I didn't make these terms up, but plainly a "polarized" distribution is one where you are at the extremes of hand value (either very strong or very weak). From googling, "range merge" seems to mean additionally betting hands of intermediate strength for value, with the intention of punishing bluff-catchers.

This is frequently described as some kind of advanced wizardry. I can't really figure out why.

Yes it's a reasonable thing to do, but it's not really quite right. I mean, you have to consider the action on previous streets too. So you say you have to prevent him from check-raising profitably with 53o, but you can't just consider this street in a vacuum, like he just has a random hand. He had to call a raise or something preflop, so that costs him money too in order to make that bluff. Hence you don't need to call/raise often enough on the flop to prevent pure garbage from making money naively considering the flop action. You need to call/raise only often enough to prevent pure garbage from making money by calling preflop and then check-raising this kind of flop (and playing other ways on other flops potentially). So tighter than "prevent 53o" is probably okay.

For an extreme version of this there is a game in our book where one guy is made and the other is drawing and on the river in a pot of six bets the made hand only has to call like 2/7 of the time when the flush comes in because of this concept.

Most games now are played pretty far from equilibrium, I think. Also intuition seems to basically be always wrong about GTO stuff until the computers spit it out.

I'm tired of poker, mostly. I did it for a number of years in various ways. With online gone and the burden that even traveling to the WSOP puts on my family, I'm basically just semi-retired now. I working on my PhD in applied math, doing stuff that has nothing to with poker. I may still occasionally do poker-related research, and might show back up at BARGE, etc. But as far as "I play poker for a living," that's over.

Well, I mean the obvious trope is that she will return to Westeros with her Faceless Man skills and play some key part in the resolution of all the plot lines. It wouldn't surprise me if it gets a little subverted, such that she winds up causing the deaths of people she didn't mean to or something.

If I had to make a market on who kills Danaerys Targaryen, it seems Arya would be the frontrunner but not the favorite.

I figured it might take some time. I'm not going to work out any games for anyone though, maybe just provide hints. So if someone posts a game and other people want to jump in and work on it, please do.

We worked this game out, but it's not terribly interesting. You just wind up with an infinite sequence of thresholds and their associated bluffing/calling regions. The even numbered ones are smaller regions (for check-raising) and the odd-numbered ones are bigger. When there's no cap, there really isn't "the top of your range." The sequence of thresholds converges to 0, but any sufficiently small interval is completely contained in a region; maybe it's the check-raise-and-reraise-674-times-but-fold-to-the-1349th-bet region or something, but it's always clearly defined what to do with a particular hand.

More stuff about exploitive play, probably.

Computer science.

Way to go?



"All hands end. Not all hands really live."

Well yeah, it's hard to go from street to street with distributions and have them be unexploitable. But no, it's probably not close to GTO to only flat out of the BB in NL, because we could gain equity by 3-betting sometimes. How often to do it and how to play all the flops and turns and rivers after that is a big question, and one that noone has solid answers to AFAIK.

Looser than people think is right at the moment. If you play perfectly on the later streets, you can play a lot more hands. That doesn't mean that you (or I) should play a lot of hands, because we all probably suck at playing postflop (compared to GTO).

Awhile back, David Sklansky asked about AONLHE as played in the last WSOP. He is interested in a useful rough bound and not expecting AONLHE to get solved.

Thoughts on how river overbets might/might not be a part of GTO play? What size river shove would definitely be "too big" to be part of any GTO strategy?

Thanks for doing this! Very interesting so far.

In chapter 21 of MoP you talk about preflop sizing and come to the conclusion that raises should get bigger as we progress to later position (2x in EP, 3x on button).

I've always heard general disagreement with the idea, and I don't know of anyone at mid/high stakes who uses those sizings (often the reverse).

Do you still believe the strategy is correct? If not, what do you think the correct strategy should be (assume 100bb stacks for convenience, but I'm also interested in deeper stacks if you care to expand).

Also, what sizing do you think we should be using in the SB? I see everything from 3.5x to 2x.

I thought GTO only made sense for HU poker since for 2 player zero sum games, Nash equilibrium is equivalent to minimax. What does it mean in terms of poker with 3+ players?

1. Do you think the NE for HUNLHE ever has a player bet (or check-raise, or reraise) all-in on any street? If yes, what would an example of this being clearly correct?

2. Do you think the NE for HUNLHE ever has the SB 4-bet preflop?

3. Tom Weideman strongly believed HUNLHE is significantly easier to solve than HULHE, a rather counter-intuitive claim. However CS has made far faster progress towards NE for HULHE than for NL. Was Tom wrong or are poker researchers simply not using the best algorithm to tackle NLHE?

4. In the HULHE game tree, what is your best estimate on its distribution between pure versus mixed decisions? In other words, from all the nodes in its game-tree, how many decision points would be played using a pure strategy and how many would be played using a mixed strategy? e.g. 90%/10%, respectively?

5. In HULHE, from a NE standpoint, are blocker hands the only hands that should be used to check-raise bluff on the river? If not, what would be a clear example to the contrary?

Seems that no river shove would be "too big" to be part of a GTO strat. Suppose you got to the river with the untieable nuts (eg nut flush) and you had a million pots in your stack. Now suppose you have some hands with just the bare ace. Suppose you bash with some amt of nut flushes and some amt of bare aces. Your opponent can't just fold without the nut flush, so he has to call enough to make you indifferent to shoving with bare aces. (Not very often!) Since you presumably have more bare aces than nut flushes (assumptions, assumptions), But we know that in the clairvoyant NL game (which we sorta have here for the river shove), the clairvoyant guy bets as much as he can to maximize value.

Now its likely that he wants to have nut flushes in other value betting ranges too, so it's probably not the case that he shoves all the time with these hands; in fact, since he will be mixed, his equity from regular value bets will be equal to his value from these shoves.

So I don't see any reason to believe there is some threshold where "oh this shove is too big to be optimal."

Well, 32o is worth like 32% against a random hand (ignoring clumping), and plays a lot worse than that. So assuming you have a limping distribution that protects you adequately, when would you want to limp with it? What % of the time would the button raise a decent defensive limping distribution? Maybe half? So half the time you limp, you get raised and have to fold for -$1. The rest of the time you have maybe 15% of the pot as equity (when he doesn't raise you are dominated more often, you are out of position, etc). So you need your equity from this case to be $2, so the pot needs to be $13?

OK, my rectally generated guess is that.

The conclusion of most highstakes players is that the difficulty of playing out of position is such that characterizing the game as a battle between the raiser and the blinds, where pf raise sizing is primarily geared toward making the BB broadly indifferent to calling widely (that is raising less when you are strong and more when you are weaker) is wrong.
Instead you should raise bigger when you might have to play out of position and smaller when you will have position, inviting the BB to play, and giving up less when your weaker distribution gets 3-bet.

I don't really know what's right, but at the WSOP I adopted constant raise sizes (2.xx where xx is smallish) from all positions except the SB on the advice of my better-than-I-at-NL friends. From the SB I raised to 3x.

To be honest, I suspect two things: 1) highstakes players are probably right about position, 2) they are probably not as right as they think; if we knew how to play optimally after the flop, being out of position would just be bad instead of terrible, or something. But we don't, so it isn't.

Also I'll make the following comment: NL has evolved an incredible amount. In the mid-2000s I was pretty good at it relative to the field. Then I spent a lot of time playing mixed limit games and stuff like that, while an entire ecosystem of NL spent five years performing natural selection with a thumb on the fast forward button. So at this point, I feel like NL theory and especially NL practice has kind of passed me by. You'd really be better off asking people who are really good at NL about these kind of things.

Well, Nash's theorem guarantees there there is at least one equilbrium for a multiplayer game. But such an equilibrium can't really be said to be "optimal" in the same way as in HU games, because there are often collusive ways to disrupt it. So if you play some 3-player Nash equiilibrium strategy, you don't ensure that much about your equity.

Nevertheless, many pots end up headsup very early in the hand, so you can probably take the preflop distributions as inputs and try to play unexploitably from that point forward, which is effective and useful.

But theoretically most of the cool stuff falls apart in multiplayer games.