
Enter the dummy with a side-suit, lead toward the Q-10 in his hand, and then finesse the 10 if East follows. |
3. |
Enter the dummy with a side-suit, lead toward the Q-10, and play the Queen if East follows suit. |
It has been mathematically calculated that the probability that the relevant distributions dealt to East are: 9-x or 9-x-x is 64%. A distribution of J-9-x-x is 27%. A singleton 9 is 9.9%.
Allowing "p" to represent the calculated mathematical probability that East will play the 9 if East has 9-x or 9-x-x, then the chance of plan 1 above of succeeding is .64 times p+.27. The chance of plan 2 above succeeding is .09 times p+.27. The result is that if p is less than 14%, then plan 1 is preferable. Plan 1 should be preferred unless it is suspected that East would not play the 9 form 9-x or 9-x-x at least 7 times in 50.
The chance that plan 3 above of succeeding is .64 times p+.09, which clearly does not offer the success rate as plan 1. In the case of multiple entries, the 2 should be led from the dummy on the first round of that suit. In this case, it would be more difficult for East to play the 9 from J-9-x-x. East may decide that West may hold the 10, and the play of the 9 by East could conceivably concede an unnecessary trick to South.
The above is the mathematical calculations arrived at by Mr. Albert Dormer and Mr. Terence Reese, and may help bridge players in deciding the percentages of the better play when it has been decided that a play of a card could by construed to be a deception. Falsecarding is a part of the game of bridge, and it is always good to have open options to counteract such deceptive plays.