Should you bet everything on a wager with infinite expected return?

A man offers to let you play a simple game, provided you pay a fee before the game. A fair coin (50-50 heads-tails) is tossed; if it comes up tails, the game is over and you get no money back.

If it comes up heads, you receive a dollar, and the coin is flipped a second time. If it comes up heads a second time, you get another two dollars; if it comes up heads the nth time, without any tails first, you get another 2(n-1) dollars. As soon as the coin comes up tails, the game ends, and you do not win or lose any more money.

The question is, how much would you be willing to pay to be allowed to play the game?

We quickly see that the expected return from this bet is infinite! Should we be willing to bet everything we own for a chance to play this game?

I don't think there is a precise solution to this problem since it depends to a certain extent on a psychological element.

However, note that at some point the exponentially increasing payoffs become either of no additional real value (there's only so much money you could ever spend) or non-credible (there's only so much money a stranger in a bar is likely to have, and ultimately, there's only so much money on Earth).

Suppose that M is the maximum payout we find desirable, or credible. Then the number of heads resulting in a payout of M is N = log2 (M + 1).

If we neglect outcomes with more than N heads, expected payout is (N/2) dollars. I conclude one should not spend more than (N/2) dollars to play the game.

Fixing M (and therefore N) is something of a psychological issue. M and N might be zero if we think the man is just a con artist.

Before considering this game in detail, my gut feeling was that I would pay $20—hey, I can lose $20 for a shot at infinity—but now I think that is much too high. I might go to $10 if I were satisfied that the stranger could pay off a million-dollar debt.