samooth- you talk on defining GTO strat; yaqh talks about coding shorter strategies and not about GTO.

amiright?

Let's cut it here; I can tell we're friendly n stuff and I believe I completely get your posts, but we're obv not on the same page here.. I was just trying to make the point that you can't save data by not specifying what you should do on decision nodes that are not reached in equlibrium, a complete strategy must specify all actions at every node; you come up w/ a clever example of a GTO strat that's very easy to write down, c/f every spot, where it's practically irrelevant to do the above, I can just insist that it's technically wrong to specify your strat like this + the example doesn't help in the context of holdem. Let's agree to disagree if you don't get what I'm saying, or simply think it's BS.

edit: I think some of the misunderstanding has something to do w/ us not thinking thru our post before we click and rather editing them afterwards. lesson learned, think first when chatting on 2p2

See, I wasn't trying to suggest that we save data by not specifying what to do on decision nodes not reached in equilibrium. I think my strategy (which I certainly didn't mean to imply was GTO or at all related to GTO play) does specify a play to make at every node, even those nodes which can't be reached in practice due to the strategy itself. And I'd be interested to know if I'm missing something.

But anyway, that's fine. Personally I want to hear more about durrrr vs slumbot . Feel free to PM if you want to chat offline.

you sure didnt help either. lol what a douchbag. And after that he says something that completely contradicts the 'maximizes EV' statement.

BZZZT!

Thank you for playing,

In regards to the complexity of poker, there is a great paper written by the guys at Alberta here: http://webdocs.cs.ualberta.ca/~darse/Papers/IJCAI03.pdf

The article I found that link in was originally posted in Two Plus Two Magazine but I found it here: http://www.lvrevealed.com/articles/r...c-optimal.html

In the PDF they nicely break down the the complexity of limit heads up poker. The result is 10^18 or quintillion states at the bottom of the tree with many more leaves that terminate when folding before the river. They then reduce the search space with abstraction methods down to a more manageable 10^7. A reduction of 11 orders of magnitude.

As a computer science major who has taken an artificial intelligence class, I would be interested to know if anyone is actively trying to build the complete game tree and how many orders of magnitude they are able to achieve. If they were able to apply algorithms to a 10^7 tree in 2003, maybe todays programmers are using more complex algorithms on orders of magnitude 10^8 or 10^9.

The time and space to traverse the whole HULHE complete game tree is huge. Especially if you will be doing calculations at each decision point to calculate a mixed strategy Nash equilibrium.

As for NLHE, the branching factor would be much bigger when counting possible betting sequences. Since LHE has 17 possible bet sequences when its capped at 3 raises and the limit is fixed, the amount of sequences with different bet sizes and uncapped raises is very very large.

In one betting round lets consider you can bet whole increments 1BB up to 100BB(all in). At 1BB the number of sequences would be 100 + 99 + 98 + ... + 1 = 5050 (the 100th triangular number). 2BB would be 50 + 49 + 48 + ... + 1 = 1275 bet sequences. This is because they could bet raise reraise reraise 100 times at 1BB and 50 times at 2BB and at any time fold or call. Each betting round would have 5050 + 1275 + 325 + 210 + 136 + 105 + 78 + 66 + 55 + 45 + 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1 = 7465 possible sequences if both players were limited to betting a fixed 1BB bet + 2BB bet + 3BB bet up to 100BB per sequence. Lets just estimate that when adding different size bet sequences as well we would have over 10^4 possible sequences.

NLHE adds at least four levels of magnitude over LHE and this is not even accounting for deeper stack sizes. The largest game ever solved is Checkers (english draughts). The search space for that game is 5 x 10^20. Solving checkers would be much easier to compute than poker because many checkers decisions are trivial and its easier to find a draw/win node than it is to compute the best line for a hand.

Not sure where I am going with this but I have to do homework so I am going to end on this quote from the guy who solved checkers:

How do we define "solve" here?

I want to go through this thread and draw a big red X through about half the posts which are flat factually wrong and have it become part of the permanent thread so that others will not be led astray.

Also von Neumann proved the minimax theorem which is the headsup-poker relevant game theory concept. Nash's contributions to game theory were deeper and probably more important overall to the field, but irrelevant to zerosum two player games.

^ give this man modpowers. I'm okay with seeing my post lit on fire.

Agreed, but Von Neumann is a sicko multi-field pro.

dont these two theories lead to the same outcome?

as in... wouldn't the nash equilibrium and minimax for hunl be the same strategy??

If my strategy for the button either folds or minraises, I do not have to specify anything for situations which arose from the button 4x-ing it. The strategy will still be completely determined.

Edit: In my strategy for the big blind I will have to consider the button 4x-ing it, but then I can exlude other subtrees that I will never choose.

Maybe I should not post hastily late at night.

Yep, though I do believe that it gets functionalized quite quickly, but you are certainly correct.

yep, you're right. Divide that by 4 and I was still off by 2 orders of magnitude, sorry.

Ah well. sparse stack matrices then.
But you guys are certainly correct and I stand corrected.

When the heads up LHE machines first showed up in Vegas, I had an email exchange with some pretty smart and generally winning poker friends about it, and they all agreed that if the machine doesn't adjust its style that it's inherently beatable. I tried to explain why that wasn't true and I'm not sure if what I was saying ever really got through.

At the time, I was really surprised that these, relatively high level thinking and winning poker players, could be that wrong about this topic. But I wasn't nearly as shocked then as I am now, seeing that durrrr manages to get the same thing so wrong.

If the (best) bot strategy is disclosed to durrrr in advance then he can calculate the perfect counter-strategy and win 0.1bb per game. That is the max he can win in this bet. If it is not disclosed to him then he has no chance of enumerating the bot strategy (and thus finding the perfect counter strategy) over any number of hands he can play in a realistic timescale. While he was searching for it he would be losing to the bot.

Bet size would not stop this happening. The money will be there under the right conditions.

This. I like you durrrr. You are good for poker. If you do the challenge then I would bet against you but I would prefer you found out the answer in a less expensive way.


Nobody claims the IGT machine plays GTO. Therefore you're friends are correct: it is inherently beatable.

Nash generalised von Neumann's result to n-player games. So if you look at poker in general his results are relevant.

The Min-Max theorem also assumes finite pure strategy spaces, which Nash doesn't IIRC.

Yes.

Nope. You must specify what you will do in every possible situation, even those you're not supposed to be in according to your early decisions. Sounds dumb and quite counter-intuitive, but that's the definition of strategy in game theory.

Is this actually the winrate of the perfect counter strategy against the currently strongest known fixed strategy or are you guessing? if not, which strategy are you using in this calculation as the strongest fixed strategy? Also what do you mean with per game (i assume either hand or orbit)?

Its the per-hand winrate of the perfect counter strategy vs the 2011 UofA bot.

Cool info, thanks, though I assume you meant it the opposite way. Do you (or ike, or anyone else) know how obtainable this strategy would be? Seems like a somewhat important piece of the puzzle if this (extremely unlikely) bet was to ever go down.

oops, yes, other way round - have corrected it, thanks. The counter strategy isn't published but presumably a number of academics at the UofA would know it. (Of course, its not the kind of thing you could write down on a few sheets of paper!) Details of the bot strategy itself would also allow the counter strategy to be calculated (if you had enough processing power). UofA say they can calculate such counter strategies in a day.

This definition is made this way because it is mathematically convenient ("a pure strategy is a function from the space of game states to the set of possible actions"). However nobody tells me that I have to store the strategy explicitly. The definition of the strategy at points that can't be reached is irrelevant, so I can simply say that I fold wherever my strategy isn't explicitly defined. So, again, unreachable state does not add to the memory requirements.

just went through all the thread, really enjoyed it, havent a clue what i read tho, heads pounding a bit now needs a lie down

I just bought expert headsup no limit holdem by Will Tipton. The book is massive. It is going to take me a while to get through all of its theory. It appears to go quite in depth into approximating GTO strategies for use in your normal games. Obviously it is focused on headsup though. When I get through it I will see if I will recommend it or not. I don't have high school level maths, as the book says I require, but I will see how easy/hard it is to understand anyway. I am hoping that the concepts are explained quite well and the algebra is relatively easy to understand/explained with examples. I think the book is designed to be accessible though so I am not too worried.

Question: How are people so well-versed in game theory and nash equilibria? Can you refer me to a website or book so I can read up on it?