Question on game theory...

[mvdgaag]

The latter of which?

[ratel]

1) "People who never heard of game theory" versus (2) "people who have trouble putting it into practice".

[AaronBrown]

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Nice post Aaron. One comment: You can use game theory for bluffing. You get the same EV if your opponent folds or calls by giving him the same odds against your bluffing as the pot odds. This is a game theoretical approach and it does not matter if your opponent makes rational or random decisions! Therefore you can use game theory to find an optimal strategy against random uncertainties.

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Thanks for the kind words. I agree that the game theoretic analysis of bluffing is very important for understanding poker. But it's also important to distinguish between game theory and equilibrium analysis. In simple symmetric zero-sum problems, it is true that the optimal game theory strategy does equally well regardless of the other player's strategy. But that's a dangerous insight because:

(1) It does not do equally well regardless of the other player's action, only his strategy, and only in the long run. In real poker, that's an essential distinction.

(2) Taking the same point in reverse, it may be an optimal strategy for you, without being an optimal action. In general, I think game theory provides a reasonably good analysis of how frequently to bluff, but does a terrible job of telling you when to bluff.

(3) There is more to poker than simple, zero-sum logic. Each decision is just one step in the hand, which is just one step in the game, which is just one step in a lifetime of winning poker. If you're goal is a lifetime of winning poker, don't settle for actions that are optimal for one betting round of one hand in one game.

[mvdgaag]

Thanks for the explanation. I'm still a bit confused though.

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(1) It does not do equally well regardless of the other player's action, only his strategy, and only in the long run. In real poker, that's an essential distinction.

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Aren't calling/folding his actions? The method described in TTOP makes sure you have the same EV for these actions. That they are part of a strategy does not matter in this case. I can see how it does if it affects play on later streets (see point 3).

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(2) Taking the same point in reverse, it may be an optimal strategy for you, without being an optimal action. In general, I think game theory provides a reasonably good analysis of how frequently to bluff, but does a terrible job of telling you when to bluff.

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I agree. In holdem it hardly works, because your opponents also see the board. In draw poker you can pick good spots to bluff, but randomizing your bluffs assures your opponent cannot get onto you. Therefore I also think in holdem Sklansky's game theoretic approach does a terrible job of telling you WHEN to bluff. Unfortunately there is no example of using game theory to bluff in holdem in TTOP.

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(3) There is more to poker than simple, zero-sum logic. Each decision is just one step in the hand, which is just one step in the game, which is just one step in a lifetime of winning poker. If you're goal is a lifetime of winning poker, don't settle for actions that are optimal for one betting round of one hand in one game.

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I see. Decisions on one street influence the ones on the other streets a lot. On the other hand proper EV calculations do take all streets into account (the pot for former streets and implied odds for the ones to come). Therefore I think an EV calculation that uses a distribution of actions and hands is not as ignorant as you state. And this optimal bluffing frequency was found using a simple EV calculation.
These calculations should be done from the opponents point of view.... For example he could think or even know: "This action could mean a bluff, a set, a big draw". Now if you wouldn't bluff you would have less EV. Same if you always bluff. Now the calculation should determine what the optimal bluffing frequency would be when this situation comes up. Probably it would have been another thing if you raised preflop, but since you just called you're probably on a set/draw.
Therefore this is part of a bigger plan not for just one hand or one street, but for overall strategy. There is not one optimal bluffing frequency, but there are more for different histories of the hand and board textures (it should be believable). When to bluff is a different thing, but you could answer the question: In an optimal strategy, when I've been chosing actions that could represent a flushdraw, how often should I bluff when the third flush card falls?

Reading the last part again I think there are so many variables that one needs to come up with so many different optimal frequencies that there is no point to it. I could erase the last part but I'll leave it there, because other people might be helped by it. Sure helped me [img]/images/graemlins/smile.gif[/img] Thanks

[jackaaron]

Okay, can we assume there are general conditions where a bluff could occur?

And, if so, can we also assume that the more often we take advantage of those situations in one game with the same opponents, the less often we will have those general conditions? (less often being undefined right now)

If we can make both of these assumptions, then it would seem we could make a game theoretical model that would allow for an optimal bluffing frequency (although it would temporarily break down when new players are introduced).

This is probably naïve, and is really just a question more than a suggestion.

[SplawnDarts]

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Okay, can we assume there are general conditions where a bluff could occur?

And, if so, can we also assume that the more often we take advantage of those situations in one game with the same opponents, the less often we will have those general conditions? (less often being undefined right now)

If we can make both of these assumptions, then it would seem we could make a game theoretical model that would allow for an optimal bluffing frequency (although it would temporarily break down when new players are introduced).

This is probably naïve, and is really just a question more than a suggestion.

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I think your assumptions are vaid. And I think a model could be made. However, under most circumstances it couldn't be solved for equilibrium without additional simplifying assumptions (such as those in the ToP bluffing on the end model).

Furthermore, even if an equilibrium solution was found, it's not usually what you want to be playing.

[jackaaron]

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Furthermore, even if an equilibrium solution was found, it's not usually what you want to be playing.

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Meaning you would probably want to be more aggressive (than the solution) if in a tournament setting near the bubble with a large stack (for example)?

[SplawnDarts]

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Furthermore, even if an equilibrium solution was found, it's not usually what you want to be playing.

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Meaning you would probably want to be more aggressive (than the solution) if in a tournament setting near the bubble with a large stack (for example)?

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I'm not even thinking those sorts of specifics.

Nash Equilibriums are found by assuming perfect opposition, or more to the point an opposition capable of maximally exploiting any mistakes you make. As such, it tends to suggest strategies that render your opponent's decisions meaningless ie. whatever they do, they can't exploit your play.

Real opponents aren't like that, and the maximally EV line against a given real world opponent is rarely the one recommended by game theory.

[dfan]

To make the discussion more concrete/basic, let's set up a scenario. You are considering a bluff when a scare card comes on the river. It is limit and there are 3 bets in the pot, so you are getting 3 to 1 if your bluff works. If you have an idea of how often your opponent will fold to a bluff, it is easy to determine your best strategy.

Opponents Strategy // Your Best Strategy:

Never fold // Never bluff
Fold 20% of the time // Never bluff
Fold 25% of the time // Doesn't matter
Fold 30% of the time // Always bluff
Always fold // Always bluff

So if you can estimate reliably the fixed probability that your opponent will fold to a bluff in this situation, your decision should be based on simple math, not game theory: if his fold proportion is above 1/(pot odds +1) then bluff. Otherwise don't.

However if you assume that your opponent's folding % is NOT FIXED and changes over time as he intuitively or otherwise adjusts to your bluffing % then game theory becomes very relevant. It says that if a bet from you in this situation is a bluff at any frequency other than 1/5th of the time, your opponent will have available a calling frequency that will result in less money for you than if you had bluffed 1/5th of the time. (It is 1/5th because when you bet 1, you are giving your opponent 4 to 1 odds to call.) So bluffing 1/5th of the time is unexploitable and any other frequency is exploitable.

But the game-theory determined bluff frequency is the optimal frequency only if you assume your opponent will also adjust optimally to your play. If you believe that your opponent will not quite adjust enough, say maybe continue to fold too much, then you should bluff more often than the unexploitable rate. And vice versa. So you probably want to use approximately the unexploitable bluff frequency vs very good opponents that you will play with for a long time. But if there is not going to be this cat-and-mouse game where your opponent is making intelligent adjustments to your play then the decision to bluff or not bluff should be simply based on pot odds and your estimate of the probability that your opponent will call in that particular case.

Of course this discussion ignores the extra complications introduced by metahand considerations such as how your bluffing frequency affects how much you get paid when you do hit, etc. And I should also say I'm no game theory expert so take the above with a grain of salt or at least watch for corrections that are sure to come if I screwed it up.

[Shroomy]

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Furthermore, even if an equilibrium solution was found, it's not usually what you want to be playing.

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Meaning you would probably want to be more aggressive (than the solution) if in a tournament setting near the bubble with a large stack (for example)?

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I'm not even thinking those sorts of specifics.

Nash Equilibriums are found by assuming perfect opposition, or more to the point an opposition capable of maximally exploiting any mistakes you make. As such, it tends to suggest strategies that render your opponent's decisions meaningless ie. whatever they do, they can't exploit your play.

Real opponents aren't like that, and the maximally EV line against a given real world opponent is rarely the one recommended by game theory.

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I totally agree with your post, but I think that you should know what the optimal strategy (roughly) is, and when you deviate from it (which is usually) you should have a reason for doing so, typically to exploit a situation or mistake by one or more of your opponents.

And for us mere mortals it is wonderful to know that you can retreat into "game theory" mode and not be exploited to heavily by a superior opponent.


btw .. what is and what is not game theory is debatable.