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Old 03-01-2017, 03:24 AM   #1
Walra
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Nash equilibrium in HU AKQ poker. Theory question.

Hi. Quick theory question. Hopefully in the right place...

Let's assume that we know our opponent's range fold/call/raise range exactly. We know their strategy and they know ours. How would one figure out the "GTO" line to take with each of our hands? We want our range to be balanced such that our opponent cannot gain an advantage by changing strategies (a Nash equilibrium).

Solving non-gto (exploitative) is easy, just take the the line that is max +EV with each hand vs range. But that leads to A>B>C>A type counter strategies. I am having trouble figuring out what a "GTO" approach would be.

Consider a simple game with only the cards:
A,K,Q.

I would assume the "GTO" range for opening/defending AKQ would be some weird mix of raising Aces, Kings and folding Queens. However, I can't think of a reasonable way to figure out the equilibrium without using a computer/brute force.

Bit of a snooze-fest for people who don't find this stuff interesting, but input would be appreciated. Thanks!

P.s. I know Kuhn-like poker is already "solved". More interested in the actual process than results.
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Old 03-01-2017, 09:19 AM   #2
Joe Knott
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Re: Nash equilibrium in HU AKQ poker. Theory question.

I tried to solve this game (no removal effect, the input is the frequencies of dealt cards) and it's far from easy.

The game (both player can either bet or check, fold or call) has 4^3 * 3^3 = 1728 unique pure strategies. This can be represented by a matrix of strategy vs. strategy values (this is called a normal form game).

The first step is to eliminate strategies that are dominated by other strategies (obvious stuff like never fold an ace), this very easy to implement and reduces the game to like 1/15th of the original game.

The difficult part is to solve the rest, in fact solving any non-trivial normal form games that are larger than 2*2 requires some kind of linear programming. One of the most powerful algorithms is simplex algorithm, it's not very easy to implement, but there are multiple solvers online, so you can just plug the matrix here (http://levine.sscnet.ucla.edu/games/zerosum.htm).
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Old 03-01-2017, 08:50 PM   #3
Walra
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Re: Nash equilibrium in HU AKQ poker. Theory question.

Thx Joe makes alot of sense,
Better dust off the linear algebra books...
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