sigh, you are not the first one itt who guesses / states nonsense and does not see why. you also are not the first one itt who thinks he would not be understood when everyone told them they were wrong.

if something "overwhelms my brain" (really?) then it is the simple fact that you say "I obv know about Nash Equilibriums" and follow by asking a dumb question noone ever would ask who knew something about Nash Equilibria. says a lot about your intelligence and about brains being overwhelmed, not what you wanted to say though.

can't be arsed to give you an explaination, i responded patiently and gave you a wiki-link. if you read (and with god's help understood) that you would not have to keep asking dumb questions, and saved yourself and us some time.

Man in my posts im explaining everything I state.. and my posts are in no obvious disagreement with your wiki link. If you think im wrong and ask dumb questions explain.. so far you have provided 0 explanations so its pretty clear who is wasting our time here. Don't wanna fight just seeking the truth yo.

Peace and Love

there are several posts by different posters itt explaining the point.

this. otherwise, it's not a Nash Equilibrium. there are many posts itt saying exactly this. did you miss all of them? or do you know better?

well obv i think i know better.. give me a source that can be trusted that proves me wrong. But be precise don't just give out a link on general Nash equilibrium stuff. And make sure you understand what im saying exactly.

why not provide a "source that can be trusted" which supports anything you say? i do not really care if you understand, just do your homework on your own.

Since most people here agree and you apparently don't, I think it would be fairer if instead you tried to prove your point (stating exactly what you mean, defining what you consider a game, a strategy, and so on), and we could then point out what we think are mistakes or inconsistencies in your reasoning.

Or we could just say that everything's been said in this thread and move on.

Given that gto is defined as a pair of mutually countering strategies, is it not possible that there are parts of the game space that gto never gets to/is undefined for?

Say for example that gto button raises are 50% 3bb, 50% limp, or whatever, then whilst there's nothing to be gained by raising to 4bb, it's far from clear how to optimally respond.

Obviously this doesn't mean that you can exploit a gto bot by raising to 4bb, but rather that presumably making a 'gto bot' actually involves even more work than just finding the pair of gto strategies.

somewhere itt there was a discussion about the computational power a gto-solution for different games would require, and one poster stated something like:

(nlh)tournaments >>>>>> plo > nlh > lhe

anybody has a take on this? would a tournament solution look that much more complicated? i imagine the biggest difference would be the use of tEV instead of cEV? or do tourney-specific parameters like fieldsize and pace need to be included for the solution (and are included as "bubble factor" in tEV anyway)? a gto-solution would work on a "single hand" base, not on a "tourney-as-a-whole" level, no?

would the additional combinatoric effort needed to get a solution for plo not be offset by the limitation to maximum pot bets?

if we wanted to invent a ilaribot which is restricted to bet pot if betting, would the computional power needed be significantly lower than needed if finding a complete solution? if you rated that bot's perceived game strength, would you expect it being practically beatable by humans? how would a nlh-ilaribot compare in relative game strength to a plo-ilaribot?

In order to be a complete strategy set a bot would have to have counterstrategies for every possible strategy of its opponent. Just because a certain strategy is never used doesn't mean the bot doesn't have a counterstrategy for it. Yes, I realize this is kinda hand-wavy.

You are right, but the strategy in the parts of the game tree that is never reached against a GTO opponent is irrelevant for the equilibrium. From an exploitative standpoint however it matters, so some GTO strategies are better than others against suboptimal opponents.

tEV/ICM/Bubble factor are just approximations. You would have to consider the whole tournament as one big game.

This is not quite right.

Suppose S is a NE and call the strategies for the respective players S1 and S2. It is relevant how player 1 plays on parts of the game tree which aren't reachable under S because it mustn't be so bad that player 2 can deviate from S2 to lead player 1 into a part of the tree he plays badly.

The second possibility is that part of the tree isn't reachable under (S1,Q) for any player-2 strategy Q. In this case player 1 can indeed do anything on these parts of the tree (call the modified strategy P) and (P,S2) will still be a NE so it is 'irrelevant'.

But there are many refinements to NE which tend to give much 'better' strategies in that they enure sensible play in parts of the tree which are not normally reached. e.g. a NE can fold the nuts in some cases but stricter forms of equilibrium won't.

I don't think it's quite right to say that the strategies are undefined in the parts of the game space that aren't ever reached at equilibrium -- a strategy by definition must say how to play in every spot in the game. However, there are multiple equilibrium strategies that only differ by play in those unreached regions. People talk about refinements of the Nash equilibrium concept to weed out the less desirable ones.

http://en.wikipedia.org/wiki/Trembli...ct_equilibrium
http://en.wikipedia.org/wiki/Perfect...an_equilibrium

if S is a NE then no player can win by deviating from his strategy. a strategy, by definition, must reach every part of the game tree.

so if S is a NE, no player can win by deviating from his strategy at any part of the game tree.

guess it's just an example, but i doubt that folding the nuts is part of any gto-solution.

Jerrod, I enjoyed your response very much.

Thank you for the thought and effort you put into all of your many posts.

A strategy profile certainly doesn't need to reach all nodes in the game tree. e.g. folding 100% preflop. It's still a valid strategy.

To see that folding the nuts can be part of a NE, consider 3-card (AKQ) poker with a cap of 100 bets/raises. If we have the A and oppo puts in the 100th bet we can fold and still be in a NE provided we call the 100th bet sufficiently often since oppo will have already given up lots of EV by putting in so many bets with K or Q by that point. The occasions where fold to the 100th will only contribute a small -EV, hence we'll still beat him for at least the value of the game, which is all a NE wants.

.

@franxic

In game theory, a strategy for a given player must be defined on all information sets where that player is to act. Reachability doesn't matter (See Osborne & Rubinstein 'A Course in Game Theory' p203 for a definition of 'strategy').

Regarding the fold-the-nuts example, yes you have understood the idea. The strategy I gave will be a NE even though it folds the nuts some of the time. This is a drawback of the NE concept, and a major reason why people are interested in refinements.

Then his losses from checking behind hands which are worse for checking behind will be no less than his gain from his attempt to gain from this card-removal effect.

You are confusing "exploit" in the sense of "change strategy to try to take advantage of what the opponent is doing" with "exploit" in the sense of "succeed at having higher EV against the opponent's strategy."

Look, here's another way to think about it. GTO strategy assumes that it is going to be exploited as much as possible. It is the strategy which performs best against an opponent who exploits it.

do we agree that a gto-solution would be defined on all information sets where a player has to act? also, wouldn't that exactly be the reachable branches of the tree? i don't see the difference to be honest, and it does not contradict what i said imo.

i think that "ever folding the nuts" and playing a gto-strategy are direct contradictions.

if you ever fold the nuts then your opponent can easily exploit this by profitably bluffing more often. this should be true for any poker game under any circumstances, and does not fit into the definition of a Nash-Equilibrium.

Player A is playing strat "a" and players B is playing strat "b". Let aNE and bNE the respective strategies when they are at NE.

I claim that it is possible that when at NE (hence a=aNE and b=bNE) "aNE" could be a function of "b" say aNE= f(b). This is in no obvious disagreement with any of the NE definition I've read.

At the NE case that would be equivalent to aNE=f(bNE), however as player A keeps playing his NE strat and player b deviates from it then by definition aNE=f(b), where b≠bNE. This means that the NE equilibrium strat of player A would now be a function of player B's strategy which would mean that player A might vary his frequencies based on player B's frequencies according to some built in mechanism (function).

Perhaps I can step in between you, since I think I see where you're both coming from. garrondo is correct. You are confusing two definitions of "reachable" in your statement above. One could mean that there is a way for the game to reach that state, but one could also mean that the two strategies in a Nash Equilibrium could reach that state. It is the second meaning that garrondo is using.

Here's a simpler counter-example to your thought that '"ever folding the nuts" and playing a gto-strategy are direct contradictions.'

Let's play Rock-Paper-Scissors for $1/round with an added fourth option: You may, instead of playing R, S, or P, choose LOSE, which offers your opponent $100. He then gets to choose whether to accept the $100 or not.

The following is a GTO strategy: Choose R, S, or P with probability 1/3. If opponent chooses LOSE, do not accept his $100.

(It is optimal, guaranteeing us the $0 that is the value of this game, and cannot be exploited, the opponent cannot do anything to raise his value.)

So, a GTO strategy can make a "bad" choice (like folding the nuts) if that branch of the game tree isn't reachable by a GTO opponent.

In my experience, these kinds of things *do* show up when you directly solve games (say, using matrix form), and if you want to maximize EV against real life opponents (who often do stray from GTO) you need to do something to ensure that you play well even when your opponent goes off the correct path. However, if you solve games using fictitious play (see my previous post in this thread) this is less of an issue. Anthropomorphizing the process: in building the GTO strategy pair with fictitious play you essentially experiment with every line, so nemeses to strategies even on "bad" branches are calculated and used in your final strategy.

Then as everybody else has already said, you need to read more, or more carefully. The definition of a "strategy" (hand waving a bit) is actions based *only on your hand and the point in the game tree.* More precisely, probabilities of opponent actions ARE NOT PART OF A STRATEGY.

link please

You're still not saying what a game is, what a strategy is, and what a Nash equilibrium is in your opinion.