You are missing an important element of the game theory, and the missing part is what makes the stakes matter.
You can't just make decisions based on expected value of the bet, because when you run out of money, you can't play anymore. A bet that has a positive expected value, but has a good chance of knocking you out of the game, is not worth playing if your baseline expected value per hand is positive. Running out of money (either your actual money, or the amount of your stack in a tournament) means guy can't make any more bets, so you have to factor that risk in.
This is why it is different without real money. Does the game end when your bankroll runs out, or are you playing such that you can refill your stack? That will change the game theory.
> You can't just make decisions based on expected value of the bet, because when you run out of money, you can't play anymore.
Yes, variance can be a bitch, but that goes back to my original point, if you can't afford to go all in on a flip with a pocket pair vs a range that's most likely to be 2 overcards (i.e. TT+), where you should have at least 55% equity in a heads up scenario, you can't afford to sit at that table.
This isn't as relevant for recreational players, but for this exact reason professional poker players should have a bankroll of _at least_ 200 max buy ins for a cash game, and even more for tournaments.
Even pocket aces will lose at least 18% of the time when going all in pre-flop in a heads up scenario (more multi way). If you can't afford to take that chance, you playing at the wrong table. Of course, most people would be willing to take these 80/20 odds even with money they can't afford to lose, but that's a whole different scenario.
You can't just make decisions based on expected value of the bet, because when you run out of money, you can't play anymore. A bet that has a positive expected value, but has a good chance of knocking you out of the game, is not worth playing if your baseline expected value per hand is positive. Running out of money (either your actual money, or the amount of your stack in a tournament) means guy can't make any more bets, so you have to factor that risk in.
This is why it is different without real money. Does the game end when your bankroll runs out, or are you playing such that you can refill your stack? That will change the game theory.