# Optimal line of play in a false carding situation

This is a slightly revised version of an article I posted in rec.games.bridge in 1987.

## The situation:

The board has xxx of trump; there is no problem with entries to the board. On the first round the declarer leads from the board and plays the A from his hand. The LHO has 9x or Tx in her hand. According to the books (as they were in 1987) LHO has must make an obligatory false card. The reasoning is that if the declarer has five trump headed by the AKJ (the only relevant case) then he will take a winning finesse on the second round of trump unless he can be bamboozled into playing for the drop.

The book play is not best however. The relevant cases that I considered are (a) declarer has AKJTx, (b) AKJ9x, and (c) AKJxx. (The three cases must be considered in conjunction.) If the declarers objective is to bring the suit in without loss, best play on both sides runs as follows:

### On the first round:

(a) LHO should false card with 9x or Tx with probability p, 1/4 <= p <= 1/3.
(b) LHO should play T and 9 with equal probability.

### On the second round:

(a) If LHO has played small, RHO should false card with probability q >= 1/6 from 9xx or Txx.

(b) If LHO has played 9 or T, RHO should false card from 9xx or Txx with probability r, 1/2 <= r <= 2p.

## Declarers play:

If declarer has AKJTx or AKJxx declarer finesses on second round.

If declarer has AKJ9x then

(a) Declarer finesses the 9 against

Round 1: xxAx
Round 2: xx
(b) Declarer finesses the J against
Round 1: xxAx
Round 2: xT
(c) Declarer plays for the drop against
Round 1: xxAT
Round 2: xx
Declarer's play is minimax -- opponents cannot improve by altering their falsecarding. However declarer can gain against erroneous false carding as follows:

### I. Declarer has AKJTxx

If LHO false cards from 9x with p<1/18, declarer should play for the drop if LHO plays the 9 on the first round.

### II. Declarer has AKJ9x

(a) If RHO false cards with probability q<1/6 from Txx then the declarer should play for the drop on

Round 1: xxAx
Round 2: xx
(b) If LHO false cards with probability p<1/6 from Tx then the declarer should finesse the J on
Round 1: xxAx
Round 2: xx
(c) If LHO false cards with probability p>1/3 from Tx then the declarer should finesse the J on
Round 1: xxAT
Round 2: xx

### III. Declarer has AKJxx

(a) If LHO false cards with probability p>1/2 from 9x or Tx the declarer should play for the drop against

Round 1: xxAx
Round 2: x9/T
(b) If RHO false cards with probability r<1/2 from 9xx or Txx then the declarer should play for drop against
Round 1: xxA9/T
Round 2: xx
(c) If r>2p then the declarer should play for the drop against
Round 1: xxA9 or Round 1: xxAT Round 2: xT Round 2: x9
Since the books say that the false card from Tx is obligatory and do not mention the false card by RHO it is probably more profitable to assume that cases II.a, II.c, III.a, and III.b apply in duplicate or tournament play. Here are the relevant percentages:
 Hand: A B C AKJ9x 1144/2300 1209/2300 1027/2300 AKJxx 845/2300 923/2300 793/2300

Column A is the probability of bringing in the suit without loss if both sides play correctly. Column B is the probability if the declarer assumes that opponents are following 'book'. Column C is the probability if declarer assumes that the opponents are following 'book' and they are, in fact, playing correctly.

Note: I have assumed in the analysis that the location of the 8 does not matter and that the 9 and the T may be treated as equals.