Yes, unlike Blackjack, in which the maximally profitable move never changes (assuming a sufficiently randomized deck) in non-iterated poker, there could be long-term changes in patterns of play, which would mean that the list of most profitable plays would have to be updated periodically. But stats on actual play would be your only way of achieving some exactitude. GT questions like "how often should I check/bet/call/fold in this situation" are meaningless, because in non-iterated poker the correct answer can only be "always" or "never."

Creating a near-perfectly non-iterated game, as you have noticed, requires some diligence: in theory, it can only be achieved with an effectively infinite player pool, and perhaps also by releasing historical information in a carefully controlled way, with some random mixing of long-term and short-term results.

(I'm not sure how it could be done perfectly, but even an an imperfect non-iterated game can produce meaningful results: the prisoner's dilemma only works as a coercive device if it is, to some extent, a non-iterated situation, in which external factors such as retribution are taken out of the equation. But it does not have to be perfectly non-iterated in order to work a lot of the time. In poker there is (I think) a large grey area, where the game is neither perfectly iterated nor perfectly non-iterated, and different forms of the game - such as Zoom, and tournament play, versus cash play - contain different levels of iteration: any reduction in iteration reduces the strategic content of the game, and the usefulness of normal GT calculations.)

I think that the play of non-iterated poker could solve, or answer a particular question, which is, what is the correct tactical play in a particular situation if you remove all strategic issues. Unlike purely mathematical calculations which may treat a hand as an isolated event, the optimal play in non-iterated poker incorporates the human element, averaged over all players, because that play is (paradoxically) determined entirely by historical data, rather than calculation. So the data produced by non-iterated poker in analyzing a single hand in isolation may be more accurate than the mathematical calculation for the same situation.

At the same time, non-iterated poker is a seriously inferior form of the game, even though in most respects it looks just the same. The value of any move you make has no residual value: if you lose when you bluff there is no "advertising value" gained, so the number of bluffing situations capable of enhancing long-term profitability is reduced, which leads to conservative play But at that same time, whenever those bluffing situations do arise the paradox is that you must then bluff 100% of the time, which is ultra-LAG.

The situation regarding non-iterated poker is also somewhat clouded by the fact that averages also work over time: that is, in non-iterated poker you can re-coup the loss of a failed bluff by repeating that bluff 100% of the time, if the average result for that bluff in that situation is positive, but you cannot recoup the value of a failed bluff through "advertising value" if that bluff is on average unprofitable. In both cases, profitability can only be measured over time, but in one case there is strategic variation, in the other, there is not.

In non-iterated poker there is no advertising, and just like an economy which does not permit advertising, it is therefore dysfunctional and inefficient, which limits but does not completely negate the value of any results it produces. A similar observation may be made of mathematical calculations and poker problems involving a single hand, ignoring all long-term considerations: the results are imperfect, and may or may not be useful in actual play.

Another problem in discussing these matters is that the term "strategy" has multiple meanings: I started thinking about these issues in order to clarify for myself the distinction between strategy and tactics, which in common speech are more or less interchangeable, and could in many instances be replaced by "policy". The answer (which satisfied me at least) is that strategy can only work by varying your actual plays over time, while tactics relate only to the immediate situation and, in the absence of any strategic considerations or prior knowledge, the optimal tactical move is a constant, as it is in a sufficiently randomized game of Blackjack.

A policy which simply says "ignore strategy, and consider only tactical matters" is, in common speech, still a "strategy" since it sets the long-term agenda: but since that "strategy" does not involve any variation, it is not a strategy in GT terms, or at least, it is not one which requires mini-max or other GT calculations to solve. In normal GT-strategy calculations there must be variation over time, and again, there is room for misinterpretation, because someone might consider that an unvarying policy of bluffing 10% of the time in a particular situation represents a static rather than a dynamic playing strategy: but in fact it is not static: it varies 10% of the time.

In tournaments there is a degree of non-iteration involved, which actually produces some apparently strategic variation related to bubble play, stack sizes etc. But those changes could also be seen as simply representing variations in the tactical situation, with all such situations being equal, and treated in exactly the same way each time they arise. So you might vary your play from one part of a tourney to another, but not vary your play between occurrences of the same situation in different tournaments: that is, you will always choose to open raise all-in when short-stacked on the button with a particular strength of hand at a particular stage of the tournament, which in other situations you would have some choice in how you played, or could randomize in some way. It is the automatic nature of such moves which makes them low on skill content - while not affecting their profitability - since they can be made by reading them off a list, just as they can when playing Blackjack. While we might not realize it, we may in fact be in almost Strategy-Free, All-Tactical situations in poker fairly often.

In such cases, where there is no knowledge deficit involved, and the correct play is known to all competent players, "skill" is replaced by self-discipline, and poker courage.. Perhaps that is all there is in poker when played between experts: poker courage, discipline and the fear factor are the intangibles of the game, which are outside the realm of classical game theory, and relate to the psychological side of the game.

Sorry about the length and any redundancies or omissions, or misuse of GT terms: as far as I know, this is the first conversation which makes extensive use of the term "non-iterated poker", so some problems with terms are to be expected.

As I already mentioned, the idea not so practical. A friend of mine, more simply, challenged me to come up with a concrete hand where this idea would apply and I found it not so easy. I was actually hoping you could do better than I did.

I'm not sure I understand. There can't be any k in those first three equations because those three equations are for any hand distributions F and G and are independent of the specific problem that slowjoe posted that involves k.

0------v------c---------b---1

Pot=1, Bet=s

EV(check with v) = 1 x Prob(player 2 hand worse than v) = (1 - G(v))

EV(bet with v) = 1 x Prob(fold) + (1+s) x Prob(calls with worse) - s x Prob(calls with better)
= (1)(1-G(c)) + (1+s)(G(c)-G(v)) - s G(v)
= 1 - G(c) + G(c) - G(v) + sG(c) - sG(v) - sG(v)
= 1 - G(v) + sG(c) - 2sG(v)

EV(check with v) = EV(bet with v)
1 - G(v) = 1 - G(v) + sG(c) - 2sG(v)
0 = sG(c) - 2sG(v)
0 = G(c) - 2G(v)
G(c) = 2G(v)

This is saying that our worst value betting hand must beat 50% of our opponents calling range.

Oh, I see, F and G are the CDFs for the distributions. Yes, this will work.

Well, playing GTO is still the only way to ensure that you achieve your game equity, since a group of computer programmers could come along and let loose a bunch of players designed to exploit the exploitive strategy that you have determined based on some kind of statistical analysis. (By the way, something like this actually happened and it had to do with chinese poker.) Claiming that an action is "correct" because it supposedly maximizes against the field of unknown players is basically just defining what "correct" means.

Well now I don't know what you're talking about. How can reducing the possible avenues of exploitation (don't know your opponent, don't play with him more than once) reduce the usefulness of optimal strategies? It seems blindingly obvious to me that this would enhance their value, since you know nothing about the opponent you face, that you can just play unexploitably and collect money.

But now your game is somehow intimately tied to the provision of historical data or something...because what you want to do is say "if I exploit the field, that's correct." But why should any historical data be available at all? Suppose that there were some trust mechanism which I can't conceive of, but suppose it anyway, such that hands weren't even revealed at showdown; the pot was just awarded to the showdown winner and neither hand was shown. This is even MORE "non-iterative" to use your term, since you get the absolute minimum of information about your opponents. Again this environment is better for GTO strategies.

I argue that historical averages of what people do tell you nothing about what is "correct" because most people are hosers. If you are good at exploiting the hosers, you will make money in poker. But there is no insight gained from taking a big pile of hosers and averaging out what they do and figuring out what strategy exploits. Because sooner or later you will want to play the next harder game, or the one after that, and now the players will hose less and your exploitive strategy will be wrong.

The limit of that process (tougher and tougher players) IS just GTO.

I snipped more below, but I'll make one more comment. Everything about advertising and the psychological side of poker and repeated play against a single opponent and moving your strategy and poker courage is just "exploitive play." That's a topic of some interest to a lot of people, but not me so much.

Suppose a toy-game [0,1] vs [0,1] uniform distributions
N (many) rounds of betting
Only Pot-sized bets are allowed
Very large stacks (enough for say N pot-size bets)

I am curious about the structure of the strategy for the first player's very first action.

Is the first player's strategy for his very first action going to be:
a) bet worst hands as a bluff | check medium hands | value bet better hands | check best hands. i.e. exactly 4 intervals with alternating bet/check actions. Let's call that a BKBK structure (B=Bet, K=Check).
b) or are there going to be many alternating intervals BKBKBK... with the number of alternating intervals being larger for larger N.

The reason I ask is that if it were (a) then it seems like the whole problem might be solvable analytically by some recursive means. But I think it's (b) which makes it hard or maybe impossible to solve analytically.

Did you publish or post anything about your Ultimate Holdem solution anywhere? Would be very interested to read about it.

No, it's just like the same solution that the Wizard of Odds has on his site. There's nothing interesting involved, mostly brute force.

can you tell me what is polarizing and merging ranges completely with details?

In HU nlhe the flop comes Q84r. Intuitively I believe that the optimal strategy for the button would be such that the bb couldn't profitably c/r hands with very little equity e.g. 53o.

If I then adopted a strategy loose enough so the above condition holds do you think as an attempt to play closer to a Nash equilibrium this is reasonable?

Or do you think it is likely that the games played today are so far from a an equilibrium that most intuition has very little power at predicting what a Nash strategy would even look like?

For example the Nash might be to get it all in for value with second pair+ as the BB which would necessitate playing weak hands aggressively also.

A second question is why don't you play much poker any more and do you see yourself having more time for it in the future?

Thanks for this AMA and answering my PM.

What will happen to Arya Stark?

Are you regretting doing this AMA yet?

Seriously though, Im interested in what our action regions would look like in the full-street [0,1] game if there is no betting cap. I wont ask you to write out all the action regions, but I'm specifically curious about what happens with the very top of our range.

With a 1 bet cap we simply bet the bottom of our range as a bluff and the top for value, and have a c/f and c/c range in between. With a 2 bet cap, we now c/r the very top of our range and b/c a region directly beneath that. With a 3 bet cap we now want to b/3b the very top of our range, and c/r/c a region directly under that. I'm guessing with a 4 bet cap we would want to c/r/cap with the very top of our range. So this pattern comes out of betting out with the very top of our range for odd betting caps, and checking with the intention of raising for even betting caps. This reserves the ability to put in the final raise for the very top of our range. But what happens if there is no betting cap? What determines if we would prefer to bet out or check?

If you had another 50 pages to fill when writing MoP, what would you have put in it?

If you were writing MoP II, what would you put in it?

What would you say to someone that strictly trying to play GTO?

How do we deal with the fact that hands dont end
like for exampel sb raise 2xbb we need to defend 50% lets say we 3bet 0 % and just call now we are letting him "freeroll"


What do you think the gto defending range would be with 20bbs in bb vs sb in a heads up game?

I think this turns into an auction game (see the section on auction games in chapter 19 of our book). Then there wouldn't be too much complication. But I haven't thought it through carefully or worked it out.