[Cascade]

xcurt, on Apr 10 2010, 03:04 PM, said:

Mbodell, on Apr 9 2010, 08:44 PM, said:

xcurt, on Apr 9 2010, 05:30 PM, said:

I worked through most of the game theory of this, but it's too much to compute the exact Nash equilibrium because it's dependent on the falsecarding (unnecessarily pitching an honor) strategy of both East and West -- and they cannot both falsecard at the same time lest the DT become good.  Nonetheless, the conclusion that emerged was that against most mortal defenders who probably don't falsecard enough, if no honors appear before T12 the finesse is probably the better play at that point.
I know this sounds weird since if declarer closed his eyes and played for the drop he would get favorable odds, but that's the Monty Hall problem in a nutshell -- two people can get different odds calling for the same door, if they have different information at the decision point.

That seems wrong to me. The numbers where you don't know which diamonds they play favor the drop. The numbers where you know they didn't play an honor favor the drop even more strongly. Thus I think you have to play for the drop unless you are thinking they false card a lot (and even then it may still be right to drop).

A priori there are Choose[9,5] (126) ways to give West 5 of the {8 diamonds and the SJ}. He can't have 5 small diamonds (6 cases, the hand plays itself on a showup line), and he can't have {SJ, Dxxxx} (15 cases, East would have been forced to play an honor). The remaining 105 cases are 55:50 for the drop.

9C5 = 126

West's holdings

♦ xxxxx = 6 cases - squeeze

♦ Hxxxx = 2 (honours) x 6C4 = 30 cases - an honour showed from east with Hxx

♦ HHxxx = 6C3 = 20 cases (west might have false carded)

♠ J ♦ xxxx = 6C4 = 15 cases - east must have played an honour

♠ J ♦ Hxxx = 2 (honours) x 6C3 = 40 cases (east might have false carded)

♠ J ♦ HHxx = 6C2 = 15 cases - west must have played an honour.

Without any information the drop is:

70:50 on (eliminating the 6 cases where we made on a show up squeeze)

If they never false card then there are only two possibilities west HHxxx or west J Hxxx

and the drop is:

40:20 on

[Cascade]

Cascade, on Apr 9 2010, 12:38 AM, said:

Mbodell, on Apr 8 2010, 08:53 PM, said:

So I ran 2 simulations of 1,000,000 matching hands to get the numbers for what is best.  If you assume that you haven't seen either of the diamond honors then the drop wins in around 52.42% of the time [this is pretty close to the 11:10 number that xcurt calculated].  If you assume nothing about the diamond honors (which is obviously fine as both will not have dropped no matter how you arrange them so it isn't like not making this assumption is unrealistic as in none of these is the ♦T high) then the drop wins around 55.59% of the time [this is pretty close to the 5:4 number that I got once I corrected for the spade jack in my last post].

Does anyone want to tell us what was the full hand at the table?

I'm also including my deal code in case anyone sees an obvious mistake.

Not caring about the diamonds (~55.6%):

south is "KQT2 Q2 AQ2 AQ32"
north is "A43 AKJT3 T3 K54" [space]
sdev drop [space]
main { [space] 
 [space] [space]set cw [clubs west] [space] 
 [space] [space]set dw [diamonds west] [space] 
 [space] [space]set hw [hearts west] [space] 
 [space] [space]set sw [spades west]
 [space] [space]if {$cw!=4} { reject }
 [space] [space]if {$sw < 2 || $sw > 3} { reject }
 [space] [space]if {$hw != 2} { reject }
 [space] [space]if {$sw == 2 && [west has JS]} { reject }
 [space] [space]if {$sw == 3 && [east has JS]} { reject }
 [space] [space]accept 
} [space]

proc write_deal {} { [space] 
 [space] [space]variable drop
 [space] [space]if {[west has JS]} {
 [space] [space] [space] [space]drop add 1 [space] 
 [space] [space]} else {
 [space] [space] [space] [space]drop add 0 
 [space] [space]} 
# [space]formatter::write_deal 
# [space]puts stdout "================================="
} [space]

deal_finished { [space] 
 [space] [space]puts "drop works [drop average] percent of the time" 
}

Caring about the diamonds (~52.4%):

south is "KQT2 Q2 AQ2 AQ32"
north is "A43 AKJT3 T3 K54" [space]
sdev drop [space]
main { [space] 
 [space] [space]set cw [clubs west] [space] 
 [space] [space]set dw [diamonds west] [space] 
 [space] [space]set hw [hearts west] [space] 
 [space] [space]set sw [spades west]
 [space] [space]if {$cw!=4} { reject }
 [space] [space]if {$sw < 2 || $sw > 3} { reject }
 [space] [space]if {$hw != 2} { reject }
 [space] [space]if {$sw == 2 && [west has JS]} { reject }
 [space] [space]if {$sw == 3 && [east has JS]} { reject }
 [space] [space]if {$dw == 4 && [west has KD] && [west has JD]} { reject } 
 [space] [space]if {$dw == 5 && [east has KD] && [east has JD]} { reject }
 [space] [space]accept 
} [space]

proc write_deal {} { [space] 
 [space] [space]variable drop
 [space] [space]if {[west has JS]} {
 [space] [space] [space] [space]drop add 1 [space] 
 [space] [space]} else {
 [space] [space] [space] [space]drop add 0 
 [space] [space]} 
# [space]formatter::write_deal 
# [space]puts stdout "================================="
} [space]

deal_finished { [space] 
 [space] [space]puts "drop works [drop average] percent of the time" 
}

Don't you need this:

if {$dw == 5 && [east has KD] && [east has JD]} { reject }

in the first case.

Since if west has 5 diamonds and east has both honours the ♦10 is good.

And don't you need

if {$dw == 5 && ([east has KD] || [east has JD])} { reject }

in the second case.

Since if west has five diamonds then east will be forced to throw an honour to guard the spades.

The first case gave me 58.3%

The second case gave me 73.2%

Whoops I just noticed that in the second case we also need:

if {$dw == 4 && [east has KD] && [east has JD]} { reject }

Since both west with HHxx and west with xxxx will lead to a diamond honour being played.

Now in the first case

58.3% is close enough to 70/120 and

after correcting my simulation the second case gives

66.7% which is close enough to 40/60

[nige1]

Thank you, Cascade. Please would you explain how to run your code?

[Cascade]

nige1, on Apr 11 2010, 01:39 AM, said:

Thank you, Cascade. Please would you explain how to run your code?

That code is not mine.

It is using a program called Deal I think - you can google it.

I usually do my simulations with dealer.exe which is the same program that BBO uses for hand selection in the web version.

[gnasher]

It's available from here:
Thomas Andrew's Deal