st pete game? by pete sampras? And it's ducats by the way.
There is NO fair entrance fee unless you look at the weighted winnings.
I have been doing this kind of work for over 20 years.
In Feller (vol 1, I think) there is a weak solution (convergence in probability)
BUT that isn't advisable unless you're the gambler.
Since the limsup of the ratio of winnings to cummulative entrance fee is infinite with probability one.
And the liminf is finite (with probability one) the same value that Feller obtained in his Weak Law.
The Bernoulli's asked this question 295 years ago.
You're a little late to this GAME.