sure, they do come up, and to the extent they do this toy game can help inform our view of what good play would look like (given the assumptions hold true). i think its very important to note the limitations of the game though (esp wrt villain only holding true bluff catchers), and to be careful not to over-apply the concept. one thing to keep in mind is whether why/how we got to this river spot is consistent with the assumptions of the model or goal of being balanced; if we played prior streets assuming villain was over-folding, he might be left with a strong range otr (ie has some hands that beat our value hands), or there might be an exploitative strategy that is best given our profile of villain. also note that in the toy game villain's option to check-raise is irrelevant since he would rationally never choose to do so.

in the context of this toy game, no. hero will choose a strategy that maximizes his range's ev, which will use a bet size* that allows him as many bluffs as possible while keeping villain indifferent. essentially, the hands in his betting range will (on average) have an ev equal to the entire pot, and thus he wants to maximize the number of hands he can bet with (since all other bluffs will xb and lose). when going all-in, he can bluff with ((spr/(spr+1)) times the number of value hands) (ie hero's % of bluffs equals villains break even equity on a call). i think people check rivers mainly because they don't think the assumptions hold or because they expect villain to play irrationally (betting with pure bluff catchers, etc.).

*hero can also bet a larger size and villain will simply fold 100%, yielding the same result.

1) If V's line to this point, is overall satisfactorily GTO, then it doesn't matter what Hero does -- bet or not, bet/fold or not -- Hero's EV is at best 0.

2) If hero has knowledge of V's line, then unless V's line is actually GTO, there will be an exploitive line that Hero can employ which will be +ev relative to any GTO line taken by Hero.

By the way, "0EV" is a bit of a misnomer in my last post. When we're at the river, there's dead money in the pot already. This sub-game is not zero-sum; it's positive-sum. (Although counting the money paid to get to this sub-game, it is zero-sum.) So if Hero is last to act, knows GTO, and knows Villain is playing GTO, he should be able to calculate exactly what the EV is of checking behind. Then he always bets if the EV of betting against a GTO Villain is greater than the EV of checking behind against one, and probably randomizes between betting and checking if they're equal.

The problem tho is that it's not GTO to find yourself otr and decide to take a "GTO line" once there.

GTO isn't a sum of independent GTO lines street by street.

This to me seems to be an important point. In order for the villain to get to the river with just a bluff catcher, it implies that the game is being played for thin value. GTO just reduces the value to zero.

CMV's point on knowing the villain's range brings up another question. On-line, one can record and tabulate thousands to millions of hands to come up with precise range. Realistically, can a live player generate a near approximation of this same precise range? If so, how would they do it?

So I took a 4 day temp ban for trying to point out flaws in OP's reasoning.

I understand that the OP is a mathemetician and he is genuinely trying to help, but him saying learning GTO is a waste of time and wont maximize profits vs low stakes is extremely unfortunate. GTO underpins all of the math in poker. If you arent strongly rooted in the math, I think it is going to be hard to win.

I have a mathematics degree from a prestigious and competitive university and I have over 300K in online earnings, and I am currently beating 2/5 for $80/hour. Most people in my room know me as one of the best players.

When OP wrote the original thread 3 years ago, there were 3 pages of new threads per day. Traffic has now slowed to a crawl. There is under a handful of new threads. Meanwhile, live poker is growing. My room is full 24/7. You can"t have a serious discussion about poker math without mentioning GTO.

I am sorry but a primer on GTO has to be completely re- written but i want to clear up a common misconception, namely, the difference between "unexploitable" and GTO.

The whole game in the main form is zero sum, which naturally implies there are no dominating strategies. But there are dominating strategies in many of the subgames. For example, a 30bb shove over a raise and a call is unexploitable because it is +ev. There are many, many other dominating strategies, the most well known is the 3way flush draw (see Morton's theorem and implicit collusion).

But the full game is zero sum because both players will have the same opportuniy on future hands. So if i dont shove my AK as squeeze, I am losing money to the people who do.

GTO is not unexploitable. If I was prisoner A and I knew that Player B was a game theorist, I have a read that he will always choose to stay silent, and his GTO play is the worst play.

I am going to leave it at that for now and hope i dont get infracted for posting my opinion.

You did not take a temp ban for voicing your opinions.

You took a temp ban for being unable and/or unwilling to provide proof of your statements, which appear incorrect.

There is a difference.

There is and will never be a proof in yours or my lifetime. Everything anyone is saying about where these inflection points lie is just pure conjecture for 3+ player games. So saying you need to prove x is akin to saying don"t talk about it.

I can go through OPs arguments line by line and refute most of what he said but it would be much easier to simply have someone who believes that GTO exists for 3+ player games as it should for any zero sum imperfect information game. He doesn"t even believe it exists, so how is he qualified to be the authority on the subject. John Nash already proved they exist, so where is OPs proof that he is wrong?

What if B knows A is a game theory expert as well.

Then what is the correct, GTO, play?

This. In an interesting coincidence, while you were gone2+2 Magazine reposted an old article by David Sklanksy in which he shows a mathematical shortcut that falsified the point you were called out on in the other thread.

Also in that thread I told you that you either needed to accept your error there, provide evidence of your assertion with actual math, or stop talking about GTO. That requirement still exists, and you will be further infracted if you continue without either admitting your misunderstanding or providing mathematical support to show that it's not an error after all.

Heh, now you see why they call it the prisoner's dilemma

The premise of GTO is that neither player knows each others strategy. If you knew exactly what your opponents stategy was, you can adjust and play a different strategy. In poker we call this having a read. Our infornation is no longer imperfect!

But the GTO play in the prisoner's dilemma is to defect (talk). That is the only strategy that is unexploitable. You are truly indifferent to what your opponent does (well, you would like it if he stays silent, but no matter what he does, you cannot obtain the worst outcome).

Should have given you 4 days to get a basic understanding of GTO as it relates to poker, you can look at all of the times I quotes you and said "no" to figure out where you're lacking.

I'm skeptical of this, but if it's true then that's great! Should give you a leg up on finding a basic understanding of the issue at hand.

Might be more practical to look at this way, but this is incorrect





I do agree with you, to an extent, that GTO is a useful thing to study even for live poker though.

I don't really get what you mean by this. Since we're starting off after the cards are dealt this isn't necessarily true right?

I will give CMV a bit of credit for his analysis of 3way action. Nash says there are at least n- 1 optima for every player in the game, so that a 3 way pot has at least 2 and maybe more GTO solutions.

The biggest thing CMV missed in his first thread is what is actually being optimized. The word optimum has a precise mathematical meaning. Its the inflection point between 2 competing forces and its used in just about every scientific discipline. In astrophysics it could mean max power vs min weight in a rocket ship.

In poker, it means max profit with minimum variance. It is the point at which you gain the most profit with least risk. If you had an infinite bankroll, then studying GTO really would be a waste of time

Yes, sorry im writing fast a lot of the time so feel free to correct me. The prisoners dilemma is such a great little thought exercise. If you knew player b's strategy, you can adjust your strategy and obtain a better outcome. Thats the main point and how it relates to poker.

There are some inflection points that are somewhat easy to guess at. For example, it would take logarithmic computing time to calculate what a GTO opening frequency in a 9player game is over a 100k hand sample size.

But if you have played a lot of poker, you kmow what you can raise with and where, so if you played strictly GTO your VPIP would be somewhere around 16/12.

In the strictest sense, If i knew you were playing GTO, i essentially know your cards. As a game theorist, knowing your strategy = kmowing your cards. That is why no strategy in poker is unexploitable

What is it about GTO that makes people talk nonsense? I include myself in that as I was recently wrong about something, but holy hell.

This is literally the first sentence on the Wiki article for "Nash equilibrium":

No, each player is assumed to know the PURE strategies of the other player. Everyone plays the pure strategies perfectly and the game always reaches an equilibrium. It is always a higher frequency than the pure strategy.

A simple example:

Pure strategy vpip 9players =12/10
Equilibrium strategy 9player = 16/12

I can't prove it

What lol. Player B's strategy is "always betray". You have the choice of either betray (which is the Nash equilibrium) for -2, or stay silent for -3. It doesn't help you to know that Player B's strategy is always betray. That's the whole point of a Nash equilibrium.

How does this follow at all? If I'm playing GTO, I raise preflop with some subset of hands. You then know that I have a hand in that subset, but that's all.

[citation needed]

I feel like if the Wiki article didn't mean "each player is assumed to know the equilibrium strategies of the other players" then it wouldn't say that.

Also you don't even understand the Prisoner's Dilemma, so yeah.

Thanks for writing this up, Vernon. Very interesting read, just as the COTM was.

I will point out one thing, which I also mentioned in that thread a while back: just as expanding a game from 2 players to 3 players changes the landscape in non-intuitive ways, expanding from a one-shot game to a repeated game does too. In a repeated game, strategies are allowed to consider what's happened in previous rounds, and all sorts of complicated equilibria become possible, even in a game as simple as the prisoners' dilemma.

To me, this just highlights how absurdly, ridiculously arcane the concept of a GTO strategy in poker is. When I see the term used in threads now, I just assume the person meant something about being balanced and let it go.

My brain doesn't work fast enough, nor recall past events quickly enough, for me to be able to play a perfect GTO strategy against anyone of the 9 other players at the table when I get it HU vs. any of them, going into the Flop. Therefore, I am destined to be an average Joe Poker Playa'.

I have a hard enough time trying to put my V[s] on a range, based upon their betting patterns, the board & the possible combos they could be drawing to or made hands they are holding.

However, I do enjoy the banter, even if I don't understand the majority of it.

The most basic basic basic way to examine your own play. Theoretically speaking, you have to play 16/12/2 vpip in poker just to break even. You cant just play 88+AQ+ and win theoretically. That constitutes a 12% frequency and GTO proves its - ev to play that tight.

Mix in some suited connectors and other hands.

Plays as loose as you can and if you can manage to break even at 16/12 you have the potential to become a winner

OMG LMFAO