Quote:
Originally Posted by Garick
A GTO strategy is, by definition, unexploitable. Whether it is a guaranteed winner is unknown, but unlikely, due to rake.
So I almost put something like what I'm about to say now, in my OP, but decided not to. My point here is going to be, in this quote, the second sentence is correct, but the bolded not only misses the point but is wrong. And I'm sure if Garick did it then many other people reading this thread also did it.
The entire notion that GTO is unexploitable by definition is a myth. The reason it persists is because it's true in 2-player games. But this is exactly what fails in multi-player games. When you are playing a game with 3 or more players, GTO is exploitable.
The definition of GTO is that it is one player's strategy in a Nash equilibrium. And as I pointed out in my OP, a Nash equilibrium by definition is a set of strategies for all players such that one player cannot exploit the
collective NE strategies of the other players. In 2-player games, there is only 1 other player, and so this reduces to saying that GTO is unexploitable. But when there is more than one other player, this is a grossly wrong oversimplification.
If I'm playing a 3-player game with a known Nash equilibrium, and I know that my 2 opponents are playing their NE strategies, they are, by definition, both playing GTO. That means they--as a
collective--are playing unexploitably. But they are not necessarily playing unexploitably
as individuals.
That's exactly what I tried to demonstrate with my 2 toy games in the OP. In the coin game, there is a Nash equilibrium where one of your opponents always plays H and the other always plays T. So you will take -1 every time. There is nothing you can do to exploit that collective strategy to improve your EV. However, it definitely still matters to the individual GTO players what you do:
-If you play 50/50 H/T, you will give each opponent an EV of .5.
-If you play H, you force the H player to take -1 while giving the T player +2, and vice versa.
In the second scenario, what's the difference between you and the other H player? There is none! You are both playing the same strategy, which is GTO, and you are both losing. If you would call this situation "unexploitable" from your perspective, then you must have a very different definition of what it means to exploit someone than I do.
The poker toy game is similar. P can't improve his EV by unilaterally deviating from the NE. The combined strategies of A and B are keeping his EV maxed out at .9. However, if P overbluffs, would we not say that A's strategy of always folding is now being exploited? He's losing compared to the NE, right? What else can it mean to be exploited?
Not only that, but let's say P starts overbluffing, and A recognizes this and starts calling 100% of his hands. Now if B sticks to GTO it is he who will never call, and the collective strategies of A and P are forcing B's EV down even though he is playing GTO!
So when you say "GTO is unexploitable", either you're only talking about 2-player games, or you're wrong, or your definition of "unexploitable" is different from mine and doesn't make sense to me.