Sure. In the regular [0,1] game, the probability of having a hand between 0 and x is x. Now for one of the players, let the probability of having a hand between 0 and x be x^2. I seem to remember that this game has a neat analytic solution that doesn't get hairy, unlike some other possible densities.

I really don't know much about HUD stats, others are probably more qualified to comment on that.

Jerrod, no questions to ask at the moment but much thanks for doing this.

I'm not a GT or math wrangler, so I hope I don't mash the terms, but a non-iterated game is one in which each game is a discrete event which has no significant connection with any past or possible future games with the same players. In poker that can be done by playing fast-fold poker (ie, Rush, Zoom, etc) with all players completely anonymous, so that no one ever knows who you are, or what you did before, and vice-versa: you don't know who your opponents are at any stage.

If you want to work out the implications of playing non-iterated poker, which are quite interesting (to me anyway) then you may wish to do so before you read my GT-amateur take on it, which follows.

In the game of Twenty-one, single-deck Pontoon is the iterated version, while Blackjack is (or should be) a non-iterated game, in order to prevent strategic play via card-counting. That means that in Pontoon your optimal play may change over time, but in Blackjack (if we use enough decks and a deep cut, or use an electronic deck, or shuffle after each hand) the optimal action in each situation is completely predetermined and known to all: you can read them off a list provided by most casinos. Variation from those plays is guaranteed to be sub-optimal, which means that in BJ there is no strategy and the only useful information available is tactical, that is, pertaining only to the current hand. The tactical situation is always the same, in other words, and there are no strategic factors involved, except perhaps BR management, which is extraneous to the game itself.

The interesting thing is that if you play non-iterated poker the same principle applies and all strategic variation disappears: you must do the same thing each time you are in the same situation. So, you either bluff 100 percent of the time in a particular situation, or never bluff, with the only criterion being whether or not bluffing is, on average, profitable in that particular situation. Bluffing part of the time cannot be correct, because you cannot recoup your losses later by gaining a rep for being a bluffer, because no one knows what you did. The same applies to every possible action in every possible situation. Variation simply cannot be optimal in a non-iterated game.

So non-iterated poker could and should be played like BJ, that is, by reading the correct tactical play off a list: the only difference is that in BJ the plays can be worked out mathematically, but in Poker they must be compiled from the stats of fully anonymous fast-fold poker, because of the human factors involved. In an un-raked game you could not lose, long term, by doing that, and you would almost certainly make a profit, because many players would ignore the tactically correct play, just as they do in BJ, and that would be donating money to the other players. .

My interest in this idea arose when the notion of fast-fold poker occurred to me, a few months before Rush poker appeared. I published the idea here on 2+2 as SFAToker, for Strategy-free, all-tactical poker, and then a few months later, Rush appeared. Coincidence? Yes, absolutely, though for a few hours I speculated that there might be a connection. I wasn't the first to come up with that specific idea, which was no surprise, because it's obvious, once you see it.

One practical use of the concept is that the stats from playing SFAToker would answer the question of what the best tactical move is in any particular situation: that might be very different from the strategic play, but then in many cases it might not. Playing a strong tactical game - that is, exploiting every tactical advantage - is a "strategy" followed by some very successful players, perhaps most visibly by Tom Dwan in recent times. Perhaps playing SFAToker "by the book" with no strategic variation would not be easy at all, because it would often involve high-variance moves with a small +ev. It would be very easy to start taking a more conservative and slightly less profitable line. I think that is what happens to many poker players over time, as they lose their poker heart, and taking a high-variance, tactically strong approach becomes too stressful. What they may think of as a tight-aggressive strategy, might simply be a weakening tactical approach.

Jerrod's book, Mathematics of Poker, mainly discusses and solves game theory optimal strategies of toy poker games. The game theory optimal strategies are "non-iterated" by your definition. The past events are irrelevant to the strategy. Also, Zoom Poker is not anonymous. Even in Bovada's anonymous poker, you may still want to be concerned with the past 5 hands you had just played with Player #1.

As far as I know the optimal strategy of NLHE likely involves some mixed strategies like limping AA 2% of the time on the button. But yes, the correct frequencies of a given action wouldn't change throughout a match in an optimal strategy.

Variation would exist not because of the worry of future events, but because of the random distribution of hand strengths in a given hand. Our opponent may only be able to bluff us 70% in a given spot because we would have a balanced range of strong hands and weak hands.

The plays in poker can also be worked out mathematically. It just hasn't been done yet. These plays would ignore the stats of fast-fold poker and any human factors would be irrelevant.

Your idea of SFAToker sounds like trying to find the GTO strategy you would play against any opponent. This is actually probably quite different from Tom Dwan's exploitative approach to poker, in which he looks for specific weaknesses in opponents and looks to exploit them strongly. He's constantly famously quoted as not even believing poker has a GTO strategy.

Also, it seems strange to me that you'd want to label your all-encompassing strategy as strategy-free.

I have a feeling you may learn a lot by reading Mathematics of Poker??

I'm interested in your thoughts on multiple bet sizes in NL Hold'em... and how it relates to information hiding. Maybe a guesstimate on how it would be practiced?

Most of your book was spent on symmetrical games where both players have equal strength distributions. Any interesting thoughts when one players range is stronger than the other?

How does the fact that bluff (or semi-bluff, strictly speaking) have some equity change the bluff to value ratio (alpha)? Let's say, all my bluffs on the turn will turn into monster hands 20% of the time and 80% of the time will be extremely weak hands on the river.

Thanks for doing this AMA.

Have you solved [0,1] vs [b,1] where 0 < b < 1?

Can it be solved analytically?

If Zoom poker was played anonymously, for cash, and with large player pools, then it would be non-iterated. In a tournament, other factors such as bubble play and stack size come into play, which muddy the waters. It only becomes completely non-iterated if players are shuffled after each hand, so Bovada's anonymous games don't fully qualify: it has to be fast-fold poker.

I think you are missing the point, which is that in the non-iterated version of poker, variation of any kind is completely pointless, and the only correct frequencies are 100% and zero, as they are in Blackjack: variation in play would be as pointless and unprofitable as sometimes randomly choosing to split fives, or to not split eights, rather than simply making the most profitable play every time.

What the stats from play of anonymous Zoom cash games would provide is a baseline of the most profitable tactical play in a neutral strategic setting. Whether or not that would have any value in actual or simulated play, or anything else, I don't know. It's an interesting situation though: a form of poker in which normal GT considerations play no part at all.

This post has a good question, and maybe some misunderstandings. But I'm glad you asked it.

First, the broader field of game theory encompasses a lot of stuff, and some of it is related to things like "trying to figure out your opponents' strategies and maximize against them." In general, though, and maybe this is partially our fault, when people write about game theory with regard to poker, what they are talking about is specifically the problem of finding equilibrium or optimal strategies -- a process that can roughly be described as maximizing while preventing yourself from being exploited.

This is because apart from the problem of exploiting your opponents and the problem of playing defensively, there just isn't much else. Additionally, before game theory was much of a part of poker thought, lots of people were working on exploiting their opponents. So "game theory" came to be more or less synonomous with the "defensive" mode of play.

First I'm going to just assume headsup play for now, there are technical complications with multiway play that I don't want to get into. I'm also just going to use the following shorthand. Playing game theory optimal/equilibrium strategies is "playing defense." Estimating your opponents' strategies and trying to exploit them is "playing offense." The first important fact here is that playing defense ignores your opponents. Defensive strategies are exactly the following: they maximize against an opponent who exploits you maximally.

All the stuff we wrote about in part 3 of our book is basically about that: suppose you are playing against a super-opponent, who will know what your strategy is. Not your cards, but your strategy -- here strategy is a term of art, but think of your blackjack strategy card. So whatever you are going to do with all your hands, he will know and play to exploit you as much as possible. The strategy that does the best against him is the defensive strategy.

Now you may think that trying to play this way is insane, but I assure you that I really actually try to do this to an extent that many players would be skeptical of, and there is some pretty decent empirical evidence that defensive strategies are extremely strong against a wide variety of opposition.

I snipped a bunch of stuff here, where you seem to have advocated the view that if you had a game where opponents were anonymized and randomly shuffled every hand such that the game was still fair (buttons equalized etc), then the "optimal" play would be play that maximized vs the field on average, and that such plays would effectively be the "solution" to poker. Setting aside the different definitions of the term "optimal," the game evolves over time.

When TJ and Tom wrote that the fourth raise preflop was always aces or whatever they wrote, that was bad advice. But it wasn't as bad then as it is now! So now the solution is constantly moving as the way people plays evolves. At the WSOP this year, in the $1500 NLs, people opened for 2.5x most of the time. They made reasonable-sized reraises and played kind of decently. They made mistakes, sure, but overall they played like people who were not clueless. In 2006, people frequently opened to 5x the blinds, raised half their stack preflop and then folded to a jam, did all kinds of terrible things. When we talk about game theoretic "solutions" to games, we are talking about a timeless solution, that isn't dependent on the trends in play, isn't dependent on our opponents, is inherent in the structure of the game.

If you bet multiple amounts, you need to have multiple hand types of different strengths at each bet amount. There are complications of this due to card removal, but in general you would bet a given amount s with some "virtual nuts" hands that are willing to commit your whole stack, with some "regular value bet" hands that aren't so strong but can catch raise-bluffs and get value from calls, and some "bluffs" that are, you know, bluffs. Then the fact that you bet a particular amount doesn't really give away your hand because you're still balanced against opponent actions (ie if he increases his frequency of doing whatever, you have a mix of hands that prevents this from exploiting you).

Well at some point you get an auto-check phenomenon where the weaker distribution automatically checks to the stronger distribution, or an auto-bet where the strong distribution automatically bets. I don't have a lot of cool things to say about this though.

It's also clear that the more polarized your distribution is on the river, the more profit you make (compared to the pot or bet sizes). So in that sense "stronger" isn't necessarily the perfect word, but different shapes of distributions with the same overall equity have radically different ex-showdown equities from river betting. I think anyone who plays poker probably understands this intuitively; it's why A7o is so hard to play on later streets.

There isn't an easy, clear answer, in real games semi-bluffing with different hand types is kind of card-removal driven, and bluff-to-bet ratios aren't so clean, because there are many possible river cards and you can't make the ratios all line up for every river when deciding your turn distribution.

But if you were trying to work out some rule-of-thumb strategy, you might want to use the multi-street bluffing frequencies, but allocate semi-bluffs to "value" and "bluff" according to their frequency of being those things. But you have to be careful with this, say there's only one clear draw (Ks8s4d2c); if you have too much of your value tied up in "a spade comes," you'll be unbalanced. I think stuff like this is close enough for government work.

not a theory question but a question on theory itself.

Do you have to understand poker theory to be a successful player?

I can't remember if I did that particularly; since I don't remember that it was a cool result, it almost certainly wasn't. It seems highly likely that it can be solved analytically, though. Also you might want to consider the [0,1] vs [0,2] game. In your game, X's worst hand is the same as Y's. In this other game, they both have the same BEST hand, but Y has a bunch of worthless hands in his distribution.