g-rh@cca.CCA.COM (Richard Harter) (01/02/87)
References:
Recently I posted an erroneous line of play (Cohen was right,
I was wrong). Having gone through a full mathematical analysis, here
are the results:
The situation:
The board has xxx of trump. On the first round the declarer
leads from the boar and plays the A from his hand. 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 AKJTx
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.
--
Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.]