It doesn't look like it converges.
Fraction of People that win after one play
Fraction of People that win after two plays:
Fraction of people that win after three Plays
It looks like the system should eventually work for many more people then it doesn't but for the people it doesn't work for they'll lose big keeping the average winnings zero.
After about 20 plays starting at 1$ the bet to win has went up to 1 million dollars. About one in a million people will still be playing. For this one in a millionth person that had over a million dollars to bet, I can't help wonder if the first 10 bets really mattered that much to them. However, if they started at something meaningful say $1000 then they are much more likely to end up betting one million dollars. In fact if they started at 1000 dollars they'd have about a one in 1000 chance of still playing by the time the bet got to 1 million dollars. The moral is, if you start out betting something meaningful there is a one in 1000 chance you'll lose big.