Will you and Bill Chen write a sequel to Mathematics of Poker?

Have there been any big breakthroughs in the acedemics of poker research since your book was published?

Can you please elaborate on "as a mathematical topic GT is basically a dead end these days"?

The [0,1] distribution naturally lends itself to independent distributions, where there is no card removal. You could try to use a joint density on [0,1] x [0,1], which would be what you're describing (ie card removal), but such a density would lose most of the nice properties of the uniform density that make solving these games relatively easy.

I think solving a particular game where the distributions are some arbitrary independent measures on [0,1] is possible, but writing down a general method for solving these games algebraically is non-trivial. (Translation: I don't remember running into a river we couldn't solve, but writing a fast non-iterative solver is hard.)

On the third question, what is the nature of the map? If you are mapping to arbitrary measures on [0,1], then of course the solutions will be exact because they are isomorphic. If you are mapping to some continuous distributions without card removal, then the solutions are kind of broadly similar but the details are quite different. This is because card removal breaks all the symmetry of the solutions.

I don't see how it could be. I mean, the other guy has to act first.

I can see how it could be kind of difficult in practice, when the other guy shoves and I have some non-nut hand for half the pot and don't know whether I should call or not. But the cold-blooded assurance of an optimal strategy would render that moot.

Unless there is some other reason you are thinking of, I didn't think very hard about this.

I'm not sure what you mean by "non-iterated."

Matt Hawrilenko and I used to play LHE without replacement (ie, we would just deal down from the stub and set the board cards and dead hands aside). That was pretty fun. Obviously the most important thing was tracking what was left in the deck. Let's just say I wouldn't play that game against Andy Bloch.

Well there are a lot of possible models; a simple one is that the players can throw their hands away and get another hand from [0,1]. You can also do games where there is an "open draw" that maybe promotes some hand types. So like there is a round of betting, then some Bernoulli event that is visible to everyone, and if it comes up true, then [.75,.95] becomes the nut range and everything else slides down, or whatever.

Haven't looked at it but I will try.

Bill and I met at like ESCARGOT or BARGE in '99 or 2000. We hit it off pretty well; he started backing me in tournaments at the time. I was just getting into poker. Later (01 or 02), he found out that I used to be pretty good at math before I became a screwup. He started talking to me about these simplified toy games he had been working on, which eventually turned into many late-night phone calls of length typically reserved to long-distance relationships. That turned into the [0,1] series on rgp, and we started to work together on all kinds of things.

There had been discussion of Bill and Andy Latto writing MoP for a while; the book they envisioned would be like much more technical, like a real math book. At some point I got inserted into this conversation as well. Then we started talking about doing it seriously. Andy had to bow out for personal reasons, and Bill and I started working on the book, but my vision for the book was something more accessible to non-mathematicians. We started out with some idea of splitting the writing but after a while it just seemed best for the book to largely be written in my voice. So the writing is mostly mine. The topics covered are like a mixture of things we worked on individually and things we worked on together, but Bill surely contributed more on that front.

The book was written over a period of like 18 months from late 04 to mid 06. So that was a financial disaster, since you could make like $40 per $215 SNG at the time.

There are a lot of toy games you can solve, but honestly I think the best direction to go after working through the examples in the book is to start using computers to solve games, writing iterative solvers, familiarizing yourself with the literature on the computer science problem. If you must, though:

[0,1] vs [0,1]^2 (analogous to lowball draw 1 vs draw 2)
games with a "draw" as above
games with a "draw" and a "secondary draw" which gets there but you dont see (like a flush draw and a gutshot)

Koller et al's paper was important in the beginning.
Some of Tom Ferguson's papers are of interest in solving toy games.
UAlberta's people have good papers.
Gilpin/Sandholm etc at CMU are doing good stuff too.

Highly unlikely. Effort to reward ratio is just too high.

Yes, there is work being published all the time on the algorithmic problem of solving poker. There is an annual competition, and check out an earlier post for a list of places to find papers etc.

Well, it's just that it's basically done as a field of research. Applying it to specific problems is still worthwhile, but (although I don't know for sure), I suspect that hardly any mathematics PhDs are being handed out for research into the structure of game theory; just like noone is writing dissertations on measure theory. It's not that measure theory isn't important and noone uses it...it's just that there isn't too much more to find in there.

Same question:

What's the largest game you've solved?

- Andrew

I solved Ultimate Texas Holdem last year.

I haven't done much work myself on solving midsize games (ie, significantly smaller than a real poker game but much too big outside of a computer), since you need input distributions for that, and since I don't play much any more, there's not so much value in my guessing at those and trying to solve the ensuing games through serious computation.

I've gotten interested recently in three-handed [0,1] games. I'm sure someone else has worked on them. Are you aware if they've been published?

Also, I've been thinking about modeling draws in toy games. How does this sound to you:

Each player is dealt an unsigned integer (e.g. in the set {0, 1, ..., 255}), represented in binary (so the representation is a member of {0x00, 0x01, ... 0xFF}). The draw is another unsigned integer. There is a round of wagering, then each player's hand is XORed with the draw, then a second round of wagering. Community-card games could be modeled by XORing all hands with the same draw.

How much value do you see in this in making multi-street games computationally tractable? Would enough insight be transferrable to complex games like hold'em to make the effort of solving them worthwhile?

I don't know of anything published about three player [0,1] games. Part of the problem is that such games (and all such multi-agent games) aren't guaranteed to have a "solution." So if you find some equilibrium for such a game, you would need to investigate the existence of other equilibria and come up with some meaningful way of characterizing the different equilibria and deciding what to do about it.

Your proposal is a fine way of modeling draws; the question on "will it transfer" is hard for the above reason, and also for the reason that the better that your model approximates the actual shape of draw distributions, the better the approximation will be. Clearly there is some insight you could gain, but it will sort of lack clarity compared to corresponding solutions and insight from HU games.

Another thing that solving these games will teach you is that soft collusion (where two players don't share hand info, but play some kind of joint strategy against a third player) happens all the time and often accidentally, and if you are going to play in tough three player games you need to understand it pretty well.

Is it ever possible, as part of a nash equilibrium range, to take a '-ev action' with a hand? e.g 'I shove as a bluff here, even though he calls slightly more than enough to make my bluff breakeven, it means I get paid off more with my value hands, so overall it is +EV?

Here's the important property of an optimal/equilibrium strategy:
When played against the opponent's fixed optimal/equilibrium strategy, neither side can unilaterally improve EV by switching from one action to another.

So if the opponent is playing optimally, then no, this can't occur, because if you could improve against his fixed strategy by, say, checking instead of bluffing, say, then your strategy is not optimal.

However, against any old suboptimal strategy the opponent employs, this might occur. That is, suppose he calls far too loosely. Then your (fixed) optimal strategy loses extra on bluffs but makes it up and more on value bets as you describe.

If you can improve your optimal/equilibrium strategy against some particular strategy your opponent is employing, then switching is just exploitation.

I don't understand this at all. Could you provide a brief description?

Also, can you say anything about interpretation of HUD stats? They seem to be a classic Bernoulli distribution, but skewed. It seems to me that the standard deviation might be different in each direction.