Tuesday, August 4, 2009

Favorite card combination follow-up

Yesterday I posted a card combination essentially similar to this one:



It appears that there's nothing to do here except hope for a 3-2 split. But say you start with an honor from dummy, what is RHO supposed to play holding J9xx? From his perspective, you'll have the ten 2/3 3/4 of the time, and if he never falsecards with the 9, you'll be able to pick up all his stiff 9s (when you have the T).

In fact, RHO should drop the 9 1/3 of the time. When you hold the T, you can do no better expectation-wise than to fall for this falsecard and he'll gain a trick.

But, if LHO has the T, you will score an "impossible" trick.

Note that leading from your hand can't work: you must force RHO to play before LHO.

But, this layout also creates a great opportunity for a defensive Grosvenor Coup: you cash the K and see the 9 on your right and ten on your left. But when you play the Q next, RHO shows out!


  1. What is your experience with the actual rate at which someone will make a falsecard relative to what is theoretically correct? Assuming the player is aware of the falsecard situation. Obviously in most games most players are unaware of basic falsecards. (I think only a tiny percentage of players would know to play the 9 from J9xx even if they knew declarer had the Ten.)

    I tend to think that people who recognize the falsecard situation make it more often than is theoretically correct. (But, since I don't recognize nearly as many as I should, I could be way off.)

  2. It took me a while to realize that your conclusion is correct. Here’s a more verbose way of looking at it. It’s getting into Millenial Celebration territory…

    If declarer has ATxx and the defenders know this, then the purpose of East’s falsecard from J9xx is not to win a trick with that particular holding. Declarer should always cash the queen next, winning against J9xx and losing to the three times less likely stiff nine. The purpose is to protect the holding of the stiff nine. Otherwise, declarer could read the nine as being from J9 or 9 and could afford to cash the ace next, picking up all three combinations.

    Therefore East must falsecard at least a third of the time with J9xx. But if East only plays the nine from J9xx, J9 or 9, then declarer can afford to lead the second round from dummy, and if East plays low, to finesse the then. Therefore East must also mix it up from 9x and 9xx.

    I guess East can afford to falsecard all the time from all these holdings since he’s going to lose them anyway, but I haven’t verified that this doesn’t expose some other holding. If I’m right, then East can falsecard from J9xx anywhere between one third and all of the time.

    Now, if East knows that his partner has the ten, each play of the nine from J9xx loses that holding with no compensation. Not good.

    If East doesn’t know how has the ten, it follows that there’s no point in falsecarding more than one third of the time. It doesn’t gain anything if declarer has the ten (since the holding of the stiff nine is already adequately protected by this), but it loses whenever declarer has the ten.

    The remaining question is whether East should falsecard less than a third of the time because of the uncertainty over the location of the ten. If he “under-falsecards” and declarer has the ten (and declarer knows East’s strategy), then declarer will switch strategies and cash the ace at trick two. Declarer will gain against Jxxx-9 in the proportion that East under-falsecards.

    Meanwhile, when declarer doesn’t have the ten, declarer will lose against T-J9xx in the proportion that East under-falsecards.

    Each of these specific singletons are equally likely in their respective parallel universes. But declarer has the ten 3/4 of the time (not 2/3, I think). Therefore the federation of declarers in both parallel universes will gain on average if East under-falsecards. The conclusion is that East should indeed falsecard exactly one third of the time.


  3. thg: I'm not observant enough to have drawn any conclusions.

    Soren: exactly right. In particular, I agree on the layout above declarer has the T 3/4 of the time, not 2/3. (I moved the top honors around from the original post where declarer had 2 spots and 2 honors instead of 3 & 1.) This does not affect the optimal play, though.