Random Game Theory Stuff

Starting this thread as a place to talk about different game theory concepts without derailing other threads.

This seems to be a very developed field with right and wrong answers but most people in the poker world misuse or misunderstand the terms and concepts of this field (myself included). I'd like to use this thread to get the correct definitions of terms and to better understand game theory concepts in general.

On January 28 2015 12:02 Romm3l wrote: remember optimal strategy is defined as the strategy that minimises the expectation of a perfectly adapting opponent.

First of all, what is optimal play?

Is this just shorthand for "game theory optimal play"?

Because it seems "optimal play" could also be interpreted as the maximally exploitative play -ex. taking maximum advantage of a fish's weaknesses. I'm pretty sure we usually mean the former definition but wanted to point out it could be confused with the latter.

Right now I assume that "optimal play" is shorthand for "game theory optimal play" which is a strategy that is at "nash equilibrium"

http://en.wikipedia.org/wiki/Nash_equilibrium


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On January 29 2015 17:02 Romm3l wrote: because minimising a perfectly exploitive opponent's expectation is the strict definition of optimal play. http://en.wikipedia.org/wiki/Minimax

The creator of GTORB had this to say about defining GTO Strategy (I'm assuming we're using this interchangeably with "optimal play" in the way that you have:

"One of the most common misconceptions that people tend to have regarding GTO poker play comes from the idea that somehow the key element of a GTO strategy is its "unexploitability" or "balance" and the belief that any unexploitable strategy is inherently GTO.

The conditions required for a strategy to be GTO are much stronger than simple unexploitability (although of course any GTO strategy must be unexploitable), and in a practical sense, the elements of GTO play that are generally going to be the most valuable to try and use in real world poker games are the elements that have nothing to do with unexploitability. By focusing on unexploitability people minimize and miss what is actually the a huge part of the value of understanding GTO play." - http://blog.gtorangebuilder.com/2014/...-so-much-more-than-unexploitable.html

Interested to hear your thoughts after reading and if you would change your definition or not.

GTO = Optimal = Nash

I think "optimal" is a bit confusing because it doesn't necessarily mean "the best" in this instance, but whatever, we're stuck with it.

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I think much of the other confusion actually comes from different usages of "exploitable". It can mainly mean one of two things:

1. Exploitable means using a strategy that has weaknesses. In this sense, a weakness merely means "not perfect", and we usually define "perfect" or "optimal" as the GTO strategy. So when we use this definition, an "exploitable strategy" is actually just the negation of the GTO strategy by definition. This logically entails that all GTO strategies are actually unexploitable by definition. This does not mean that a strategy cannot be beaten or will never lose money on average.

2. Exploitable means a strategy that can lose money. This logically entails that an unexploitable strategy cannot lose money. For heads up rake-less poker, in which an even number of hands are played and starting stacks are reset, the GTO strategy is unexploitable. For multiway poker, or for hu poker with an uneven number of hands (one player gets the sb more often), the GTO strategy is not unexploitable. This should be intuitively obvious. No matter how good you are, if you play the big blind in a heads up game every hand you'll lose money.

I don't really think using either 1 or 2 is necessarily wrong or right. But I think I'm a bit partial to 2 since it seems to fit the more colloquial definition of "exploit". We just need to drop the notion that GTO play is "unexploitable" for it to fit. Definition 1 also doesn't quite work when we're playing multiway poker. The Nash equilibrium is the state where no one player can deviate and improve their expectation. But two players still can both deviate (collusion). So it doesn't seem right to say we're "unexploitable" when two players can be colluding against us and winning a ton of money against us.

That link actually clarified a lot of things to me I was confused about how could only be a virtually infinite amount of balanced shove ranges on the river by changing sizing yet only one GTO, good to know now.

2. Exploitable means a strategy that can lose money. This logically entails that an unexploitable strategy cannot lose money. For heads up rake-less poker, in which an even number of hands are played and starting stacks are reset, the GTO strategy is unexploitable. For multiway poker, or for hu poker with an uneven number of hands (one player gets the sb more often), the GTO strategy is not unexploitable. This should be intuitively obvious. No matter how good you are, if you play the big blind in a heads up game every hand you'll lose money.

From a math point of view, that's not what exploitable means. Whether a strategy can or can't lose money on average doesn't necessarily have to do with GTO (although if you are restricting it to a situation that is symmetric it should) . If we were playing a headsup poker game where the button didn't move, even though someone is stuck with the big-blind every hand they can still play in a way which is game theory optimal. That way would simply be the strategy that guarantees the highest EV if the opponent is playing in the best way possible to beat that strategy. The strategy is GTO (and also unexploitable, which is one of the necessary components of GTO) but in this case it can still be losing because there is an inherent advantage in the game that the button gets to take advantage of.

*EDIT*

Which was the point of your post. Sorry. That the mathematical definition of it is not how most people would usually toss around the word.

I don't think there's an official "mathematical definition" of exploit like there isnt an official definition for "game theory optimal". It's mainly just a word poker players throw around.

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Been reading the GTORB blog a bit and I gotta say it seems completely wrong to me somehow. Speaking mainly of the Brain Teaser #6. Seems like the author is arguing that we're unexploitable because our opponent can't adjust to us to make more money, even though we're not GTO because we can adjust. This however doesn't account for the fact that villain could already be using the best possible exploitative strategy.

On January 30 2015 02:00 NMcNasty wrote: I don't think there's an official "mathematical definition" of exploit like there isnt an official definition for "game theory optimal". It's mainly just a word poker players throw around. ------- Been reading the GTORB blog a bit and I gotta say it seems completely wrong to me somehow. Speaking mainly of the Brain Teaser #6. Seems like the author is arguing that we're unexploitable because our opponent can't adjust to us to make more money, even though we're not GTO because we can adjust. This however doesn't account for the fact that villain could already be using the best possible exploitative strategy.

Brain Teaser 6 is saying that on both cases, in the shove and the bet/bet scenarios we have a perfect balanced ratio of bluffs/value thus even if villian knows perfectly our range he cannot exploit us, however the bet/bet line results in better equity.

Nash Equilibrium (NE) means that players are playing best response to each other. So If I'm playing x, villain's best response to that is y, my best response to villain's y is again x. (x,y) is NE.

OP's link gives a definition of unexploitable which goes like this: If I play x1, best response to that is y, which is part of (x,y) which is NE, then I'm unexploitable. As the example demonstrates, the hero is better off by switching to x from x1, knowing that the villain will play y. That is, x1 is "unexploitable" but is not a best response to the best response of x1, which is y.
I don't know if this is the "correct" definition of unexploiatbility.

In 0-sum games like poker, NE is always also a minimax solution. That is, if I minimize my loss against the "best" strategy that my opponent can choose, that strategy x will be my part of NE.

It may happen that I am playing x, part of NE (x,y), villain is playing y1, so that y1 is a best response to x, but x is not best response to y1. That is, I'm playing my part of minimax/NE but actually pass on making money against villain. If I switch to x2, which is best response to y1, we will not be at a NE with (x2, y1), because now the villain can switch to y2 and start making money against my x2. So if I choose to "abandon" my NE strategy and try to exploit the opponent, I may open myself to exploitation. The cycle of adapting to each others' strategies may or may not bring us to NE, depending on how we adapt.

On January 30 2015 03:27 Baalim wrote: Brain Teaser 6 is saying that on both cases, in the shove and the bet/bet scenarios we have a perfect balanced ratio of bluffs/value thus even if villian knows perfectly our range he cannot exploit us, however the bet/bet line results in better equity.

Right, so he's a using weird third definition of exploit, where it specifically means changing your strategy. If we already have the best possible strategy against any given strategy, and given that our opponent's strategy is non-optimal, we're already exploiting our opponent. There's no need to change strategies in order to satisfy the definition of exploit if we already have the best one.

On January 30 2015 07:48 pluzich wrote: OP's link gives a definition of unexploitable which goes like this: If I play x1, best response to that is y, which is part of (x,y) which is NE, then I'm unexploitable. As the example demonstrates, the hero is better off by switching to x from x1, knowing that the villain will play y. That is, x1 is "unexploitable" but is not a best response to the best response of x1, which is y. I don't know if this is the "correct" definition of unexploiatbility.

It seems wrong. Its basically saying that if our opponent's best strategy against us is the Nash strategy, then somehow, our strategy, regardless of what it is or what its EV is, automatically becomes unexploitable. I mean if our opponent is both taking advantage of the fact that we're using a non-optimal line, and is making money off us, but we can still be unexploitable, I'm not sure what "exploitable" could possibly be.

gt course i plan to watch

On January 30 2015 09:10 NMcNasty wrote: Show nested quote + On January 30 2015 03:27 Baalim wrote: Brain Teaser 6 is saying that on both cases, in the shove and the bet/bet scenarios we have a perfect balanced ratio of bluffs/value thus even if villian knows perfectly our range he cannot exploit us, however the bet/bet line results in better equity. Right, so he's a using weird third definition of exploit, where it specifically means changing your strategy. If we already have the best possible strategy against any given strategy, and given that our opponent's strategy is non-optimal, we're already exploiting our opponent. There's no need to change strategies in order to satisfy the definition of exploit if we already have the best one.

I dont know whats your confusion.

That example is to demonstrate that an inexploitable play (balanced range) =/= GTO there are many inexploitable lines but they are not GTO because they yield lower EV overall

When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced.

On January 30 2015 17:31 NMcNasty wrote: When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced.

imagine c/r river with nuts to 'balance' your c/f when betting is more +ev (=putting hands in wrong ranges)

or even using 'non-optimal' sizing on any street


e: nvm what i said

:sitting here, confused:

On January 30 2015 18:07 dogmeat wrote: Show nested quote + On January 30 2015 17:31 NMcNasty wrote: When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced. imagine c/r river with nuts to 'balance' your c/f when betting is more +ev (=putting hands in wrong ranges)

The reason we do that is to prevent ourselves from being exploited by having a weak checking range. Its not that we have two unexploitable options but one is better. One option is clearly exploitable.

On January 30 2015 17:31 NMcNasty wrote: When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced.

i think you understand concept of exploitability wrong

imo one is exploitable if his opponent has incentive to change his strategy to gain ev
unexploitable strategy with highest ev is 'gto'

On January 30 2015 18:50 NMcNasty wrote: Show nested quote + On January 30 2015 18:07 dogmeat wrote:   On January 30 2015 17:31 NMcNasty wrote: When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced. imagine c/r river with nuts to 'balance' your c/f when betting is more +ev (=putting hands in wrong ranges) The reason we do that is to prevent ourselves from being exploited by having a weak checking range. Its not that we have two unexploitable options but one is better. One option is clearly exploitable.

here one could c/r w/ nuts to protect his range while it might be better to bet and c/c with some other hand etc. goal is to not allow our opponent to profitably bluff atc (and even this isnt true for some scenarios like if we get to ther river w/ polarized range and we are just c/folding our 2 street bluffs, etc)

On January 30 2015 17:31 NMcNasty wrote: When our "inexploitable lines" are yielding less EV, that means our opponent is gaining EV. In other words he's exploiting us, both in absolute terms (he's making money overall) and in relative terms (he's making more than what he would if we were playing GTO). So I'm not sure how hero would be unexploitable at all except for some vague notion of being balanced.

No, you are confusing unexploitability with GTO, thats what the example is explaining.

He has a perfect ratio between bluffs and valubets on the turn, a clarivoyant opponent who knows our range exactly to the last combo cannot exploit us on the turn, however a line that contains a bet on the turn and bet on the river also perfectly balanced yield us a better overall equity.

The opponent didnt exploit us, he didnt deviate from GTO play in order to increase his EV, that is exploiting

anyone can help me solve this simple toy gaming? I feel a retard lol

The pot is 1, my range is {As, 2s, 2h, 2c, 2d} Your range is {Ks} Highest ranked card wins You are forced to check I can either bet 2, or check behind What is my GTO strategy?