Question/discussion on multiway pots and nash equilibria - how do we approach it ?
Posted by Benzzz
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Benzzz
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Mid Stakes
Question/discussion on multiway pots and nash equilibria - how do we approach it ?
Hello all
I have just read a post on how GTO multi-way can harm us in cases and it has disturbed my understanding/approach.
I was wondering how you guys approach multi-way scenarios with regards to study and improvement?
By considering the below information it would seem that applying/following preflop equilibrium solutions solved by a solver like monkersolver for say btn sb bb as a guide when playing with unknown opponents may in fact be useless ?
I have pasted the text below:
In 3-handed situations the entire premise of GTO starts to break down,
because a decrease in one of our opponents EV does not necessarily
mean an increase in ours. In fact, it is often the case that an
opponent who makes mistakes can actually decrease our EV even if we
continue to play GTO. The easiest way to see this is to start with a
simple example of a 3 handed push/fold equilibrium in a short stacked
scenario.Suppose we are 3 handed and all players have 15BB and are playing
shove/fold poker in a rake free cash game (15BB is a bit to deep for
this to be a great idea, but that doesn't matter for the sake of the
example). The equilibrium solution for this game is reasonably simple
and can be found here.Basically the button shoves about 29% of his hands, the small blind
calls with 14.5%, and the big blind over calls with a very tight range
(9.4%) and calls if the small blind folds with a wider range (14.8%).
If the button folds, then the small blind shoves a very wide range
(46%) and the big blind calls 28%.Lets assume the hero is in the small blind. If you put that scenario
into an analysis tool like CardRunnersEV (I ran CREV with a 1 million
hand monte-carlo sample which is pretty good but not perfectly
accurate, particlarly because CREV rounds to the nearest 1BB/100) you
can easily see the expected value for each player when playing the
Nash Equilibrium strategies. They are:Button EV: 19 bb / 100 (HERO) Small Blind EV: -11 bb / 100 Big Blind
EV: -8 bb / 100If all 3 players play GTO, on average each player will win 19bb / 100
in the button, lose 11bb / 100 in the small blind, and lose 8bb / 100
in the big blind, netting to 0bb / 100 break even play.Now we know from the Nash definition that if any player starts from
the Nash state (where all 3 players are playing Nash) and changes only
his own strategy, that he will reduce his EV. Lets assume that the
button is a weak tight player and does not shove nearly enough. We
know this has to decrease the buttons EV, but nothing about the
definition of a Nash Equilibrium guarantees that the button's change
in strategy won't also decrease the hero's ev.If the button only shoves: 55+, AJ+, KQ, KJs, QJs, JTs the EVs become:
Button EV: 15 bb / 100 (HERO) Small Blind EV: - 17 bb / 100 Big
Blind EV: 2 bb / 100The hero's EV is down 6bb per 100, even though he is still playing the
GTO strategy. The hero's EV decreases by more than the buttons EV,
even though the button is the player making a mistake! If you imagine
that every player plays GTO in all positions, except for the one fishy
player who is too tight on the button, what happens to the hero's
winrate? He wins 19bb / 100 on the button, loses 17bb / 100 on the
small blind, and loses 8bb / 100 on the big blind, for an average of
-2bb / 100. Playing GTO poker in 3+ way scenarios can lose money if there is a fishy player at the table who is not playing GTO.If you imagine that the Big Blind player is a smart reactive player it
can get even worse! The condition that the big blind must lose EV if
he changes his strategy away from the Nash Equilibrium strategy no
longer applies once there is a fish on the button. The Nash condition
is only relevant when ALL players are playing Nash. Now that the
button has changed his strategy, the big blind player can change his
strategy as well to increase his profit and to reduce our hero's ev.
If the BB tightens up his over-calling range he can further reduce the
hero's EV by almost another 1BB / 100 when the hero is in the small
blind.In 3-way pots with a fish a GTO strategy can lose and furthermore, a
smart reactive player can adjust his strategy to make the GTO strategy
lose even more.It is important to note that the above are not due to ICM, they appear
even in cash games. In SnG situations where ICM is a factor there are
even bigger and more obvious instances where the presence of a fish
can make a nash strategy -EV, but the fundamental issue in both cash
games and ICM cases is the same.
Using any postflop scenario that has been solved by monkersolver as a basis for learning, as if one of the players is deviating greatly it can potentially be reducing our ev drastically to achieve the opposite of what we set out when solving the scenario in the first place ?
I have been treating my multi-way solutions with monker solver the same way that i had any head-sup solutions that i had solved for.
Any discussion around the approach to multi-way scenarios would be appreciated !
Here is the link to the text i quoted:
http://blog.gtorangebuilder.com/2014/03/gto-poker-outside-of-heads-up-what-it.html
Loading 13 Comments...
Well you basically conclude that playing a GTO strategy vs any other strategy is not maximizing EV. That is true for any scenario, not only multiway pots. Using GTO ranges for HU scenarios for example is not maximizing EV either, since no one is playing GTO preflop.
Saulo, i appreciate your replies. Thank you.
As I understood is that we are trying to achieve a given amount of ev in a strategy - if both players in a hu match played perfect gto their ev's would be net 0 over both positions, that being said they aim for an ev of 0.
If one was to be achieving a negative ev in the headsup case when aiming for the ev of 0 that would be undesirable like in the text multi-way where the sb nash losing 17bb and not 11. Here it would seem pointless to even use the nash solutions as a framework ?
This thought process also doesn't seem to consider that when the BTN shoves a tigher range than he should, even tho we make less money by calling with the same GTO range from before, we face the opportunity to shove vs the BB more often, which then should help increase the EV of our overall SB strategy
Surely this (the fact that sb has more opportunity to shove as a result of btn folding more) is included within the EV results? I mean the solver solved for the EV's of the entire strategy of each individual playing nash and then it solved again with sb and bb playing nash and the button not.
There is no known multiway solution that guarantees plus or 0 EV in all multiway pot situations.
The other players can collectively play in ways that would cause the "system" player to lose.
Not saying we should not study multiway with technologies aids, though.
This is no different in practice from heads up pots. When you hold a bluffcatcher and call against an opponent that never bluffs he's also making a mistake that reduces your ev. Multi way it's exactly the same. That doesn't mean that the solution is useless, just that you have to be careful in which situations to use it and in which situations to deviate. Unless you play some very hard games you should study gto mostly to know how to deviate and not how to play it anyways.
Not true. You must not just look at the EV for one single combo. Our overall EV with a GTO strategy stays exactly the same, as we are winning the pot more often uncontested if V. is never bluffing.
And once I'm in nitpicking mode - let's continue. :D
That is not true either! All we know from the definition is that he can't increase his own EV by changing his strategy. It does not necessarily decrease his EV. It can stay the same as well - as long as his opponents are sticking to the GTO strat.
And let's move on - I'm running hot:
Nope. What he (correctly!) concluded was that playing a GTO strategy vs. any other strategy is minimizing potential loss! And that principle seems to lose it's validity in multiway pots.
To finally add something more "constructive" I'll have a look at the model and come back. :)
BigFiszh
OK, I've taken a look at the original model you posted and entered the model in CREV. Honestly, it looks like some kind of combowise rounding error.
Pretty sure, if you enter the correct split "Nash combos" for each position, it will show that SB's EV will remain stable.
I mean, look at it - the overall EV of SB is -0.12bb vs. -0.19bb. And then look at the EV in the original ("Nash") model. Just trim it to BTN vs. SB - and see that 33 is a losing call (let alone that BB could overcall) and instead KJs, A5s and such were +EV calls (which are missing). That let me think that the combos are just not perfect "GTO" which leads to the difference when we change the BTN strategy.
I haven't looked too closely at the model, but I'll make a different point about multiway pots, not relying on "GTO".
Multiway pots are very tricky beasts. The Mathematics of Poker gives an example of a simple toy game where three people play a game. Two of the Villains explicitly announce their strategies. One player is a super nit who checks all his hands and only calls bets if he has the nuts. The second player is a maniac who bets everything and calls every bet.
These strategies are, taken individually, very bad strategies.
It turns out that the third player loses in this game no matter what strategy he applies. The details are in the MoP. But the gist is that, in effect, the two people are colluding against the third player. He gets shafted no matter what he does.
Very good question imo.
I would say it is unfair to expect GTO strategy to not suffer in the face of assymetry.
The asymmetry being the weak tight player only plays weak tight on the button.
If he was to be playing weak tight in every position, i would expect both of the remaining players to profit from his mistakes (though the profit share may or may not be equal because of the inherent asymnetry of a 3 handed game )
So if it was a 4 handed game
A B C D seated around the table in that order clockwise.
B and D were playing weak tight in every position and are playing the same strategy.
A and C are playing the GTO solution.
Then i would expect A and C to profit in this game and equally share that profit because of positional symmetry.
If A and B were playing qeak tight in every position and are plauong the same strategy.
And C and D are pkaying GTO solution
Then I would expdct C and D to profit though not equally because D has 2 weak tight players on his left while C has only one. Positional assymetry.
To get another thing clear: GTO never "guarantees" profit. It ... no I will go one step back:
GT means Game Theory. It's the entirety around maximizing profit (which is equal to minimizing losses). GTO is the abbreviation for "Game Theory Optimal" and actually means the state where EVERY player "minimizes his potential loss" (= Nash Equilibrium), by definition the state where nobody can deviate from his own strategy to gain more profit (this is NOT equal to win less / lose more, not yet!).
So, when two of three players are playing suboptimally, we are never talking about "GTO" anymore! By definition we are not, because GTO demands any player to play "optimally" in terms of GT.
Now, that said, back to my first sentence: a GTO-strategy does not guarantee wins. If we have the choice between -40 and -20, then our "optimal" strategy is -20. For example, even in a HU-game, where we are ALWAYS playing the BB (oop), we are forced to lose. No different way. So, GTO / Nash Equilibrium will show a negative EV for BB - still it's the GTO-strategy (within the range of possible choices, meaning if BB is not allowed to leave the game obv. :D).
That said, what belrio announced is not contradictory to GT - it's just that the "perfect" strategy still is a losing strategy.
Yes, I agree with the clarification. As I said, I'm not really talking about GTO, but about the weird nature of multiway pots.
There is a "meta" level solution which is better in that game: "the only winning move is not to play" :P
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