Problem: EV++ and risk tolerance in a multiple decision scenario.

[pococurante]

I would probably stop at $4k, possibly 8k. It's enough money to matter to me, and I'm just not comfortable with losing that much money at once.

Odds of being able to win 2k: 83%
4k: 69%
8k: 58%
16k: 48%
32k: 40%

While the odds are on your side, it's hardly "free money" to continue playing. At some point, getting great odds is less important than staying without your bankroll... for me that's about 5k.