Interesting posts here!
I thought snowie was a gto bot which by definition should not be loosing to anyone. Does the screenshot above mean that snowie is not optimalized well enough yet?
I don't really get what you're saying about multiple equilibria. Would you mind explaining that a bit more to a fish like me? 
Originally posted by DeLau
I thought snowie was a gto bot which by definition should not be loosing to anyone. Does the screenshot above mean that snowie is not optimalized well enough yet?
Although Snowie claims to be GTO, I now realize that I object to the phrase 'GTO' on principle, because it implies that there's a unique equilibrium strategy. Maybe there is, maybe there isn't. Snowie is a neural network that has played a gazillion hands both against itself and against opponents created to try to exploit it. All it does is try to maximise the EV of its strategy against the strategies that it has seen. This probably leads it to play an equilibrium strategy, but unless you're playing a two player zero sum game, there's so much more to it than just finding an equilibrium strategy and claiming it's 'GTO', whatever that means.
I don't really get what you're saying about multiple equilibria. Would you mind explaining that a bit more to a fish like me?
Luckily I've been thinking about this, so here's some examples. Remember, I'm a novice at this, so be nice to me.
Let's consider a 6max table where there are four players playing some rubbishy strategies and two other players who have to decide between two possible strategies, Aggro and Snowie. If they both play the strategy Snowie, they win at 8bb/100 hands, and if they both play Aggro, they win at 4bb/100 hands. If they play different strategies, the player who plays Aggro wins more bb/100 than the player who plays Snowie. Ofc, this is slightly tongue in cheek, because the real situation is astronomically more complex, but it gives you something to think about.
Game 1
---------
Aggro wins at 6bb/100 hands against Snowie's 2bb/100 hands. In the table the first number is the winrate of the player who chooses a row, and the second the winrate of the player who chooses a column.
If both players play Aggro, switching to Snowie unilaterally will reduce his winrate to 2bb/100 hands. This is therefore an equilibrium strategy pair.
Similarly, if both players play Snowie, they have no incentive to change strategy, so this is also an equilibrium strategy pair.
Voila! Multiple equilibria, as promised.
If they play different strategies, the Snowie player has an incentive to turn Aggro, and vice versa. Where will they jump to? Who knows! That's the tricky point. You need to know about the dynamics of this game when it's played repeatedly to find out what happens. How should we model the dynamics? Good question. Feel free to try to answer.
Game 2
---------
Let's change the winrates of the players when they play different strategies to 10 and 2bb/100 hands.
If they both start playing like Snowie, they both have an incentive to start playing Aggro. Once they're playing different strategies, the Snowie player has an incentive to start playing Aggro too so that he's not getting pwned any more. In this case (Aggro, Aggro) is the only equilibrium. However, notice that they'd both make more money if they played (Snowie, Snowie). This game is identical to the classical Prisoner's Dillema game. If you've seen this, you'll know that the prisoners would be better off keeping their mouths shut, but the equilibrium strategy is for both to confess.
Game 3
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Finally, let's look at what happens when the winrates are 10 and 6bb/100 hands. Now, if the players start off using the same strategy, they each have an incentive to change. Both (Aggro, Snowie) and (Snowie, Aggro) are equilibrium pairs. This game is identical to the classical Game of Chicken. (Two cars drive head first at each other, and they can Drive On or Swerve.) I know that there's also a mixed equilibrium strategy for this game where each player could play either strategy with some probability. (There may be mixed strategy equilibria for the other games too, but I haven't chekced.)
What do I conclude from this? Well, as I've said before, there's more that game theory can suggest to us than 'play GTO', it's not clear what 'play GTO' even means, and frankly, the whole thing makes my head spin. I'm planning to try to look at some toy dynamic poker games, where I add in some model for how the players switch between strategies, and I think I have an undergraduate student starting a Maths of Poker project soon, so if he's any good he can have a crack at it.
I just played some Adrenaline Rush on F T. OMG. Four handed. 10bb cap. Possibly the most retarded idea ever.
Here's the GTO solution.
Wait. Didn't I say something about GTO solutions of nonzero sum games? 
Originally posted by jbpatzer
Game 1
---------
Aggro wins at 6bb/100 hands against Snowie's 2bb/100 hands. In the table the first number is the winrate of the player who chooses a row, and the second the winrate of the player who chooses a column.
If both players play Aggro, switching to Snowie unilaterally will reduce his winrate to 2bb/100 hands. This is therefore an equilibrium strategy pair.
Similarly, if both players play Snowie, they have no incentive to change strategy, so this is also an equilibrium strategy pair.
Voila! Multiple equilibria, as promised.
If they play different strategies, the Snowie player has an incentive to turn Aggro, and vice versa. Where will they jump to? Who knows! That's the tricky point. You need to know about the dynamics of this game when it's played repeatedly to find out what happens. How should we model the dynamics? Good question. Feel free to try to answer.
Not sure if I'm on your level here but I'll give it a try. If the game is played repeatedly, I would expect the choice to be at random until there is a reasonably high probability (90 or 95%) that the winrate of (snowie, snowie) is higher than the (aggro, aggro) combination (in this case). It's no model but I'm no mathematician either 
You explained the concept very well btw, awesome post!
Originally posted by DeLau
Originally posted by jbpatzer
Game 1
---------
Aggro wins at 6bb/100 hands against Snowie's 2bb/100 hands. In the table the first number is the winrate of the player who chooses a row, and the second the winrate of the player who chooses a column.
If both players play Aggro, switching to Snowie unilaterally will reduce his winrate to 2bb/100 hands. This is therefore an equilibrium strategy pair.
Similarly, if both players play Snowie, they have no incentive to change strategy, so this is also an equilibrium strategy pair.
Voila! Multiple equilibria, as promised.
If they play different strategies, the Snowie player has an incentive to turn Aggro, and vice versa. Where will they jump to? Who knows! That's the tricky point. You need to know about the dynamics of this game when it's played repeatedly to find out what happens. How should we model the dynamics? Good question. Feel free to try to answer.
Not sure if I'm on your level here but I'll give it a try. If the game is played repeatedly, I would expect the choice to be at random until there is a reasonably high probability (90 or 95%) that the winrate of (snowie, snowie) is higher than the (aggro, aggro) combination (in this case). It's no model but I'm no mathematician either
You explained the concept very well btw, awesome post!
I'd expect a player's choice at each iteration of the game to depend on his opponent's play in the previous iteration of the game or in the previous few iterations. In this simple game you know what strategy your opponent played in each of the previous iterations; in real poker, it takes a fair few hands to build up a picture of how your opponent is playing, whilst he, if he's thinking along the same lines, is doing the same to you. That's a complicated dynamic that depends on the timescales on which each of you is gathering and acting on information. This sort of thinking is what people who attempt to 'play GTO' say that they're aiming to avoid because they are trying to play 'optimally'. Once you realize that this simplistic view of poker is wrong (and yes I realize that a 'GTO strategy' would be complicated and almost impossible to find, but it's a simple intellectual framework), you get back to the sorts of ideas that the more traditional player tries to implement (i.e. hand reading, estimating frequencies and ranges, exploiting your opponent, etc). This sort of thinking is not some sort of opposite to 'GTO'; it's a valid way of thinking about the game within the context of game theory. This is why poker is the king of games and won't be dying out any time soon. 
And I'm glad you liked what I wrote. Thanks. 
And another thing that I thought of.
'GTO' strategies are based on the assumption that everyone plays rationally and tries to maximise their profit. In a two player zero sum game, their loss is your gain, so you can consider playing one of the equivalent, and possibly unique, equilibrium strategies if you can find one. No problemo.
But look at this game again.
Game 1
---------
We assume that if I play 'Snowie' and he plays 'Aggro', he looks at this table, sees that he's better off playing 'Snowie' and changes strategy. However, in complicated games, the players don't actually know the payoffs from each strategy. At high stakes poker you may say that the players do know how their strategy fairs against other strategies and the game is a complex quest to exploit your table. Maybe it is, maybe it isn't. However, at lower stakes poker, those guys just play however they want to.
Going back to the game above, let's say you're playing 'Snowie' and he's playing 'Aggro'. You say to him, 'I say old chap. Did you know that if you switch to playing 'Snowie', we'd both increase our winrates?'. Does he say ' I say. That's splendid! I'm a rational player. I'm going to play 'Snowie' too.' or does he say 'Aggro! Aggro! Duuuuh! Fuck off asswipe!'. What do you do then? Sigh and play 'Aggro' too I suppose.
Maybe we need to look at the assumption of rationality too. 
Have you read The Selfish Gene? I have been surprised at how much game theory there is in Darwinian natural selection.
Originally posted by Tim64
Have you read The Selfish Gene? I have been surprised at how much game theory there is in Darwinian natural selection.
This video is also pretty interesting, includes some game theory -->
Originally posted by outspan
Originally posted by Tim64
Have you read The Selfish Gene? I have been surprised at how much game theory there is in Darwinian natural selection.This video is also pretty interesting, includes some game theory -->
I have read The Selfish Gene, and I don't have time to watch the video, but I know that game theory is a key part of the mathematical theory of evolution, and that evolutionary game theory, which as far as I can tell involves converting games like the ones that I've discussed above into differential equations, is a central part of modern game theory. I specialize in modelling with differential equations, so of course I have some idea of how this works.
A while ago I took the simple AKQ game, converted it into a system of differential equations and used it as an example in one of my lectures. Depending on how you set it up, the two players can either converge on the equilibrium solution or endlessly cycle around trying to exploit each other without ever reaching equilibrium. You can also set it up as a set of discrete equations, at which point you can get chaotic behaviour if you pick the right parameters for the dynamics. I had an undergraduate project student look at this a couple of years ago. You can also try letting each player take some sort of weighted average over previous hands to try to calculate his opponent's strategy and adjust accordingly, but I haven't tried this yet. And this is all for the AKQ game where one player has to choose a bluffing frequency and the other a calling frequency. You couldn't make it much simpler, and yet the results are still frighteningly complex.
Oh Tim, you didn't want to do that, seriously(well maybe you did). 
This could get nasty..or good, I mean really good...!!
OMFG. How can I not have a value bet here?? Poker Snowie. 
:f_confused: 
Apparently when Villain bets the turn, he has the nut flush and the nut flush only in his range. That would explain it.
I'm losing the faith here. And yes, lol @ my turn and river bet sizing before anybody says anything.
This one's more interesting. Iirc, he was a fairly tight 3better, and I should probably fold on the flop if I'm going to give up on a heart turn, and shove the turn otherwise.
I've started playing a bit more now on reguar NL50 tables and had a reasonable start to the month. Can't win anything in these wretched AI four handed January Challenge MTTs though. They're clearly rigged.
http://pt.calameo.com/read/00078372186d246c6560f
Oooooo! Big in Brazil. Cover story, but sadly you have to pay to see the article that I wrote about poker in Portuguese, in which, of course I am fluent.
Originally posted by jbpatzer
http://pt.calameo.com/read/00078372186d246c6560fOooooo! Big in Brazil. Cover story, but sadly you have to pay to see the article that I wrote about poker in Portuguese, in which, of course I am fluent.
Of course.
I'm offended that the idea of me speaking a language other than the Queen's English is so laughable....although when I said 'fluent' I probably should have said 'able to get someone to translate for me'.
Our internet died at home yesterday, and is yet to be revived. It's amazing how much we've come to depend on it. It's almost like having no water or electricity.
Originally posted by jbpatzer
Our internet died at home yesterday, and is yet to be revived. It's amazing how much we've come to depend on it. It's almost like having no water or electricity.
I rember some1 fighting with me when that happend here.
Now y would any1 want 2 speak more than 1 taal. the words "bet", "raise" and "fold" works.
Originally posted by mineriva
Originally posted by jbpatzer
Our internet died at home yesterday, and is yet to be revived. It's amazing how much we've come to depend on it. It's almost like having no water or electricity.
I rember some1 fighting with me when that happend here.
Now y would any1 want 2 speak more than 1 taal. the words "bet", "raise" and "fold" works.
Writing your reply in Portuguese isn't going to impress me.
hey jb, happy new year. Just popped in to see who was still alive in the blogosphere that is the PS.com forums. hope ur killin' it. Bom Dia good sir!
which you can understand, "of course" ![:]](https://forums.pokerstrategy.com/wp-content/uploads/wpforo/emoticons/pokerstrategy-legacy/basics/pleased.gif ":]")
