Game Theory and Poker

[JaredL]

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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.

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There are equilibrium refinements that rely on players assuming that other players are rational. One of them involves forward induction, which conveniently is discussed in my next article currently up. For the standard Nash Equilibrium that's not the case. When you show that whatever strategy n-tuple you've found is an equilibrium you start by assuming that players 1 through n-1 play their equilibrium strategy and show that a best response for n is her equilibrium strategy. When you assume that 1-(n-1) are playing their equilibrium strategy, you don't assume anything about rationality, just that they play their equilibrium strategy. Rationality comes into play only when assessing whether or not n's strategy is a best response.

It's like that in poker as well. You assume that your opponents are playing some strategy and then if you are Ray Zee you will always play a best response to what they are playing. In solving games you put yourself in player n's shoes looking at what the other players are doing and respond accordingly. In poker you actually are that player n. The only difference is that in poker the opponents aren't rational and won't be playing a best response to what you're doing. That doesn't affect the analysis.

You are correct that it's not technically a game theory problem because our opponents aren't rational, but we don't care about them because we are (attempting to be) rational. The thought process is identical.

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edit #2: just want to clarify from the first paragraph that eventually you will need to assume that players 1 through n-1 are rational but only one at a time when you are showing that their NE strategy is a best response to the other players' strategies. So you basically assume one at a time that they are rational. In poker we're always trying to be the rational one which is why the thinking is so similar.