Yeah, you're still confusing zero sum with zero EV to a certain extent. All zero sum means is that for you to win an amount, someone else (or a group of someones) has to lose that amount.

Whether your personal expectation is positive, negative or neutral is irrelevant to that distinction.

I would've learned that years ago, no idea how it got it in my head that personal expectation was involved.

Thanks.

im not quite sure what you mean by "being played for thin value," but this game doesn't imply anything about thin value. the structure of the river game is actually the opposite - our value bets are assumed to win 100% of the time when called. you can extend the game to include earlier streets, and the villain will still be calling on earlier streets in order to prevent us from being able to bluff too much (his call % will be lower on earlier streets bc of the fact there are more streets to play, though). if you change the % of the time we win when called on the river to model thin value bets, you will just see that we can bluff less often.

an important point about the game is that the value of the game to the betting player is not 0, even when both players are playing gto, since when the bettor bets he expects to capture the whole pot, on average. if the bettor never has to x/f (ie he doesn't have "too many" bluffs), then the value of the game to him is the amount in the pot. nothing about gto guarantees that any player will be able to break even or that one player cannot profit, and this game is structured such that the bettor will profit. i think people have brought up the argument that if one player's gto strategy has negative ev (ie the best he can do is minimize his losses), then he can just choose not to play the game. this is sort of circular though, since that is just adding a rule/assumption to the game that players can choose not to play, and therefore break even. you could just as easily add a stipulation that someone has a gun to the player's head and is forcing him to play. games dont necessarily have to model real life to have informative value.

even with a database you can't "know" the villains range. this is simply a challenge in applying game theory to poker, and to playing poker generally.

i think the disconnect is just related to when you are defining the game or sub-game to begin. using HUNLHE as an example, if you define the game as starting after blinds have been posted, then yes, both players have positive expectation because they will each capture some of the dead money in the pot and will never vpip a hand if it has -ev. if you define the game as a single hand of HUNLHE before blinds are posted, then (as you noted) btn/sb has a positive expectation and bb has a negative expectation (equal to btns positive expectation; the game is zero sum). if the game is a full round, where each player plays each position once, then both players have an ev of 0 (absent rake). as CMV noted, sub-games of zero sum games do not have to be zero sum, so if you look at a river sub-game in isolation, it will be positive sum and (usually) +ev for both players (ev could be zero for one player).