Game Theory and Holdem

[QTip]

I'm almost finished rereading TOP. I remember being somewhat fascinated by the concept of game theory before. However, as I'm thinking about it this time through, the idea doesn't seem that usable in holdem as compared to other games where less information is available to players.

I've never used the concept in a game before, so this is just thinking about things that probably aren't even practical at this stage in my poker life. This is most likely because I just play holdem at this point.

At any rate, in holdem, the fact that the final card is revealed to and used by all seems to make this concept almost unusable in holdem. I guess the way to put it is that information is so readily available in holdem that a good player is almost always better off using his judgement.

Does this make sense?

[Octopus]

No. An example. If I know you will always check behind on the river with overcards UI (and that you will always fire twice with them heads up,) then I can call profitably with stuff like bottom pair on the turn. If you will bet even a small fraction of the time, I can not (all depending on pot size, of course).

It is certainly true that in many cases, the correct mixed strategy is so close to always doing the same thing as to make the concept irrelevant. This is especially true against the worst players. But as you move up and spend your time playing against strong observant players, mixing your play will become required. True you can probably not make the necessary calculations over the table, but as with all things, thinking about them away from the table will give you good intuition in making the right choice. (Edit: if that is what you mean by judgement, then how can anyone disagree? /images/graemlins/smile.gif )

(Do you buy any of this?)

I been thinking a lot about that Game Theory chapter myself, and how to apply the concepts.

Here's one specific situation where I think it could work:

Coming into the River, you're on a flush draw, say spades, but there are also two of another suit on the board, say diamonds.

Depending on the pot odds, you pick a number of random diamond cards, add them to your draw, and bluff if they come out.

Of course, there are problems. Your opponent might have diamond flush himself.

I hope this thread stays alive, as I'm very interested in learning more about Game Theory as applied to Poker.

[Octopus]

More generally, the correct probabilities (for bluffing, etc.) depend on your opponent's estimation of what your hand range is (and probabilities thereof) and on how well he impliments his side of the game theory war. We are deep into read-based second order thinking, so it is hard to talk about in general terms.

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More generally, the correct probabilities (for bluffing, etc.) depend on your opponent's estimation of what your hand range

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That's why I believe that using random scare cards to bluff can work in Hold'm because they can conceivably make your hand without actually doing so.

[SA125]

I find it most applicable when playing h/u, somewhat applicable in playing your blind and least applicable to other positions in full ring play.

[CallMeIshmael]

Game theory is very very applicable to hold'em.


Poker can sort of be compared to chess in this situation. If anyone watches grandmaster level chess played now, and compare it to world class players from 100 years ago, they will notice a remarkable difference. Chess now is far more about not losing than it is about winning. That is, you present a strategy that isnt likely to lose, but not necessarily one that is likely to win. A non exploitable strategy.

(note: im not a very good chess player, and this statement is based on limited reading/knowledge. It could very easily be a mistake)


Poker, as of now, is still a game dominated by attacking others' mistakes (as chess was). That is, we look at a fish, who calls WAY too much, and we know to value bet the hell out of him. BUT, your strategy is exploitable. You are, in fact, value betting too frequently. The fish just isnt going to exploit it.

This is the difference between a play that is optimal, and one that maximizes your EV. The optimal play is the one that is least exploitable and the highest EV against an UNKNOWN/GENERAL, and the maximal one is the one that depends on your oponents tendancies, and the one that happens to be the highest EV against a PARTICULAR opponent.

The optimal play is in at a nash equilibrium. Its interesting how you get here.

Lets say, you are playing heads up (it applies to all games, but this is easier logistically).

1. I have a game plan X (that describes all of my plays, and therefore X would take a near eternity to fully explain)

2. I tell my opponent I will play X

3. My opponent makes his strategy Y that is the best response to X (that is, the best way he can play given my ranges at all points)

3. He tells me he is playing Y, and I make the best response to that, call it X2.

4. He knows im playing X2, and he makes Y2, etc...


This continues until we reach an equilibrium point. At this nash equilibrium, my strategy is the best response to his AND his is the best response to mine! As such, given a situation Z (ie. SB, dealt ATo), I will play the hand in the same manner as my opponent (note: this would, of course, involve mixed strategies). As such, our EVs are equal. They would both be -Rake/2.

Think about what that is saying: there exists a strategy, when your oppoenents play perfectly, you lose rake/2. This is the worst you can do.

Certainly, this is not overly necessary now, especially at the lower limits. But, as poker progresses, game theory will become more useful.


FWIW, Ive been working on some software recently to help me find these points. Ive been in contact with a professor at Cornell that is going to help me get some thoughts published. If I dont, Ill definitely post it here.

[admiralfluff]

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Think about what that is saying: there exists a strategy, when your oppoenents play perfectly, you lose rake/2. This is the worst you can do.


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But a game-theoretical optimal strategy is exploitable by your opponent playing imperfectly.

The game-theoretical optimal solution has an inherent "assumption" that it's opponent is playing a non-exploitable strategy. The opponent can then play an exploitable strategy without fear of having tendencies exploited. (If the Nash is assuming you will bluff with some optimal EV neutral frequency X, it's optimal play will include a bluff reraise of optimal frequency Y that makes your inital bet and subsequent call decision EV neutral. If you never bluff, your opponent will not pick up on this tendency. It will assume you are playing Nash optimally, and pay you off and bluff reraise you far too often)

.I don't have time to really get into now, but a game-theoretical solution is not the optimal in holdem, and can be beaten. This is because holdem is a game of incomplete information.

I do agree that as your opponents' skill increases, and the number of opponents you face decrease (you did not specify this, but I believe it to be true) correct game-theory applications to holdem multiply, and that as games get tougher more people will need to correctly apply game theory to beat them.

[CallMeIshmael]

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But a game-theoretical optimal strategy is exploitable by your opponent playing imperfectly.

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By definition it is not.

This is the concept of the Nash equilibrium.

more info here: http://en.wikipedia.org/wiki/Nash_equilibrium

Unilateral deviation by your opponent from perfect play (at which you lose rake/2), means you make more than -rake/2.


EDIT: I believe you are confusing maximal and optimal. You might not make the most you can if you play the game theoretic solution, but you guarentee you can do no worse than -rake/2. (keep in mind you told your opponent your strategy, and his best response was YN (no idea what N is), and that causes both to lose rake/2.

[admiralfluff]

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I believe you are confusing maximal and optimal

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I am not, but it is possible I am wrong.

I have loosely used the term "Nash Equilibrium," as I do not believe that a stable NE exists for holdem games. If a game-theoretical optimal approach is taken, the strategy makes it impossible for the opponent to win money ASSUMING that the opponent is rationally optimizing their strategy as well. If this assumption is false, the game-theoretical optimal solution can be subverted by an irrational opponent.

Have you read the U. Alberta papers on holdem AI? One (or more) of them touches on this.

[CallMeIshmael]

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If a game-theoretical optimal approach is taken, the strategy makes it impossible for the opponent to win money ASSUMING that the opponent is rationally optimizing their strategy as well.

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Strategy YN (his strategy in NE) isnt the act of a player optimizing his strategy. It is the mathematically determined highest EV strategy against X(N-1). This strategy pays him -rake/2.

Reflecting back, it probably isnt overly correct to use the term nash equilibrium. Because that requires both players to have optimized their strategy and play perfectly rationally, and this be in this equilibrium. This isnt what Im talking about. This is more the idea that a NE exists, and one player can play that strategy.

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Have you read the U. Alberta papers on holdem AI? One (or more) of them touches on this.

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No. Ive heard about them. Do you have a link?

[istewart]

Listen to what CMI has to say. He never misses class, FWIW.

[admiralfluff]

No. /images/graemlins/frown.gif I printed them out a while ago and don't have the link anymore. It's all University of Alberta stuff. Some titles:

Opponent Modeling in Poker

Approximating Game-Theoretical Optimal Strategies for Full-scale Poker

The Challenge of Poker

Darse Billings is 1st credit on all of these papers.

I may have to eat my words, as I just found this (http://www.cs.cmu.edu/~sandholm/RIHoldEm.ISD.aaai05proceedings.pdf). It claims to have solved a stable NE for a simplified version of holdem, which is pretty cool (for nerds like me).

I miss college and going to classes. I'm surprised at how retarded I've become in half a year. I'd be interested to hear more of your ideas.

[CallMeIshmael]

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Listen to what CMI has to say. He never misses class, FWIW.

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You are a cancer on twoplustwo.com.

[CallMeIshmael]

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I just found this (http://www.cs.cmu.edu/~sandholm/RIHoldEm.ISD.aaai05proceedings.pdf). It claims to have solved a stable NE for a simplified version of holdem, which is pretty cool (for nerds like me).

[/ QUOTE ]

I too am a nerd, and this is a really awesome article.

I have a midterm tomorrow, so I should go to bed. But, Ill definitely check out those other articles. Thanks!

[elmitchbo]

here's the alberta link if anyone is stil interested.

http://www.cs.ualberta.ca/~games/poker/

it's a great read.