Basic TD Math - or - A Pat Jack Is Not Always Favored Over 1 Draw

[fnord_too]

At lunch today I decided to begin analyzing the old HU with one draw situation. (I know, I haven't finished the c/r, lead, or c/c EV analysis for a pat 9 with one draw to go, opponent drew one scenario yet. I figured I'd step back to the simplest situation.)

So, I have two flavors of pat hands here, smooth and rough. The smooths are all 2356X where X is 8, 9 , T, J, or Q. The roughs are 34578, and 3478X where X is 9, T, J, or Q.

Assumption: 38 card stub, no needed cards burned beyond those in pat hands hand.

This chart will show the drawing hand, then its odds of beating the smooth and rough versions of each hand.

2347x
8 06/38 10/38
9 10/38 14/38
T 14/38 18/38
J 18/38 22/38
Q 22/38 26/38

2345x
8 07/38 06/38 (!)
9 11/38 09/38 (!)
T 15/38 13/38 (!)
J 19/38 17/38 (!)
Q 23/38 21/38 (!)

2368x
8 04/38* 06/38 *3/38 to tie
9 11/38 13/38
T 15/38 17/38
J 19/38 21/38
Q 23/38 25/38

2378x
8 00/38 06/38
9 10/38 14/38
T 14/38 18/38
J 18/38 22/38
Q 22/38 26/38


Notes:
I may have miscounted in some spots, but I don't think I did.
The next step is to figure what the best play for pat hands against one card draws. For instance, even though a pat Q is not favored against any of these 1 card draws, it may be more EV to keep it than to break it. I know how to do this calculation, but it is tedious and I did not have time at lunch.

The (!) s are there because it is better to have a rough hand against someone drawing to a 2345, this is because you kill three of their outs with a rough hand where as you only kill one with a smooth hand.

[randomstumbl]

As an interesting note, in the NL 2-7 WSOP episode, Barry Greenstein stays pat with a queen at one point when his opponent draws 1. I was always curious why he did that and figured that either the rule of a jack being the line where you're ahead of a one card draw wasn't always right or he just figured the other guy was drawing to an inside straight or something.

I'll try to look up what the exact hands where, but it might take awhile to download that ep. again.

[fnord_too]

[ QUOTE ]
As an interesting note, in the NL 2-7 WSOP episode, Barry Greenstein stays pat with a queen at one point when his opponent draws 1. I was always curious why he did that and figured that either the rule of a jack being the line where you're ahead of a one card draw wasn't always right or he just figured the other guy was drawing to an inside straight or something.

I'll try to look up what the exact hands where, but it might take awhile to download that ep. again.

[/ QUOTE ]

Just because a Q is not favored over a one card draw, it may be at less of a disadvantage than drawing one. That is the second part of the analysis I alluded to, and is a real pain to do by hand. It's kind of like there are times when you are a favorite if you stay pat but more of a favorite if you draw one. (I think there are cases of this, I would have to go back through Weisenberg's articles at Card Player, I think he talks about this situation some).

[randomstumbl]

Oh yeah, that makes perfect sense. Don't know why I didn't think about that.

[Phat Mack]

You've done some nice work here, but I'm curious about this:

[ QUOTE ]
Assumption: 38 card stub, no needed cards burned beyond those in pat hands hand.

[/ QUOTE ]

Are you assuming that Pat Hand has been pat from the beginning, and that Draw has drawn 2 cards twice? If so, I can accept the assumption, but if both hands have drawn one card twice, then I have to assume the possibility that Pat Hand occasionally has mucked a pairing card from his hand that Draw needed.

To construct an example, in 2347x vs. 2356J, the latter hand could presumably be tossing any of 12 A's, K's or Q's; any of 4 2's or 3's; or any of 6 5's or 6's. So as an egregious simplification, the chances are 6/22 that one of his tosses contained an out for 2347x.

This is a miniscule nit, I realize, but might be of interest in making close calls.

[fnord_too]

Yeah, the 38 card, none burned isn't too realistic. I think 36 or 37, 1 or 2 burned is better, but I had already done the chart before I thought about it. It would probably be like 2 - 1, pat 1 or 1 - 1, 1 -1. I don't think changing that assumption will change the percentages too much.

I really just needed something to get numbers. I am interested to see how breaking a queen against a one card draw compares to staying pat. That's a situation that comes up all the time, you opponent draws one and you have something like a queen high.

Also, these sorts of figures can help us come up with better river bluffing frequency. I think TD is one of the easiest games to apply game theory to. I am pretty sure that the play of hands is much richer than people think. Eventually, I will write code for analysis and write little TD agents to try out different strategies against each other.

[Phat Mack]

Eventually, I will write code for analysis and write little TD agents to try out different strategies against each other.

That would be very interesting.

[dibbs]

Thank you very much for this, I've actually been thinking about this situatuon a lot lately.

[Soviet Exile]

I'd like to watch this episode as well. I heard it's good. Is there a torrent somewhere?

[randomstumbl]

I believe it's still on ESPN insider or something like that. I originally watched it on their website for free; you might have to pay for access now.

As far as torrents go, good luck trying to find a year old TV show that never actually aired. A friend of mine has it on his FTP server, but I doubt he'd appreciate me giving that out.

[lucas9000]

[ QUOTE ]
I believe it's still on ESPN insider or something like that. I originally watched it on their website for free; you might have to pay for access now.

As far as torrents go, good luck trying to find a year old TV show that never actually aired. A friend of mine has it on his FTP server, but I doubt he'd appreciate me giving that out.

[/ QUOTE ]

can you ask him? /images/graemlins/smile.gif i've never seen it and i'd love to grab a copy.

[Soviet Exile]

Thanks. I think that it was available for free one weekend, but now one has to pay. If you find it anywhere, let me know.