Pre-millenial post: 27 lowball research problems

[MarkGritter]

I thought of saving this for my 1000th post, but the list was ready and I had some downtime at work...

Research problems in lowball (TD2-7, Kansas City Lowball, or Badugi)

or

"27 challenging problems"

1. (TD) Nontrivial 1st-round hand equities, hot + cold. For example, if you push all-in with 23xxx, what are your chances of winning the pot vs. a player who calls with a range of two-card draws?

2. (Badugi) Nontrivial 1st-round hand equities, hot + cold.

3. (TD) Nontrivial 1st-round hand expected values. Is it actually profitable to play three-card draws from the button? From SB? This can be attacked from either a theoretical or a measurement perspective.

4. (TD) Game-theoretic optimal 3rd round play on the button. What is the appropriate mix of call + break, raise for value, raise to induce a break, fold, call and stand pat (freeze), etc.?

5. (TD or KCL) Quantify the increased fold equity from having paired cards first to enter.

6. (TD) Describe a strategy for beating 'Sheiky draw' (one player gets six cards before the first draw but must call all bets in the first two rounds.)

Alternately, quantify the increased equity from being the Sheiky.

7. (KCL) Develop an analogue of S-C numbers for SB vs BB matchups.

The S-C number for a Hold'em hand tells you when it "cannot be correct to fold" because if you push all-in with a stack smaller than the S-C number, you guarantee a profit even against a BB who plays perfectly.

8. (TD) Calculate S-C numbers for NL 2-7 triple draw.

9. (TD) Determine in what ways the optimal last-round play in triple draw diverges from the optimal play in the [0,1] game.

10. (TD) Quantify the negative value of drawing to a flush on the first draw (compared with a non-flush.)

11. (Badugi) Quantify the notion of a "median badugi". The following definitions are arranged in assumed increasing strength:
Dealt pat badugis
Badugis made after the 1st draw drawing to any badugi
Badugis made after the 2nd draw drawing to any badugi
Badugis made after the 3rd draw drawing to any badugi
Badugis made by discarding any card if a favorite to get a better replacement
Badugis made by drawing to a specific range (e.g., 8 or better)
Badugis made by drawing optimally against a known opponent or opponents

12. (Studugi) Quantify how likely you are to get a badugi in 7 cards. What is the median strength of this badugi?

13. (TD) Describe a game-theoretical snowing frequency on the first draw. (i.e, when you stand pat on the 1st draw you should have a mix of decent to strong pat hands and complete bluffs. How often should you be on a complete bluff?)

14. (TD) Describe a game-theoretical snowing frequency on the second draw.

15. (KCL) Describe a game-theoretical optimal solution for push-or-fold NL single draw 2-7.

16. (TD) Provide bounds on the net EV gain of being in position rather than out of position in a limit game.

17. (TD) Demonstrate or refute, using collected data, the "big pots are won by big hands" maxim for TD.

18. (TD) Does any non-pat triple draw starting hand or 2nd round hand benefit from having additional players in the pot?

19. (TD) Does observation of how often a player stands pat on the 2nd or 3rd draw provide a faster or more accurate estimate of his hand range than observing showdowns? Quantify this.

20. (TD) Describe the "expected" distribution of draws by round (i.e., number of cards) for a winning player. Describe how to exploit deviations from this distribution.

21. (TD) Describe TD statistics that correlate well with win rate. (i.e, what should "poker tracker TD" measure?)

22. (TD) Completely characterize all the "degenerate" cases in perfect-information last-round triple draw decisions.

For example, I described a case in "2-7 theory: strange and disturbing" where it is correct to draw to 29T instead of 289T when the deck contains just 6's and 8's. A degenerate case involves discarding a card other than the highest one(s) in your hand, or discarding two cards even though you would be drawing live drawing one card.

23. (KCL) Your arch-enemy offer you a deal. He will play no-limit single-draw lowball with you heads-up. You may have the button 100% of the time, but you will have to pay a bigger blind predraw. Note that if you pay the only blind it's a losing game for you since he can just wait for the nuts and shove. What ratio of small blind to big blind produces a fair game?

24. (Chinese Poker) Describe the optimal strategy for Chinese Poker with 2-7 rankings in the middle. (Bonus: Would the game be improved by some requirement about the strength of the 2-7 relative to front or back?)

25. (Chinese Poker - Badugi) Invent and solve a Chinese Poker variant using badugi rankings in the middle.

26. (TD) It seems likely that showdown hand strength improves as the game contains more drawing rounds. Come up with a model describing this relationship.

As an easy first step, determine (for a limit game) how many rounds of drawing make it unprofitable to draw to anything but the nuts. Is three rounds some sort of optimum?

27. (TD) Develop a bot capable of playing winning triple draw.

[randomstumbl]

So, I'm drinking and figured I'd donk answer questions rather than play cards. Boy do we ever think about games differently. I think most of these are better math (or CS) problems than they are poker questions.

For example, I rarely stand pat as a snow on the first round because people still pay off my good hands thinking I might have a 9 and I don't get much benefit when they do continue to call trying to outdraw my pat 9's. I have no idea how to even start trying to figure out what % of the time I would need to bluff in this spot to make it fair for my opponents to keep me honest.

3.) This clearly depends more on who you're playing, how they play and how many people have folded in front of you. Three card draws are obviously good HU. In a six handed game, they're still probably fine with position(especially considering how bad most people play against someone drawing 3 cards from a rich deck).

4.) You can't solve the Nash equilibrium for one player. You'd have to solve it for all players. (...is that even the right term in a manyplayer game...) Still, my point is that the other players' actions define your optimal strategy.

I am pretty sure you could end up winning a Nobel Prize or something if you figured out a way to "quickly" solve this.

6.) Without doing any math - there is no solution that doesn't involve collusion. Think about how much the extra cards help. The only way to beat the Shieky is to get a lot of money in early with hands stronger than the Shieky's average hand. The big money maker is when someone has an okay hand and the Sheiky has to draw 5 or 6 on the first round.

Here's another good question though. NL HU Shieky draw - what stack size makes the game fair for both players. Bonus question, PL and 1/2 pot limit are probably not fair. Is there some %PL game that would be fair with infinite stacks?

11.) Calculating this is crazy easy. The later parts you have to assume what your minimum starting hand is, but it's still crazy easy.

Here's how to figure out the average pat badugi. Write down every possible badugi on a bunch of seperate pieces of paper. Find the piece of paper in the exact middle of the stack (use a computer if you're a tree hugger). It's way rougher than you would think because of how many king high badugis there are.

I guess the very last subquestion is hard, but only because you have to calculate what the optimal draws are. Though really, the hard part is when to fold, not what to draw.

12.) I'm pretty sure this is something that would be easy to program if anyone ever played studugi or cared.

18.) So you're asking if 2347 is more profitable putting 1/6th of the money in the pot or 1/2. Really? We need pencil and paper?

19.) Snowing frequencies produce a lot of uncertainty.

20.) See 19 Also, you're assuming people draw the same against all opponents. That is clearly false for good players.

21.) It seems like I should have a good answer for this. I don't.

22.) You must really like knowledge. This can't have any practical applications, can it? Lets say you have 86543 and the only cards left in the deck are 23457 of at least two suits. And your opponent has 23458. You should draw 5! How is this important? Do you really think there's some completely insane situation where these "degenerate" cases allow you to make a different decision in a real game?

[MarkGritter]

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So, I'm drinking and figured I'd donk answer questions rather than play cards. Boy do we ever think about games differently. I think most of these are better math (or CS) problems than they are poker questions.

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Undoubtedly. But I think they also illustrate how little solid theoretical foundation the games have, or even practical understanding of what works and what doesn't work in the long run.

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3. (TD) Nontrivial 1st-round hand expected values. Is it actually profitable to play three-card draws from the button? From SB? This can be attacked from either a theoretical or a measurement perspective.


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3.) This clearly depends more on who you're playing, how they play and how many people have folded in front of you. Three card draws are obviously good HU. In a six handed game, they're still probably fine with position(especially considering how bad most people play against someone drawing 3 cards from a rich deck).


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This is the conventional wisdom. Does anybody have numbers proving it for their own history? Maybe they're actually losing hands. Maybe 72 is actually more profitable in the long run than something like 862. Or 8763.

Compare to Hold'em where we have a variety of sources of information about the relative real-world profitability of various starting hands, and a large community of people using a sophisticated database program to tally their own results over hundreds of thousands of hands.

For HE, you can give an answer or a range of anwswers. For TD, I'd be interested in finding somebody with any real idea what the answer is.

[MarkGritter]

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4. (TD) Game-theoretic optimal 3rd round play on the button. What is the appropriate mix of call + break, raise for value, raise to induce a break, fold, call and stand pat (freeze), etc.?


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4.) You can't solve the Nash equilibrium for one player. You'd have to solve it for all players. (...is that even the right term in a manyplayer game...) Still, my point is that the other players' actions define your optimal strategy.


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Actually the optimal strategy is found by making you indifferent to the other player's choices. [img]/images/graemlins/wink.gif[/img]

Even a two-player solution with known hands is challenging but not, I think, out of reach.

[MarkGritter]

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11.) Calculating this is crazy easy....

I guess the very last subquestion is hard, but only because you have to calculate what the optimal draws are. Though really, the hard part is when to fold, not what to draw.

12.) I'm pretty sure this is something that would be easy to program if anyone ever played studugi or cared.


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I had to include some easy ones. [img]/images/graemlins/wink.gif[/img]

The answer to the first question about pat Badugis is well known and appeared several times on this board.

Figuring out when you're a favorite to improve is at least a little nontrivial, as is working through how the hand distribution changes over three draws.

Solving face-up badugi as asked in the last subquestion is rather hard, but might be easier than the equivalent problem for 2-7.

Enumerating all the lows for Stud/8 turned out to be nontrivial, particularly when they are broken down according to which high hand gets made:

http://www.math.sfu.ca/~alspach/comp49.pdf

And equivalent look at studugi (with and without qualifier) would be interesting.

[MarkGritter]

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18.) So you're asking if 2347 is more profitable putting 1/6th of the money in the pot or 1/2. Really? We need pencil and paper?


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Maybe. The 2347 probably benefits from the 5th and 6th players entering the pot. Does it also benefit from the 3rd and 4th, or do they take away more equity than they bring?

Can we make this argument rigorous, or just hand-wave faster?

Obviously at some pot size the 2347 prefers to be heads-up on the 2nd or 3rd round. How big is that pot size?

If we can answer affirmatively for 2347, how about 2357? 2467? 2348?

Generalize. [img]/images/graemlins/wink.gif[/img]

[MarkGritter]

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22.) You must really like knowledge. This can't have any practical applications, can it? Lets say you have 86543 and the only cards left in the deck are 23457 of at least two suits. And your opponent has 23458. You should draw 5! How is this important? Do you really think there's some completely insane situation where these "degenerate" cases allow you to make a different decision in a real game?

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It is nearly completely impractical.

Say you want to solve "face-up" triple draw. (Which is what I'm trying to write an calculator to do.) Any rules you can apply to limit what possibilities get examined will speed up the search. Many of the likely rules fail due to degenerate cases like this. Completely characterizing these cases would allow the calculator to work faster by applying rules of thumb when they are known to be sound. So if you are interested in what 'perfect' TD drawing looks like an answer would have some utility.

(Notice that the Sklansky definition of "mistake" as in the FTOP implies making drawing decisions with perfect knowledge of your opponent's hand--- which lets you make inferences about what's leftin the deck. Maybe a different definition is appropriate for draw games.)

The answer might also be mathematically interesting in its own right, as a combinatorial problem. Particularly if there is some clever way of arriving at an answer.

[randomstumbl]

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3. (TD) Nontrivial 1st-round hand expected values. Is it actually profitable to play three-card draws from the button? From SB? This can be attacked from either a theoretical or a measurement perspective.


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3.) This clearly depends more on who you're playing, how they play and how many people have folded in front of you. Three card draws are obviously good HU. In a six handed game, they're still probably fine with position(especially considering how bad most people play against someone drawing 3 cards from a rich deck).


[/ QUOTE ]

This is the conventional wisdom. Does anybody have numbers proving it for their own history? Maybe they're actually losing hands. Maybe 72 is actually more profitable in the long run than something like 862. Or 8763.

Compare to Hold'em where we have a variety of sources of information about the relative real-world profitability of various starting hands, and a large community of people using a sophisticated database program to tally their own results over hundreds of thousands of hands.

For HE, you can give an answer or a range of anwswers. For TD, I'd be interested in finding somebody with any real idea what the answer is.

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I can confirm this with my memory, that's about it. My results are good enough that I'm pretty sure I would have noticed if this was a leak. Maybe in a couple months I'll have some actual statistics.

In general, I think that most starting hands in my range are going to run fairly close together in real world play. The places that I expect to see small differences would be between 865xx and 863xx. Again, hopefully in a couple months I'll have better information.

[randomstumbl]

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4. (TD) Game-theoretic optimal 3rd round play on the button. What is the appropriate mix of call + break, raise for value, raise to induce a break, fold, call and stand pat (freeze), etc.?


[/ QUOTE ]
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4.) You can't solve the Nash equilibrium for one player. You'd have to solve it for all players. (...is that even the right term in a manyplayer game...) Still, my point is that the other players' actions define your optimal strategy.


[/ QUOTE ]

Actually the optimal strategy is found by making you indifferent to the other player's choices. [img]/images/graemlins/wink.gif[/img]

Even a two-player solution with known hands is challenging but not, I think, out of reach.

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I am not sure what you're asking then. I can't imagine there's actually some unexploitable dominant strategy. Though, take that with a grain of salt, I'm wrong a lot with these things.

[randomstumbl]

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18.) So you're asking if 2347 is more profitable putting 1/6th of the money in the pot or 1/2. Really? We need pencil and paper?


[/ QUOTE ]

Maybe. The 2347 probably benefits from the 5th and 6th players entering the pot. Does it also benefit from the 3rd and 4th, or do they take away more equity than they bring?

Can we make this argument rigorous, or just hand-wave faster?

Obviously at some pot size the 2347 prefers to be heads-up on the 2nd or 3rd round. How big is that pot size?

If we can answer affirmatively for 2347, how about 2357? 2467? 2348?

Generalize. [img]/images/graemlins/wink.gif[/img]

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I suck at actual math. From a poker perspective, you're asking when to isolate and when to not isolate. I could answer that question, but it's just not that hard or interesting.