The expected value is indeed infinite, but I don't think it follows from that that one should pay an unboundedly high amount to play the game.

Sure, over an unbounded number of games, I can expect to make an unbounded amount of money, however that's only *if* I'm able to play an unbounded number of games. It doesn't take into account how quickly my cash on hand depletes, preventing me from playing further.

For a fixed amount $n I pay to play, my chance of making a profit *in a single game* is less than 1/n. (If n is a power of 2, then the chance of profit is 1/(2n), if I n is not a power of 2, then the chance of profit is 1/k where k is the smallest power of 2 larger than n). That's just a profit on the one game though. The chance that I make an overall profit, however, is going to be tricky to analyse I'm going to have to think about that for a bit.

So, I guess it comes down to how many times I am allowed to play. Only once, then $2 (the 50/50 amount, or maybe $4 on a gamble). Multiple times: then it would be an optimisation of how much money I had and how many times I could play vs chance of making an overall profit vs how much I stand to lose.

If I'm allowed unlimited store credit, then *maybe* I keep playing until I'm on top (or I die, whichever comes first) :) Although, in reality, if somebody was offering unlimited credit on a game like this, I'd expect some sort of scam/trap/trick.