Game Theory and Poker

[sjb]

Good show. A lot of weird "game theory" claims do show up. In support of the section "Fallacious Concept 4: An Equlibrium Strategy is a Good Strategy", let me point out that, 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.

In any given hand, the EV would vary by position. In HE the highest EV would be on the button and decreases in earlier positions. But since each player eventually sits in each seat an equal number of times, those all average out.

The EV of the equilibrium strategy is the expectation when everyone else is playing their corresponding strategy in the equilibrium. If all players are playing optimally, then a non-zero EV comes from asymmetries in the game. In poker, that asymmetry is position. But since the button rotates, as you go from a single hand to repeated hands, that asymmetry is destroyed and the result is that everyone breaks even.

Maybe the most problematic thing about using game theory in poker is that game theory doesn't tell you how to win. Because game theory assumes all players are "rational" (that is, they play so as to maximize utility - in poker that means they play to make the most money possible) and "intelligent" (that is, they're at least as capable at analyzing the game as we are), game theory doesn't tell you how to win. It tells you how to not lose.

The EV of an equilibrium strategy represents the worst you could do. The best you could do depends on exactly how your opponents deviate from their equilibrium strategies. Game theory would still tell you how to analyze that game, but the opponent's strategy becomes part of the game and it has to be re-analyzed for each new set of opponents.

So, while game theory can make interesting contributions to poker theory, it's hardly the last word.

[Chipp Leider]

Actually I think the main problem is people thinking GT is somehow restricted to just optimal play, which is not at all the case.

[mustlearnpoker]

what is ev

[jogsxyz]

"Players are assumed to be rational:"

How did Lunsford come up with that? If players were rational, why bother to play? We would all lose the rake.
We hope opponents play badly, but predictably badly. And we hope to exploit their bad tendencies.

[JaredL]

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.

The paragraph about GT showing how not to lose versus how to win is good, as is the bit about opponents' strategies. The next edition will discuss how to use your opponents' strategies to put them on a hand range.

Thanks again for the feedback.

[JaredL]

[ QUOTE ]
Actually I think the main problem is people thinking GT is somehow restricted to just optimal play, which is not at all the case.

[/ QUOTE ]

GT results are based on the assumption that the players in the game are playing, in some sense, optimally. So it is restricted to optimal play.

If you're pointing out that you can use the same techniques to analyze games where players aren't using optimal strategies then we certainly agree, and that is the main reasoning behind the article series.

[JaredL]

[ QUOTE ]
"Players are assumed to be rational:"

How did Lunsford come up with that? If players were rational, why bother to play? We would all lose the rake.
We hope opponents play badly, but predictably badly. And we hope to exploit their bad tendencies.

[/ QUOTE ]

You are exactly right!

My point is that game theory results rely on the assumption that players are rational. This makes it much easier to analyze strategic interactions than using other assumptions for players' behavior and motivations. This, however, makes game theory results fairly worthless for helping you figure out the best move in an actual game against actual people.

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.

Jared

[StellarWind]

[ QUOTE ]
... game theory doesn't tell you how to win. It tells you how to not lose.

The EV of an equilibrium strategy represents the worst you could do. The best you could do depends on exactly how your opponents deviate from their equilibrium strategies.

[/ QUOTE ]
Actually an equilibrium strategy doesn't even accomplish the minimal goal of not losing when there are more than two players. If everyone follows the equilibrium strategy than the EV of each player across an entire orbit will be zero and you won't lose. But one player not following the equilibrium strategy can easily place another "innocent" player in an unavoidable negative EV situation.

There are many practical examples of this in poker. Suppose someone decides that good LHE strategy involves always raising/reraising preflop from the Button. You don't want to sit two seats to his left because he'll raise everytime you post the BB. Sure he's spewing chips but the whole table benefits from that and not just you. It doesn't make up for the fact that your BB equity is being redistributed to all the other players at the table.

Another problematic player is the small blind on your immediate right who always bets the flop into five people. Sure he's full of it but that doesn't mean you can put money in with four people behind you. You wind up folding a lot of hands you might have won.

Then there's the action player on your immediate left who straddles your big blind every orbit. He's killing your blind equity and selectively raising the effective stakes on deals where you are out-of-position.

The final example is collusion where your opponents are helping each other but not communicating with each other. This type of play does not violate the article's mathematical assumption of a noncooperative game because each player is acting independently. The player's intentions may be bad but his plays are just another possible strategy. This is why people can be concerned about a husband and wife playing together even if they are absolutely certain they are not signaling hole cards or betting plans. In fact it only takes one spouse to cheat and the other may not even realize what is happening.

[jogsxyz]

JaredL,

I just noticed that you are the mod for this section. Are you
familiar with the 3-card deck game. It has been discussed on
2+2. I lack the ability to navigate the 2+2 forum. Don't know
if the solutions are posted within the forum. I have posted
the solution to the game on my blog. Both the special case and
the general case are detailed in the blog.

http://jogsxyz.blogspot.com/

jogs

[jogsxyz]

[ QUOTE ]

Fallacious Concept 1: Game theory is very useful for poker players

Strange that I would write this in a poker article about game
theory, but this is a misnomer. Other than a few select examples
such as bluffing described in The Theory of Poker, game theory is
generally not practically useful. Even these examples are of
little use (see Fallacious Concept 4 below). One of the reasons
for this I alluded to above. The assumption that agents are
rational is clearly not valid when you are playing poker. If that
were the case the Noted Poker Authority would certainly have far
less to write about. Players seem to bet, raise, and call for no
good reason all the time.

[/ 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.
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.