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# 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 / 100

If 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 / 100

The 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: