Game Theory and Poker

[JaredL]

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I just noticed that you are the mod for this section. Are you
familiar with the 3-card deck game.

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I looked at what you wrote, seems interesting. I didn't thoroughly read it, probably will some time next week. As a homework exercise in a micro class we had to solve the 3 card game where the opponent has middle and the other player has high or low. The hi/lo guy can check for a showdown or bet. If he bets then the middle guy can call or fold. IIRC we did the version where the probability of having a high hand is p and low is 1-p, not just .5/.5.

Jared

[JaredL]

[ QUOTE ]

Most of the players are semi-rational. It's just that they may not
be on the same wave length as us on strategy. Are they consistent
and predictable? To some degree they are. At least we can make
assumptions which should produce better results than assuming
all actions by opponents are random.


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When I say players are not rational, I'm not saying that they play randomly just that they don't follow the game theoretic definition of rationality that is assumed.

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On earlier streets, streets before the river it's less game
theory doesn't apply than with our limited understanding of game
theory the simplistic models are not very useful for most game
situations. In the future we should be able to construct better
models to solve more game situations. But game theory will never
be able to quantify and assign proper values to opponents'
tendencies. That will be an exercise for each player.

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Not sure what you mean in the first sentence. As for the end, just to promote my next article, I will be discussing specifically how "to quantify and assign proper values to opponents' tendencies." This is an exercise for the player, but the best way is to use techniques from game theory.

Jared

[jogsxyz]

[ QUOTE ]
[ QUOTE ]

I just noticed that you are the mod for this section. Are you
familiar with the 3-card deck game.

[/ QUOTE ]

I looked at what you wrote, seems interesting. I didn't thoroughly read it, probably will some time next week. As a homework exercise in a micro class we had to solve the 3 card game where the opponent has middle and the other player has high or low. The hi/lo guy can check for a showdown or bet. If he bets then the middle guy can call or fold. IIRC we did the version where the probability of having a high hand is p and low is 1-p, not just .5/.5.

Jared

[/ QUOTE ]

I've also posted the solution for the general case. You may vary p, the bet size and the pot size.

[Whirly]

Just a minor whinge. I am not sure that the description of Nash Equilibrium is wrong in "A Beautiful Mind"? Isn't Nash supposing that if anyone were to alter their strategy and go for the Blonde then that would cause the others to similarly alter their strategy? Thus they are optimal if they stick with their own brunette! It's a kind of Prisoner's Dilemma situation.

[img]/images/graemlins/confused.gif[/img]

Whirly
Q. If I've got Aces, should I tell you to fold or raise?
A. Neither, just get me another beer and play the damn game.

[StellarWind]

[ QUOTE ]
Just a minor whinge. I am not sure that the description of Nash Equilibrium is wrong in "A Beautiful Mind"? Isn't Nash supposing that if anyone were to alter their strategy and go for the Blonde then that would cause the others to similarly alter their strategy? Thus they are optimal if they stick with their own brunette! It's a kind of Prisoner's Dilemma situation.

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As I understand the article, in a true Nash Equilibrium going after the blonde would not improve your results even if no one else altered their strategy. Obviously that isn't true if everyone else is pursuing a strategy of only courting brunettes.

Returning to poker, in an equilibrium situation bluffing more than the equilibrium amount cannot make you more money even if no one else compensates by calling more.

[JaredL]

[ QUOTE ]
Just a minor whinge. I am not sure that the description of Nash Equilibrium is wrong in "A Beautiful Mind"? Isn't Nash supposing that if anyone were to alter their strategy and go for the Blonde then that would cause the others to similarly alter their strategy? Thus they are optimal if they stick with their own brunette! It's a kind of Prisoner's Dilemma situation.

[/ QUOTE ]

A Nash equilibrium is a situation where given what your opponents are doing, you are acting optimally (and that's true for everyone). It doesn't matter how your opponents would respond to you deviating, they simply assume that you are doing whatever it is that you're doing. So in the example if any of the guys recognizes that the others are going after brunettes he should go after the blonde (I personally prefer the darker haired ladies but the guys in the movie like the blondes). It's possible, as you suggest, that the scenario where one individual guy is going after the blonde and the rest are after the brunettes is not a NE but that also doesn't affect the conclusion that all going after the brunettes is not a NE.

[sjb]

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Thank you for the feedback, I'm glad you liked the article.

You said:
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in all forms of poker I know of (certainly HE, Omaha, and Stud, and regardless of limit structure or high/low splits) and in the absence of a rake, the EV of any Nash equilibrium strategy is exactly zero. With a rake, the EV is negative and equal to the total amount of the rake, divided by the number of players.

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This isn't quite true. The EV (Expected Value, or average earn) of the Nash Equilibrium will be zero minus rake, but the EV of the NE strategies won't necesarily be. If the other players aren't playing their NE strategy then it could be different. In the bluffing example the bluffers EV doesn't depend on the other guy's strategy but in other cases it might. For example, the caller in the bluffing situation should call or fold with the right frequency to make the bettor indifferent between bluffing and checking. If the bettor changes her strategy to, for example, always betting then the caller is no longer breaking even with the Nash strategy and will have won a positive amount in expectation.

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Maybe I didn't state it clearly, but when I was talking about the EV of the equilibrium strategy being zero (less the rake), I was talking about the game-theoretic "value" of the game - when players deviate from the strategy dictated by the equilibrium, then of course it's possible (even likely) for the EV to go up. But then you're no longer talking about the EV of the equilibrium strategy - you're talking about the EV of some non-equilibrium. You're not in equilibrium unless both players are playing that way.

[sjb]

[ QUOTE ]
In poker other players are far from rational. The clearest example of this is a player calling on 7th street in Stud without being able to beat the other players board. A point I make in the article, and this will be expanded upon in the future, is that you can still use some of the techniques from game theory to help you determine the best play even if your opponents' are not playing optimally.

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Actually, game theory is about studying the strategies when all players are rational. If you're going to study scenarios where only one of the players is "rational", then you have to build a model of the supposed "irrational" behavior of the other players. Once you've done that, you don't need game theory anymore - you just need the (earlier) discipline that game theory was built on: decision theory.

Very much of the poker literature that discusses game theory is really discussing decision theory. When you see "assume he would raise with any high pair and call with any suited ace" or similar hypothetical scenarios, you've built a model of the other player, who's no longer a "player" in the game-theoretic sense, but now just a stochastic process that you've statistically modelled.

[jogsxyz]

There's more than one strategy available to the players. There's optimal strategy and exploitive strategy.

Let's say Bob plays a fixed strategy that's not optimal. Now Ann can adjust her strategy to exploit Bob's tendencies. This is still part of game theory.

Game theory is not a fixed strategy. Optimal strategy is a subset of game theory.

Bob plays a poor strategy. Ann can play an adjusted strategy to maximize on Bob's poor play. Then Ann can have +EV. The sum of Ann's EV plus Bob's EV plus the rake will still equal zero.

[JaredL]

[ QUOTE ]
[ QUOTE ]
Thank you for the feedback, I'm glad you liked the article.

You said:
[ QUOTE ]
in all forms of poker I know of (certainly HE, Omaha, and Stud, and regardless of limit structure or high/low splits) and in the absence of a rake, the EV of any Nash equilibrium strategy is exactly zero. With a rake, the EV is negative and equal to the total amount of the rake, divided by the number of players.

[/ QUOTE ]

This isn't quite true. The EV (Expected Value, or average earn) of the Nash Equilibrium will be zero minus rake, but the EV of the NE strategies won't necesarily be. If the other players aren't playing their NE strategy then it could be different. In the bluffing example the bluffers EV doesn't depend on the other guy's strategy but in other cases it might. For example, the caller in the bluffing situation should call or fold with the right frequency to make the bettor indifferent between bluffing and checking. If the bettor changes her strategy to, for example, always betting then the caller is no longer breaking even with the Nash strategy and will have won a positive amount in expectation.

[/ QUOTE ]

Maybe I didn't state it clearly, but when I was talking about the EV of the equilibrium strategy being zero (less the rake), I was talking about the game-theoretic "value" of the game - when players deviate from the strategy dictated by the equilibrium, then of course it's possible (even likely) for the EV to go up. But then you're no longer talking about the EV of the equilibrium strategy - you're talking about the EV of some non-equilibrium. You're not in equilibrium unless both players are playing that way.

[/ QUOTE ]

We're basically just arguing semantics. As I said, I think you're right you're just using the wrong terminology.

A strategy is a plan of action for all of the various scenarios that could come up in the game. A particular strategy is a/the (Nash) equilibrium. The EV of any strategy (including the NE strategy) depends on the strategy the other players are using. Hence the EV of the NE strategy will vary depending on what the other players are doing. So the game theoretic "value" of the game is the EV of the equilibrium strategy when the opponent plays her equilibrium strategy. There is nothing special about an equilibrium strategy other than that it leads to an equilibrium if the other players act accordingly. You can still look at its EV even if they don't.

Again, I think we're just splitting hairs on the definition of 'equilibrium' and 'equilibrium strategy'.