Badugi Dilemma

MarkGritter · #1 10-30-2007, 11:52 PM

So, if you pat he has about 8 outs (maybe 7) in 44 remaining cards, which is odds 4.5 to 1. 18% of the time he makes his hand. Let's say he gets an extra bet out of you every time. (He can't do quite that well if you play the game-theoretic optimal, but he definitely has the ex-showdown advantage when you are pat and he is in position.) So you earn 0.82 * p - 0.18 bets

Let's give him some crap like 762x. Again we'll vastly simplify stuff and assume 1 additional bet goes in on all the Badugi vs. Badugi cases and 0 bets otherwise. He has 10 outs to a badugi while you only have 9. (no Q!)

There are 1892 possible cases
In 61 of them you both make a Badugi but yours is better
In 29 of them his Badugi beats yours. (Total=90, good)
In 9*(43-10)=297 you make another Badugi but he doesn't
In 10*(43-9)=340 he makes a Badugi and you do not
The remaining 1165 cases you win with the best 3-card hand. (Check: does this make sense? (44-9)*(43-10) = 1155, close enough... there are 10 cases missing somewhere.)

So, you get 1165/1892 * p + 65/1892 * ( p + 1 ) + 29/1892 * (-1) + 297/1892 * p =
0.616 p + 0.034 p + 0.034 - 0.015 + 0.157 * p =
0.807p - 0.019

Given the assumptions above:
Patting =~ 0.82 * pot size - 0.18
Drawing =~ 0.81 * pot size - 0.02

This suggests drawing is slightly better unless the pot is very large. I think the 32A draw is actually worth more on the last round of betting than the above analysis suggests (even out of position) so I drawing is actually worth a bit more. But you'd need a game-theory solver to tell you how much it is really "worth".

Phat Mack · #2 10-31-2007, 03:19 AM

[ QUOTE ]
He has 10 outs to a badugi while you only have 9. (no Q!)

[/ QUOTE ]

I'm not sure how deeply we want to get into this, but the above is only true if they are drawing to the same suit; otherwise it should be 9 and 8. Correct? (Even if they draw to the same suit, the queen is gone, so it might be 9 and 9. hmmm...)

MarkGritter · #3 10-31-2007, 11:27 AM

[ QUOTE ]
[ QUOTE ]
He has 10 outs to a badugi while you only have 9. (no Q!)

[/ QUOTE ]

I'm not sure how deeply we want to get into this, but the above is only true if they are drawing to the same suit; otherwise it should be 9 and 8. Correct? (Even if they draw to the same suit, the queen is gone, so it might be 9 and 9. hmmm...)

[/ QUOTE ]

If Hero has 32A and Villain has 654, then, yes, either they are both drawing to the same suit or they each hold one of each other's outs.

If Hero has 32A and Villain has 732, though, then they can each have their full complement of outs, for example:

3c 2s Ad (drawing to hearts: 456789TJK)
7d 3h 2c (drawing to spades: A45689TJQK)

Phat Mack · #4 10-31-2007, 03:59 PM

Interesting. If they have at least one common card, and the opponent's common card is of the drawn-to suit, the blocker does not exist. Baysean analysis could tell me how often this happens, but I'm not sure I'll mess with it. [img]/images/graemlins/wink.gif[/img]

*TT* · #5 11-01-2007, 02:36 AM

Mark - good catch with the Q. Need to run my numbers again to see why you found dropping the Q is a favorite I cat imagine one out would change the results that much - but obviously it has.

Tilt_Monkey · #6 11-01-2007, 12:36 PM

[ QUOTE ]
In 29 of them his Badugi beats yours. (Total=90, good)

[/ QUOTE ]
I came up with 39 here - assuming that your opponent has
267 and you are both drawing to the same suit.

MarkGritter · #7 11-01-2007, 01:00 PM

Let's see... I was assuming different suits.

His badugi / your badugis that lose / running total
762A 89TJK 5
7632 89TJK 10
7642 89TJK 15
7652 89TJK 20
8762 9TJK 24
9762 TJK 27
T762 JK 29
J762 K 30
Q762 K 31
K762 none

Oops, now I ended up with 31. I don't see how to get to 39?

Phat Mack · #8 11-01-2007, 01:19 PM

[ QUOTE ]
I don't see how to get to 39?

[/ QUOTE ]

Maybe you were counting the Q.

Tilt_Monkey · #9 11-01-2007, 01:52 PM

[ QUOTE ]
Maybe you were counting the Q.

[/ QUOTE ]
Yes, I forgot about the Q. Duh.
Also if they were both drawing to the same suit they would both have 9 outs and the number in question would be 30
(Your last line: Q762 K 31 will be ommited)
Sorry for the confusion.

Tilt_Monkey · #10 11-01-2007, 02:56 PM

[ QUOTE ]
In 9*(43-10)=297 you make another Badugi but he doesn't
In 10*(43-9)=340 he makes a Badugi and you do not
The remaining 1165 cases you win with the best 3-card hand. (Check: does this make sense? (44-9)*(43-10) = 1155, close enough... there are 10 cases missing somewhere.)

[/ QUOTE ]

I could be thinking wrong about this, but I think that this ought to be:

9(44-10)=306 (44 cards in the deck - 10 cards that will give a badugi to the villain)
10(43-9)340

So you have 1892-90-306-340=1156

And then (44-10)*(43-9)=1156

I don't think this would make much difference in the overall conclussions though.