Since noone seems to reflect on my last posting, I pose a problem.
Play the game in [0,1]. I.e., choose a point at random (uniformly) in the interval, and decide, if you want to keep this value or not. If not, you get a new point, and loose everything if the new point is less or equal the old one. You can repeat that as often as you like.
What is the best strategy, and what is the average win?