3-card endings (or 4)

[ThorvaldAagaard]

BEN has a problem in the 3-cards endings and with very few cards left it should be possible to find the best line,

Before last tricks there are 2 unknown cards and they can be placed in 2 combinations (based on play, there might be only one), but the last trick is not interesting, so lets see about trick 12.

There are 4 unknown cards and they might be placed in 6 combinations (at most).

At trick 11 there is 6 unknown cards = 20 combinarions

At trick 10 there is 8 unknown cards = 70 combinations.

With the current setup for sampling BEN will find these combinations and calculate the odds for each card, but the problem occurs in this situation:

Declarer; A trick and XX in a suit
Dummy: AQX in the suit

The opponents hold Kxx in the suit together, and 3 uninteresting cards.

Now when declarer takes his trick the clever defenders will throw a small in the suit if possible, so when playing the suit and next hand follows, the odds have changed, as there now are odds for dropping the king instead of the finesse.

This is because we are calculating new double dummy result after each played card.

So it would be interesting if we instead of using double dummy could plan how the last 3 (4) tricks should be played.

[lorserker]

i don't follow why the defenders discard from the suit rather than throwing the uninteresting cards

[ThorvaldAagaard]

The problem is there is no memory of played cards.

You probably know "8 ever - 9 never" about finessing for the queen.

So when you have 11 cards in a suit missing the King, you will play for the drop if no other clues, based on vacant places.

So when you are down to xx opposite AQ it is number of remaining cards in that suit, that matters. If 4 left, the finesse can win but never lose. If the opponents have Kxx it is 50-50 but when the opponents have only Kx we are down to the 11 card problem, and play for the drop gives double dummy the best chances.

So as opponent you should discard the x if Kxx outstanding to give the robot better chances of going wrong.
This flaw is also present in GIB, where I have seen it drop a finesse it had taken once.

As I see it double dummy is dropping a priori odds, and for the same reasons restricted choice is also absent.

[lorserker]

thanks. i will think about this more

[ThePokerDude]

Yes, your example shows the weakness pretty well.
If you can discard a small singleton of the relevant suit sitting behind AQ you could "fool" the engine because
it does not eliminate the option that you had originally Kx.

Of course discarding x from Kx is clever if you need to score two tricks to set the contract.

Let's look at the two card ending with declarer having x and dummy AQ and the opponents K32 and one irrelevant card in other suit (OS), so the suit was not distributed 3-0.
east west distribution could be.
K3 - 2 + OS
K2 - 3 + OS
32 - K + OS
K + OS - 32
2 + OS - K3
3 + OS - K2

Now if declarer plays low and east the 2 the possible options left are:

a) K - 3 + OS
b) 3 - K + OS
c) OS - K3

in case of a) you should finesse, in case of b) play for the drop in case of c) does not matter.
But because of restricted choice the probability of a) is twice as high as of b), so playing for the finesse is much better.

As you stated: without a memory of played cards you cannot eliminate (or give it a low probability for) the option of somebody discarding x from Kx behind AQ and you cannot adjust probabilities because of restricted choice.

[ThorvaldAagaard]

I am working on implementing a new play engine (https://www.lamsade.dauphine.fr/~cazenave/papers/AlphaMu.pdf), and is looking forward to be able to test this deal :-)

[ThePokerDude]

The paper is pretty nice. Ther is a PhD thesis about computer bridge algorithms where the paper you mentioned is cited.

https://www.proquest.com/openview/b318635a9db6cd0ec234cfaa05635a28/1?pq-origsite=gscholar&cbl=18750&diss=y