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When playing 1/2NLHE. Given that a player has a proper bankroll. What is the right amount a player should be waiting to make per session before cashing out?

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This question is considered by way to many players as relevant to building a bankroll. What you are winning or losing in any particular session has nothing to do with your expectation. One is going to have rushes and bad runs. These things even out. What the important thing to keep in mind is that you want to play winning poker for as many hours as you can.

Most poker players tend to want to leave when they are ahead, and stay to recoup losses when they are behind. Logically this is the worst thing you can be doing. When your losing there are a lot of good arguments to get out of the game. You're perhaps losing because you're playing poorly, you're losing because they are playing better, you're losing because your being cheated. If you're really lucky you're losing because you took a couple of bad beats, and can regain your composure enough to beat the game. It's time to leave that game if you just cant get a handle on it. Not one of us should expect that we can get a handle on every session we play. Not every session is positive EV for any player.

Then just the opposite for when your winning. Never quite playing a session when you are winning, as long as you're feeling good. When you're having a good session it is because you have a handle on things and should keep playing and it is the other players that are making the mistake of staying and playing.

When a players asks the question "how much should I make in a session before I quit?," I am sure I am talking to a player that is not winning near what they should be. I am usually talking to a big lifetime loser who needs to get out of a game with a win once in awhile because it is the only way to get ahead of the game for a day.

Playing shorter sessions to lock up wins, and longer sessions to recoup losses, is not going to have a positive effect on your hourly rate or long term win. It is going to be negative on both, simply because your tendency is going to be to leave good games and stay in bad games.

The premise of your question is bad! The premise is that winning a certain amount makes your expectation higher if you get out of the game when that amount is achieved. That is playing a system, that does not work. You do not change your odds, you do not increase your expectation, you do not build bankroll.

You should really be considering what is wrong with your game were you feel it is advantageous for you to get out of a game you are beating. The toughest part of poker is fatigue, and overcoming yourself. Top pros, they learn to put in the hours, and when the game is not good to switch tables. They do not quit because they made a score. They would have trouble understanding why a serious player would even be asking your question. Not bad mouthing you just saying, this question is from the domain of the confused. It is not about winning, it is about controlling losses for players that mostly lose. These experienced old losers talk about this kind of thing, and new players consider it. But it is the stuff that is bad for your game. It almost looks like it makes sense, and sometimes it is just good to turn the worm with locking up a win after a bad run. But it is a loser's strategy, not a winning tactic.

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Normally you should keep playing as long as you feel you have an edge over the table.

If you do want to work with a certain cut-off point, it would be better to specify it in time than in winnings or loosings. For instance, you might know from experience that you can't keep focused for more than 4 hours, so you quit after 4 hours, unless you are at an extremely juicy table.

Another reason to quit a table could be if your stack gets very big and the only players that have an equally large stack are very strong players. If these are very weak players it would be an incentive to stay.

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