If I liked the game, I always played until I ran out of money or quit, or you did. I preferred to play by the game or long races.
When I say "ran out of money", I don't mean I threw my life savings out on the table and said "go for it". I have played $2 games and $100 games. What we played for and for how much was usually determined by the setting. If I knew the players, I usually knew what the maximum amount someone was willing to lose based upon previous matches (not necessarily with me). If I played someone I didn't know, I was always good for a few hundred just to get a feel of what you had to offer.
I always preferred long sessions because I was capable of running packages fairly consistently along the way and I didn't always start out "hot". The longer I played, the better I usually got, which is the opposite for many people. I've literally played sessions that lasted for more than 24 hours without a break.
The key to winning is to play "your" game and plan. If you start all the "double up" BS every time you lose, it becomes a game of "who has the bigger bankroll" and not a game of pool skill.