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.

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).

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

You bet the A 100%
never bet K
and bet Q often enough to make villain indifferent to calling your bets with his K's, depending on pot odds.
If it's a pot sized bet you bet all A's and 1/2 of your Q's to get to a 2 to 1 value to bluff ratio since they're getting 2 to 1 to call.

The player with the K should fold it half and call half at random to stop you from abusing him.

This is only true in some (very specific and not very poker-like) AKQ games. (I think you are referring to half-street, 3 cards in the deck 1PSB poker?)

Now what changes when the out-of-position player doesn't necessarily have to check? Or if instead of 3 cards in the deck it's distribution vs. distribution and the betting player is not polar?

yup, sorry, thought that's the "AKQ game" we were talking about.

The only thing I could explain with any real confidence is how to determine frequencies and sizings with PvsBC distributions. It's very easy to just start with river bet size and frequencies and work your way back in simple PvsBC scenarios.

When ranges are closer, betting is done with a merged range, there are potential slow plays mixed into the bluff catchers range, etc, I couldn't honestly tell you how to arrive at optimal frequencies and bet sizes. Sorry.

I'll tell you that sizing should get smaller as the betting range is wider and smaller when more merged/larger when polarized.

I am pretty confident that we have to use less bluffs and a smaller size if we expect the player we are betting against to have slow plays mixed into his range.

I really don't know the exact process to arrive at an equilibrium for a real poker scenario..
put into a solver?

Nash equilibrium is a very theoritical concept that makes little sense except when AI play each other and even there, sometimes, it doesn't work.

Forget all about it.

While there might be merit to what you're saying, there's certainly not without any elaborating.