I'm not a math or GT expert, nor a good player for that matter, so I'm feeling my way here. This is not about "real' poker, it's about the non-iterated version, which is playable, but devoid of strategic content, as far as I can see.

A non-iterated game is one in which each game is a discrete event, with no connection or influence from past or possible future games. Non-iterated poker - or at least, a very close approximation - is in principle easy to set up, by running a fast-fold poker game such as Zoom with all players anonymous, so that player histories are unavailable. You never know who you are playing, or what they did before, and PT don't work. So the first few weeks that Rush Poker ran, without PT, it was almost NIpoker. It could also be run as a simulation with bot players.

It seems to be a general rule that non-iterated games require an unvarying game plan, and that seems to also apply to NIpoker. In the non-iterated version of Pontoon - which is Blackjack with a shuffle after every deal, or something similar - you must make exactly the same play in the same situation every time it arises, or else you are throwing away money. In the non-iterated PD the game-plan is also unvarying, and there is always only one correct play to make which, however unpalatable that choice may be, is better than all others. Similar results (I think) apply in NIpoker.

Here is my game-plan for playing anonymous fast-fold poker, with a large player pool of random players. Whenever the most profitable action in a particular situation is known, I will take that action 100% of the time, with the only variation being that the bet sizes may vary sufficiently to avoid providing any potentially exploitable information, that is, within a range of say 10% either way. (Even if the precise, optimal bet size was known to all, that might not matter, because other players would use the same bet size to deceive, which would totally confuse the situation. But I'll leave some bet-size variation as an option, if it is needed)

If the most profitable action cannot be determined - which will often happen in a multi-stage, multi-player game - then I will revert to playing poker by presuming that my opponents are rational, average players, and do the usual guessing and hoping. Let's call players with that plan unvarying, or U-Players. Since my plan is known, and may be followed by others, let's consider how players who want to attack my game-plan could do it, and call those players Anti-U.

Let's say that a U-player knows that the most profitable action in a particular situation is to bet the pot, so he does. What does the Anti-U player do? If the most profitable response, on average, is known for that particular circumstance of cards and position then the Anti-U player must not make that move, because that is exactly what the U-player would do, and the premise is that Anti-U play can defeat U-play. If that's true, then whenever you play like a U-player, Anti-U players will be able to exploit you. So in order to avoid playing like the U-player, the Anti-U player must at all times ask the question "What is the most profitable play, on average, against an unknown player" and then do something different!! That cannot be a winning plan.

The premise that Anti-U play can succeed is therefore wrong, because it leads to a contradiction. And in any case, the Anti-U player can never know if his opponent is actually a U-player, so he is chasing ghosts.

If that reasoning is correct, then the U-plan cannot be exploited by actively Anti-U players, in a large and diverse player pool, and the U-players can simply select the most profitable play, whenever it is known.

Let's look at another game-plan, call it GT-play, in which the usual approach of varying one's play is used. So, when dealt AA the GT-player open-raises sometimes and at other times he flat-calls, in whatever ratio he comes up with. The U-player however doesn't care about being predictable - because you can't be predictable if no one knows who you are or what you did - so he open-raises 100% of the time with AA, as long as that is the most profitable play, on average.

Which player will come out ahead? Since one play is on average more profitable, the player who makes that play all the time must come out ahead, as long as there are no extra costs involved. (In real poker, there would be costs for doing that, but I cannot find any in NIpoker, so far..) The GT-player gains nothing by varying from the most profitable play: any residual value created by flat-calling with AA belongs to the field, not the individual. And if there are any Anti-U players in action, then they will always be flat-calling with AA, because otherwise they will simply be U-players, or GT-players. So there should be plenty of players who flat-call with AA, which helps the U-player: he can keep on open-raising 100% of the time, and he will lose nothing by doing that.

So the unvarying game-plan of the U-player, which at first seems so easy to exploit, may in fact be completely unexploitable, at least, with a large player pool. Non-iterated poker is bizarro, because the usual principle that variation in your play is essential, is reversed: variation from the most profitable play can only decrease your profits, as far as I can see.

afaict, a strategy for what you're calling th non-iterated version of the game is what people are usually talking about when they talk about GTO or Nash equilibrium play.

your assertion that the frequency of any player action in that solution should be either 0% or 100% is close to correct except that it happens to turn out pretty often that two actions can have exactly the same EV, in which case you might want to choose both sometimes.

I agree with yaqh. The best way to play what you call NIpoker is exactly a GTO strategy.

OP I don't really get your thing about U and anti-U players. How would the anti-U player "attack your game plan" by not taking the most profitable move in a given spot? That makes no sense.

GTO prevents you from being exploited, but that doesn't mean it's the best way to play. If you can make some estimation of the distribution of your opponents' strategies, then you can make small exploitative adjustments. Enough to exploit bad players but not so much that we become too exploitable.

That's the point: a policy of trying to attack those players who are playing an unvarying game leads to a logical contradiction, which means that the policy cannot work. If the Anti-U player makes the most profitable play (on average, if it is known, when playing against a complete stranger) then he is following the U-plan: if not then he is systematically taking other, less profitable actions.

Yes. It seems pretty loose to simply say "it's just a GTO optimum" because the GTO in a non-iterated game is totally different to the iterated game, and exactly how to achieve that optimum is the question in hand (I thought so anyway). And I suspect that most who say "GTO" are thinking of the variability in action which is essential in normal poker, such as mixing up your play, bluffing a certain proportion of the time, which have no place at all in NIpoker, because such variation must be sub-optimal. I might be wrong, but I would like more detail about how to create that GTO using GT principles in NIpoker.

Hi, thanks for the comment. I know I'm the only one calling it non-iterated poker, because I googled it, and no one here has heard of it before (unless they read my original post on anonymous fast-fold poker, which I called Strategy free, all tactical poker, SFAToker, which I posted several months before fast-fold poker first appeared online as Rush.) NIpoker seems a more logical term, because anonymous fast-fold poker is exactly that: the non-iterated form of the game. So in GT that should mean something, because the analysis of the iterated v non-iterated versions of the PD is central to a complete understanding of GT.

I think everyone else just calls it 'poker' -- I've never seen any game theoretic work that explicitly treats poker as a repeated game, and I'm pretty sure the reason for that is that the equilibrium strategy in the repeated game is just to play equilibrium in each stage game. (In other words, they always just treat one hand at a time, and the reason for that is that, vs someone who's playing well against you, you make the most money over a bunch of hands by making as much money as possible in each individual hand.)

edit: source that seems to agree
http://129.3.20.41/eps/game/papers/0402/0402003.pdf

Well the "U-plan" by definition would include strategies that account for the presence of other "U-plan" players. So the "U-plan" and the "anti-U-plan" are exactly the same thing.

GTO still includes those variations. They have nothing really to do with whether you're seeing the player again. They are about being unexploitable.

Not really. GTO strategy is the same in iterated vs. non-iterated, because it doesn't pay attention to previous hands. Any strategy that adjusts based on previous hands is exploitative.

Here IMO is the problem with what you are suggesting. First let me make sure I understand.

You are suggesting
- look at the history of how the field plays on average in all possible situations
- use that the calculate the best action in any given situation (for the sake of this discussion let's assume this is possible)
- based on this, take the best (highest EV vs the field) action 100% of the time in each situation

You then argue by contradiction that no mixed (varying) strategy can do better.

And that's true, based on those assumptions. You have essentially calculated the best response pure strategy vs the field.

But the huge flaw in all this is to assume that the field is not changing. If the field is changing, then your calculation is based on how the field played (past tense) and not how the field is playing now or is going to play in the future. At any given moment your strategy is a maximal EV strategy vs a field that is going to continue playing as it did in the past.

For example, suppose the average player folds way too much in some situation. So in that situation your determine, based on histories, that you should bet your entire range. But perhaps by the time you calculate this, the field has already adjusted and players are now calling too much in that situation. So your method is really only optimal if you assume that either the field is unchanging or you can somehow predict how the field is playing right now rather than how they played in the past.

It might help you to compare your method of creating a strategy with how a GTO strategy can be created.

You method is simply
1. Strategy = BestResponse(Field)

Here is an algorithm that converges to GTO
1. Field = some random strategy
2. B = BestResponse(Field)
3. Add a player playing B to Field
4. Repeat steps 2 & 3 many many times

The resulting Field, taken as a single mixed strategy, converges to GTO. In other words, in any situation, to decide what action to take, take action A with probability equal to the probability of a random player from Field taking that action. Keep in mind this Field is fictitious and has nothing to do with the actual real-world field of players you might be facing. This fictitious field of players is created by starting with some field and repeatedly applying your method of finding the best response to the current field of players and then adding that new player to the field.

This argument is flawed. Once you add a sufficient number of U-player's to the pool, the best action in many situations will be different than what it was when you calculated what a U-player should do in those situations.

Indeed, GT makes no assumptions on playing over and over, let alone on how the future rounds are played.
The only place where the concept is used, is when fictitious play or some similar algorithm is used to find a NE (and indeed such algorithms may fail to find equilibria or find a single one out of many under certain circumstances).

So, back to the original question, solution is to play GTO if you want to be unexploitable (or at least as much as you can be in a multiplayer game) and to play exploitably if you wish to do so to gain an edge given the information available to you.
Why OP thinks that given the "non iterated" nature of the game you shouldn't mix your game is a mistery to me.
You don't mix because observant opponents will then exploit you. You mix because you want to be unexploitable. If you don't mix you are exploitable and any opponent may exploit, be it by chance or skill.
BlackJack is a completely different matter, because your opponent is using a fixed known pure strategy, so any pure strategy best answer is as good as another and you don't need to mix.
Your opponents in NIPoker aren't using such strategy, so only way to be unexploitable is to play GTO, ie mix.

Well, it's not quite that trivial. There definitely are some games where playing in a repeated fashion changes strategies significantly.

Classic example, prisoner's dilemma
http://en.wikipedia.org/wiki/Prisoner%27s_dilemma
In the single-shot game, squeal dominates dont-squeal -- i.e. you should always play squeal no matter what the other player does. On the other hand, in the infinitely-repeated case, a stable equilibrium involving not-squealing would arise if both players did something like: "play dont-squeal every round until the other player squeals once, then start squealing every round."

Given a game G and a Nash Equilibrium for the game, the repeated game G* has a NE* given by playing the NE for the single game at every stage. And this is trivial to show.
So if what we are looking for is just a NE, look no further
And If we restrict ourselves to zero sum hu poker that's for sure what we are looking for.
If we don't restrict ourselves, a lot of interesting things may happen, but they
are mostly related to metagame/collusion issues.

But now, erasing most infos in between hands, ie making the game somewhat uniterated, will make the game more akin to the original Nash problem and thus strengthen the NE solution rather than not.

^ Yea, certainly I agree. It just wasn't immediately obvious to me that nothing interesting could happen in poker in the repeated case.

Thanks for the posts, which certainly put the issues very well. I'm not competent to judge on matters of GT, that's for sure, so I'm don't know what more I can contribute.

I take Blackloter's point about BJ presenting a very different set of circumstances, which would seem to argue against my reasoning that NIpoker could be played in a similarly unvarying fashion.

I did have a notion that long-term stats on real play of NIpoker would answer a question, which is what is the most profitable tactical move in any particular situation when all strategic content is removed. I'm not so sure about it now but if so, then that might be a useful thing to know, or to incorporate into simulations or whatever.

Yes, if you can get a precise model of your opponents and if they're not going to adapt (may be nearly the case in non iterated poker but not 100% true, especially on long time frames) then you can play a pure (non mixed/non randomized) exploitative strategy to max your ev