In theory couldn't I just employ the exact same strategy as my opponent, or a superior one, and thus be at least break-even? It seems that the fact that the hero is not doing this means he's being exploited.

An unexploitable strategy can be -EV if the game is rigged against you in some way.

Aside from that, I don't know what you're going on about?

@ both posters, rake?

Forced bets (i.e. blinds) could cause a potentially -EV situation for an unexploitable strategy.

In many situations gto loses money.

Seems like a basic logic question.

Unexploitable vs unexploitable creates no edge for either player leaving no expected value for either players. Rake being a part of the game now is introduced and creates an -EV scenario for both players.

It doesn't even need to be unexploitable, if 2 players are playing the exact same strategy regardless of how "good" or "bad" it is in a vacuum. They will always both lose in the long run due to rake.

It never takes -EV actions in a sense that it always choose a better one if available.
It can however lose money overall if the game is such that you are doomed to lose.

doomswitch.

nuff sed.

If you're playing heads up with no rake, the Nash equilibrium strategy at least breaks even against every other strategy. But if there are three or more players it's possible to be up against a combination of strategies where your NE strategy is -EV. All the NE guarantees is that, once all players are using the NE strategy, no one player can do better by deviating unilaterally. But two or more players might benefit by deviating at the same time in certain ways (whether consciously colluding or not). So in a zero sum game such as poker without rake, if two or more players deviate in a way that is mutually beneficial for them, they become +EV and the remaining players still using the NE must become -EV.

To see how this could happen in a very abstract sense, consider a simple three player game played with coins. Each player has a coin which they reveal simultaneously with the other players, showing either heads or tails. If one of the players' coins shows a different face than the other two, that player wins and the other two players must pay them a dollar (if all coins are the same it's a push). The NE strategy for this game is to randomize equally between heads and tails. This is easy to see, since if your opponents are both playing the NE, you don't benefit by deviating from the NE. In fact it doesn't matter what you do in that case, you're indifferent between heads and tails and any strategy you choose would be 0 EV against two opponents playing the NE.

But now consider what happens if opponent A always plays heads, and opponent B always plays tails. Now any strategy you play including the NE is guaranteed to lose every time. (If your opponents want to screw you over without being so obvious about it they could deviate from NE just slightly, for example player A goes heads 60% while player B goes tails 60%. Now your NE strategy (or any other strategy you choose) will win sometimes, but you're still -EV and there's nothing you can do about it other than stop playing.)

There are many situations where you loose in longterm, GTO only provides best strategy (against optimal strategy of villains) and losses less than other action.

Fold every hand = cannot be exploited + can't make any money

While all 3 players playing 50% heads/50% tails is a NE, isn't it the case that one player playing 100% heads, a second playing 100% tails, and the third player playing 50/50 is also a NE? No player improves their EV by unilaterally deviating from that. The player who plays 100% heads only loses by playing tails; the player who plays tails only loses by playing heads; and the player who plays 50/50 always loses due to the other two so cannot improve his situation at all by deviating.

This would seem to imply that in the game you've created, literally every strategy can be interpreted as being "GTO". Which of course gives another answer to the question of how GTO strategies can be losing, since I think there is only one NE that gives 0 EV to all players.

I was careless and you're right about there not being just one NE. Actually there are infinitely many, for example if A plays 100% heads and B plays 100% tails, C is doomed no matter what and any mix he plays is a NE strategy when combined with A heads, B tails. But not every combination of strategies is a NE. Everyone plays heads isn't a NE (whoever deviates to tails would profit).

While not every combination of strategies is a NE, it is true that for every individual strategy, there exists a NE that involves that strategy...

Ok right, every individual strategy (pure heads, pure tails, any mix of the two) is part of at least one NE for that game. I get confused by the terminology. I'm used to thinking of Nash equilibria as being unique, and in that context you can talk about "the NE strategy" for any given player without ambiguity. But really a NE is not a single strategy for one player, but a set of strategies where one strategy (pure or mixed) is assigned to each player. That's one source of confusion for me. I get further confused by the term GTO and I'm not exactly sure what it's supposed to mean. Without reference to a particular opponent or set of opponents, playing GTO would mean playing (a / the) NE strategy? Per that definition literally every strategy in the three player coin game could be construed as GTO, like you said.

But then if we know something about how our opponents play, it may be "optimal" to play a strategy that isn't part of (a / the) NE. Which would be impossible in this case since every strategy is part of a NE, but anyway, I really don't like the term GTO. If we can deviate to exploit a particular opponent, why would game theory suggest we stick with the NE? If GTO is just a synonym for a NE strategy, I find the latter terminology to be more clear. But then I still get confused by it.

So in heads up NLHE (at any reasonable effective stack size), at Nash equilibrium playing the big blind is almost certainly -EV. But the NE strategy for the small blind at the same stack size is +EV by the same amount, assuming no rake. What counts is the EV of playing NE for a full orbit, which for any given stack size is zero.

One scenario where our NE strategy could be -EV even without rake, is a tournament that pays multiple spots where our opponents are willing to make slightly -EV spite calls against us. These calls harm us, harm the caller (and therefore wouldn't be part of their NE strategy), but benefit the other players. If our opponents play NE other than the occasional spite calls which they only make against us, entering the tournament and playing a NE strategy could be -EV.

Game theory doesn't actually suggest any such thing.

Also, "GTO" is not really a strictly defined term in mathematics--I've only ever heard it used in poker discussions--but it usually means a NE strategy, like you said. The reason this thread exists is because GTO contains the word "optimal" which makes some people, like the OP, think that it can never be a losing strategy, which is obviously not the case.

CallMeVernon getting it correct ITT.

The issue is most poker players talking about "GTO" are confusing game theory for nash equilibrium, and confusing nash equilibrium for for minimax. The words are related but aren't always the same thing.

So are we playing heads up or not? Heads up with no rake, playing GTO you are at least even, assuming starting position is to be randomly allocated. More than one opponets then +EV might be impossable if opponents are colluding correctly.