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Comment by ar_lan | Hacker News Reader

It sounds like your idea is just calculating odds - this is a slight advantage, but most pro's have learned to do this in their heads already. Poker math isn't very complicated.

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I expected better from this site than multiple commenters spreading false info!

This is wrong. Making optimal decisions IS very complicated. Just a heads-up no-limit match between the two best players in the world is far from the Nash equalibrum (a game theory-optimal solution). In the 2017 Humans vs AI match, a bot destroyed very good humans by 14.7 big blinds per hand. This is without caring at all about what strategies others play. If you analyze other players and you find what mistakes they make, you can do better and it's even more complicated. And vs multiple players like here, it's more complicated again (optimal strategy can only be calculated assuming there is no collusion). A 6-person game was only finally beaten a few months ago! https://www.nature.com/articles/d41586-019-02156-9

Source: used to play poker (up to the main event WSOP and $5/10 level during the poker boom) and wrote a simple game-theory optimal solver for fun.

As an addendum to this very good comment, that Nature article had a couple of big asterisks.

* Bet sizes were restricted. E.g. humans and the the bot could only bet fixed bet sizes, like 1/4 pot, 1/2 pot, full pot etc. Creative bet sizing is one of the skills that distinguishes top pros.

* Stack sizes were reset after every hand. E.g. every player in the hand was given the same amount of chips at the start of every hand. How you performed previously in the session thus did not matter. Anyone who has played poker knows that this is highly unrealistic. Larger stack sizes convey an ability to bully smaller ones, and stack sizes greatly affect what range of hands you can reasonably play.

The point being, even a supercomputer running the most efficient heuristic based poker decision making programs has not yet been able to beat humans in a game that resembles what a real 6 or 9 person table would reflect.

---

Just as reference, on a four-year-old quad-core/8-thread Intel i7-based desktop with 32GB of RAM, to solve a SINGLE hand in PioSolver (the most popular poker solver) from flop through the river takes my machine about 7 minutes. The game tree alone takes up 4 GB of RAM, and in this scenario there are only two players, and each player is restricted to 3 bet sizes.

The idea that this kind of computation can be done on a phone is ludicrous.

Hmmm I could be wrong but I believe it's not true that humans could only bet fixed sizes. Instead, the AI was only pretrained with fixed sizes and had to do some kind of live search algorithm for any size outside of those values, which could be what you're referring to.

Stack sizes were reset to keep the research minimally scoped, taking stack sizes into account likely does not require a quantum leap in research.

This is getting pretty off topic, but the computation could be done online.

Yeah, seems like you're correct here.

I went back and re-read the pre-print here (https://www.cs.cmu.edu/~noamb/papers/19-Science-Superhuman.p...). On page 2:

> To reduce the complexity of forming a strategy, Pluribus only considers a few different bet sizes at any given decision point. The exact number of bets it consid-ers varies between one and 14 depending on the situation. Although Pluribus can limit itself to only betting one of a few different sizes between $100 and $10,000, when actually play-ing no-limit poker, the opponents are not constrained to those few options. What happens if an opponent bets $150 while Pluribus has only been trained to consider bets of $100 or $200? Generally, Pluribus will rely on its search algorithm, described in a later section, to compute a response in real time to such “off-tree” actions.

Good catch, and thanks for the correction.

Regarding the effect of stack sizes, I'm not certain on this, but my intuition is that there is some effect on perceived ranges of the other 5 players at the table if stack sizes vary. Since Facebook AI will not be releasing Pluribus code or pre-trained models/weights, we can't be certain, but things like stack-to-pot (SPR) ratio would seem to matter.

Of course, you could always make the argument that human players in a cash game can re-up/refill to the maximum buy-in whenever they're short, but that's another discussion altogether.

Do you still have that GTO solver kicking around? How'd you build it?

I do! But it would take a few days to prepare it to make it a usable github repo. I'll try to get it sometime. It's written in C++.

If you don’t mind, it’d be awesome if you could email it to me (in profile). We’re both engineers here, I don’t need anything super polished. :)

Just interested to see your approach as I’m in the middle of writing my own my own.

maybe this is just my ignorance showing, but really? people can just computer large factorials and binomial coefficients in their heads fast enough to participate in the game?

I had always thought that knowing the odds exactly just wasn't that useful because there was a lot of missing information and it was more important to read one's opponents.

It's usually not anywhere near that complicated. To use an example from the article he folded a QJ against a QT with 89J on the board with no flush possibilities. In this case there are not very many hands that can be beating him: the pocket pairs AA, KK, QQ, JJ, 99, 88, stuff that makes two pair like J9, J8, 98, and the straights QT and T7. Overall this is going to be close to the chance when holding 77 that your opponent has a higher pair, which most professional players will "just know" is about 3-4% because they memorize tables of such facts[1]. This means he should believe he's a pretty substantial favorite and why the fold was so unexpected.

[1] https://poker.stackexchange.com/questions/1203/pocket-pair-s...

It is that complicated if you’re trying to factor accurate ranges, card removal and bet sizing. He’s obviously not doing this because he’s playing an exploitative style. In otherwords, he’s guessing perfectly based on his perceptions.

For the mathematically perfect, they spend hours studying solvers which takes considerable processing power to generate the trees. A human player can calculate aspects of it but they can really only memorize what the solver would do. And some high stakes online players are using custom software to augment their decisions.

Depending on the bet size, the solver could call the turn a % of times. And depending on blockers, unblockers and the bet sizing, the solver could advocate to call/raise/fold.

Unless you are playing at the absolute elite levels you do not need to perfectly calculate your own range or another persons. But yes, the better you get the more complicated the math is.

> It's usually not anywhere near that complicated.

Top pros play millions of hands and if you truly have a knack you remember... everything.

"Read one's opponents" isn't about reading a person's soul sitting across from you. That whole aspect of poker of picking up tells plays a much smaller role than simply understanding what hands your opponent is likely to have in each possible specific situation or knowing your opponents playing tendencies, such as how frequently do they raise, call, fold, and in what cases would they do those actions. Tells can still be important but most experienced players will have learned to control their emotions to the point where it's far more reliable to rely on your understanding of the game than to rely on some psychological meaning behind someone splitting an Oreo cookie before making a decision.

As far as computing "large factorials and binomial coefficients in their heads", I don't know what makes you think that that's what poker math entails. It's nearly all just basic statistics.

> As far as computing "large factorials and binomial coefficients in their heads", I don't know what makes you think that that's what poker math entails. It's nearly all just basic statistics.

honestly I've never played much poker, but I remember doing lots of poker odds problems from my discrete math class in undergrad.

In practice,

a) most situations that come up in poker just aren't that complicated -- if you're good at mental math, you can compute the exact odds for the most common situations in your head with some practice

b) there are some simplifications/heuristics you can apply if you're willing to be off by a small margin of error -- e.g. "if you have x outs, you have approximately 2x% to hit one by the river for each card remaining" is accurate enough for most players in most situations

c) this math ends up being extremely useful in practice, because "reading" someone (which is usually just logically deducing reasonable possibilities for what is in their hand) is only useful if you can then use the information to determine whether it is profitable to make/call a bet, which requires knowing your likelihood of winning.

Texas hold'em is straightforward to calculate odds for. There are 52 cards in the deck. Two are in your hand, and after the flop three are on the table, leaving 47 unknowns. That means if you have N outs (cards that will improve your hand), the chance of hitting one on the turn is N/47, which is about 2N per cent. Likewise the chance of hitting by the river is about 4N%.

The hard part is calculating the implied odds: there you need to play out scenarios of what your opponent might hold, and what cards would cause them to bet or call with a losing hand.

Tells are overrated in poker. It's all about implied odds and grinding it out.

There are a total of 169 starting hands and at most half are relevant. Most people you can put on a range of 40 or fewer hands. All the math you need can be inferred from about 100 things you need to memorize, most of which are fairly intuitive.

only recently have very powerful computers gotten better at poker than good humans. a phone probably can't do it yet. but a phone could be used as a way to do various kinds of digital spying or communicating with other people etc.

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