Consider the following situation:

Cash Game: Hero’s stack is 30. Single villain makes an all-in bet on the turn.

Pot = 13, Bet = 10, hero equity = 1/3.

What is the GTO strategy or is more information needed?

What would you do?

The only available options here are call or fold.

Assuming all prior actions that lead to this decision point are at least 0 EV and that there is no rake, then the EV of this decision for both options are:

Fold : 0

Call: 1/3*(23) - (2/3)*(10) = 1 unit

Since calling has positive EV and that is the maximum among all available options in this scenario, the GTO play is to call.


Call.

Is that a GTO determination? I would think so. Why EV=0 for prior actions? What about a mixed strategy?

Clearly the 33% equity is only an estimate and if villain is betting at GTO frequency to make you indifferent, does it matter what you do?

Clearly it’s a positive EV case but consider this. You are risking 1/3 of your stack (10 out of 30) for an expected gain of only 1 if your equity estimate is correct. So, might it not be better to wait for a better situation for a more positive gain?


Just trying to get some basic GTO discussion started.

GTO or exploit, that is the question (or one of the questions)

imho there is more information required to properly answer this and that is namely our preflop range, our current hand and the board if any. poker is a squential imperfect information game where you can't look at a sub-game in isolation and get a optimal result. Parallel sub-games effect how we should be playing in the current sub-game and vice versa even if we could never reach that sub-game from our current sub-game.

In this video they cover a concrete example of what I mean. The example starts at about 6:45 and ends about 17:00 or so in the video

2019 poker math added to 1934 poker math. It didn't negate some of it.

I am not sure what you mean by a GTO determination.

A Nash Equillibrium solution is the strategy set where no player can unilaterally improve his or her expectation by choosing a different strategy.

In the provided example if hero were to fold then that would be 0 EV (with the same assumptions as before).

If hero were to call it would improve hero's EV to 1 unit. Thus, folding can not be a part of the nash equillibrium strategy, becuase we could improve our EV by calling.

Now one could argue we don't know if villain's play is the most profitable so technically we don't know that we are in a true equillibrium strategy.


At least 0 EV. Basically you have to have arrived at this point so all of the decisions had to have an EV of greater than or equal to 0 at the point you made them, otherwise you would have just folded.

If you are in the BB or SB then your overall EV just has to be better than -1 or -.5 bb, respectively. I guess in theory that -EV could be spread out over different -EV decisions but having a situation where your average EV is say -1.5 and then your average EV on the next decision is so positive it erases your current negative descion seems highly unlikely or at least so rare it is inconsequential to consider it.

To put it another way if the average EV of your current decision is say -2 units now, it is very unlikely the summation of the average EV of future decisions for the rest of the hand is going to be positive or counter balance that -2 units.


You could mix if the EV of your decisions are equal, but in your scenario, on average, calling is strictly better than folding.


The parameters as stated don't create an indifference scenario for our calling hands.

I am also not sure what you mean by the equity is just an estimate? Do you mean vs individual hands we have a true amount of equity whose weighted average is 33?

Or do you mean we have range inaccuracies due to hidden information and so we can't accurately calculate equity?


Your risk isn't 10 it's 20/3 since you don't always lose all of your 10 dollar investment. But even though you lose 20/3 your reward compensates the risk of the investment and adds 1 unit to your stack on average.

You can't be at a Nash Equillibrium by avoiding strategies that are positive EV because you could always change to those strategies to increase your EV, so you were never at equillibrium by definition.

You definitely seem to know more than me on this subject but I disagree with you here.

I think you can find a GTO/equillibrium for the game as presented but I agree it could not be considered an equillibrium strategy for the entire game of poker.

basically in a real world situation you need to look at more than just equity estimations. equity estimations are important but only part of it. the main problem is: where did that estimation of 33% equity come from? If it came from OP estimating the villain's range then we've got a lot of issues. It is better if hero gives us his preflop range and the board. Assuming a balanced preflop and flop strategy on the part of the hero we can solve most turn and river situations(especially allins) without needing to know the opponent's range - just the estimated evs for actions(random roll outs are good for this). In that video I linked to previously they cover this in “Subgame solving” - https://youtu.be/2dX0lwaQRX0?t=2071 which starts at about 34:00. Note “unsafe subgame solving” vs “safe subgame solving”

to make the opponent indifferent to bluffing we need to call 1/(1+(bet/pot)) or in this case roughly the top 70% of our range sorted relative to the board. if hero's current hand falls within that top 70% he should call and if not then he should fold.

Thank you for the link I look forward to viewing that video.

My argument is as presented the information here can constitute a game that can be solved. It's basically a version of the 0-1 game from MoP where equity can be used as the part of the distribution your current hand beats.

Originally I thought that we could solve an equillibrium solution with just the information given, but agree that is not possible with the game as given because villain's shove is a subgame that can have implications in the larger 0-1 game.

I agree with your conclusion that neither th e information given or even a full solution to the 1 street 0-1 game I am talking about would be a solution to the full game of poker.

If we somehow know villain is playing his part of the NE strategy then we have enough information to know that this combo is a call at equilibrium because GTO strategies always take the highest EV action. However, we don't know villain is playing at equilibrium. For all we know we are at the literal bottom of our range and should be folding at equilibrium, but we have so much equity because villain is a bluff monkey.

I think a lot of the confusion here is just semantics. We have to decide what qualifies as a "GTO solution" and whether that's different than just a regular solution.

Maybe a clearer example of this is when we hold 22 and someone shoves preflop, and then accidentally exposes their AK. Calling is clearly the correct play as its just a straightforward EV calculation where we know we're at least 50%. However the Nash solution (prior to hole card exposure) would likely be having us fold our 22 (assuming its not a short-stacked tourney spot).

So does the hole card exposure change the GTO solution based on the new information? Or is the GTO solution just always the GTO solution, where the hole card exposure merely creates a subgame?

I think the poker community is entrenched enough with the idea that GTO solution = Nash solution (without range estimates or premises) that I prefer the latter framing.

So to answer OP's question the "GTO solution" would be unknown (as it usually is), but what you should do is obviously call, because that's just what the math forces as a regular EV solution.

Yeah, I think an important point is that GTO strategy doesn’t use range estimates or premises, at least not in the way we do for standard EV analysis. It presumably relies only on the information set of the particular hand – what do I have, what is the board, what are the stacks, what happened on previous streets, what is my position.

Player Characterization. Who cares. Previous Hands. Not interested. Folds Every Hand. So what, etc.

So, when I proposed a specific equity estimate I added an element that doesn’t exist for a GTO strategy. But, I would think a “GTO player” (one who tries to be GTO) almost always makes a card equity estimate in some form (and maybe an estimate of fold equity, if applicable), one of the many seeming GTO enigmas.

I disagree since you literally can not come up with an optimal solution to a squential imperfect information game by looking at single subgame in isolation. This has been well documented by a rather huge amount of formal academic research on the subject.

You don't really need card equity estimations to find the correct solution. That is actually what is known as "unsafe subgame solving". It is labeled unsafe for a reason. That reason is that you are making major assumptions about how the villian played in order to come up with that estimate. Most people incorrectly think that the best route is to try and devine some range based on the player's perceived playing habits and then shape their range around that but you'll rarely be very accurate because that naturally comes with a huge margin of error. The best route is actually to take a balanced approach and simply adjust things like value bet and bluff percentages slightly based on those perceived habits(e.g. they fold too often, call too often, or not enough, etc.).