Oops! that'd be a $2^(n+1) payoff. and stuff. or something. Fudge it until
you get a divergent expectation value. (obo... I should've been an
astrophysicist; they're happy if orders of magnitude are correct to an order
of magnitude).
In order to limit my exposure to clerical error, I'll refer everyone to:
http://einstein.et.tudelft.nl/~arlet/puzzles/sol.cgi/decision/stpetersburg
which has a good (and presumably proofread) discussion of St. Petersburg and
variants.
Interestingly enough, the page above concludes with a discussion of playing
against a bank with finite resources (which removes the divergence) and with
an expected profit (by only charging the player 80% of the expectation
value).
>Note that the expected value of the player's profit is 0.2e. Now
>let's vary the bank's resources and observe how e and p change.
>It will be seen that as e (and hence the expected value of the profit)
>increases, p diminishes. The more the game is to the player's
>advantage in terms of expected value of profit, the less likely it is
>that the player will come away with any profit at all. This is mildly
>counterintuitive.
Perhaps mildly mathematically counterintuitive, but very reasonable based on
horse races, startups, and similar ventures.
Very similar mechanics explain the distribution of surnames in isolated
populations. One hopes the same isn't true for civilization.
-Dave