Can anyone explain the Nash equilibrium solution....the easy way..

A NE exists when no party in a strategic situation can improve their standing by changing their own strategy individually, unless another party changes theirs first. So if you were playing some sort of game with an opponent and were in a NE, then no matter how clever you are in adjusting your strategy you can't improve your expected results (though, you can make them worse in may cases).

That's the easy way, though a little sloppy.

Let's say you start with some strategy. You tell your opponent your strategy. He comes up with a strategy to counter yours and tells it to you. You come up with a strategy to counter his and tell him what it is. You continue doing this until there's no adjusting left to do. Then you have a NE.

In poker there are too many possible strategies to compute a NE (if a NE were available to us, it would be somewhat silly to continue playing, imo). Keep in mind a strategy tells you what to do in every situation. I forget where I heard this, but someone claimed that it would take computers 200 years to come up with NE for HU limit hold'em. And that was after factoring improvements in technology.

It's likely that all the poker games played today will be solved. But it's a safe bet that we'll all be dead by then.

You should also note that even if you knew the NE to hold'em it would not be correct to use it. For example, the NE suggests bluffing in certain situations. If your opponent never folds in those situations, then bluffing is unprofitable, even if it is unexploitable.

To put it another way: NE is like thinking on level infinity (no one can "level" you); if your opponent is thinking on level X, then you want to be thinking on level X+1, not level infinity.

if you did have a NE solution to holdem, this strategy would be profitable against any other strategy right?

or could a no NE strategy be profitable against an NE strategy even though this strategy could be easily exploited by a different no NE strategy?

Are you talking about the 2-player case? In that case, an equilibrium strategy would guarantee at least breaking even against every other strategy in expectation (although it wouldn't maximally exploit the other strategy). Conversely, no non-equilibrium strategy can be profitable against an equilibrium in expectation.

I am pretty confident that 99.9% of all poker players should play a GTO strategy if they knew it. Probably all players should.

Um, no. The very first thing players learn is how to exploit loose-passive players in NLHE. All that bet/folding is massively exploitable yet very profitable in the right circumstances and very easy to learn.

Actually a Nash Equilibrium strategy would probably look much more complex with a lot of balancing, possibly different bet sizes on the same board, playing the same situations differently, etc. And of course it would do a lot worse against typical fish than the typical exploitative strategy does.

Kill Everyone has a NE pushbotting solution for MTT's. This of course assumes that your actions are restricted to push/fold/call, but for simplified situations (as often occur in MTTs), Nash Equilibrium solutions are possible.

It should also be noted that Kill Everyone also discusses exploitative strategies as well. For example, it is often correct to shove ATC from the SB into the BB, as most BBs do not call wide enough, yet the NE solution is to shove ~60% of hands (depending on stack sizes).

In most realistic non-zero-sum situations, Nash equilibrium solutions are traps. Players can usually do better by cooperating, which may require trust or binding side-agreements.

For example, it is arguably a Nash equilibrium solution to kill every stranger you see. If everyone does this, it's hard for an individual to do better (maybe run and hide works, I'm not sure).

If you treat poker as a zero-sum game, then Nash equilibrium solutions are not interesting. But if poker were a zero-sum game, people wouldn't play.

Think of the game of rock - scissors - paper.

If you play any strategy other than 1/3 each, then I can play a strategy thout beats you. But once you discover my strategy, you can change to one that beats mine. So we are not in an equilibrium.

But if you play 1/3 each and I play 1/3 each, neither of us has any incentive to do anything different. That is a Nash Equilibrium.

Since rock-scissors-paper is zero-sum, that is also a minimax solution, a global equilibrium.

Suppose instead some third party would pay the winner $3 for winning with rock, $2 for winning with scissors and $1 for winning with paper; and nothing for a tie.

If we can communicate and cooperate, say by agreeing to split the winnings, we would designate one person to play rock and one person to play scissors. We would always get $3, $1.50 each.

If we cooperate but cannot communicate, we would each play rock and scissors with 50% probability, to have a 50% chance of winning $3 for an EV of $1.50, $0.75 each.

Communication without cooperation makes no difference.

If we cannot cooperate or communicate, the Nash equilibrium is to play rock 6/11 of the time, scissors 2/11 and paper 3/11. Our EV is 6/11 = $0.55. This is a trap in that it is less EV for each of us than if we cooperate. If we just play 1/3 for each shape, our EV is $0.66.

You can think of the Nash equilibrium as starting from the cooperative, no communication solution of rock 1/2 and scissors 1/2. We flip a mental coin, but if it comes up scissors, we know we are going to lose. So sometimes we play rock instead. If we cheat enough (and we assume the other guy is doing the same as us) it becomes profitable to play paper. As we cheat with paper more, it actually becomes profitable not to cheat, that is, to play scissors. In the end, if we flip scissors, 1 time in 11 we switch it to rock and 6 times in 11 we switch it to paper. 4 times in 11 we don't cheat, although in fact we are really cheating so many times we end up acting honest.

Yeah, well that doesn´t mean it´s right and also we don´t know any GTO strategy. If you did you would be foolish not to use it. Since I don´t know anything about GTO I can´t say that you would lose alot of value against fish or not, wouldn´t surprise me if you would lose less than one would intuitively(is that a word) think. However against any decent midstakes reg and better you would more likely to be better of with a GTO strategy. If that is not enough, compare your winrate at NL5k and higher today and what you might achieve with a GTO strategy there and you have your answer. I am surely a losing player at 25-50$ but if I played GTO I wouldn´t be and think about how many people that applies for.