Is St Petersburg really a paradox of infinity?

In the St Petersburg game, you keep on tossing a coin until you get heads, and you get a payoff of 2ⁿ units (e.g, 2ⁿ days of fun) if you tossed n tails. Your expected payoff is:

• (1/2) ⋅ 1 + (1/4) ⋅ 2 + (1/8) ⋅ 4 + ⋯ = ∞.

This infinite payoff leads to a variety of paradoxes (e.g., this).

But note that the infinite payoff is by itself paradoxical. If the expected payoff is infinite, it seems that it’s worth being tortured for a decade by the most effective torturers in the world for the sake of playing the game. And yet this is paradoxical!

However, the paradox here is not actually a paradox of infinity. For there will be some cut-off version of the game—a version where you get to toss the coin some predetermined finite number of times and if you don’t get to heads then you get nothing—where the value of the cut-off game exceeds the disvalue of the decade of torture. And it’s even more paradoxical to think that in the cut-off game it makes sense to pay that much to play, since the cut-off game is dominated by the infinite game.

This line of thought supports Monton’s thesis that we should neglect small probabilities.

Nano St Petersburg

In the St Petersburg game, you toss a fair coin until you get heads. If it took you n tosses to get to heads, you get a utility of 2ⁿ. The expected payoff is infinite (since (1/2) ⋅ 2 + (1/4) ⋅ 4 + (1/8) ⋅ 8 + ... = ∞), and paradoxes abound (e.g., this.

One standard way out is to deny the possibility of unboundedly large utilities.

Interestingly, though, it is possible to imagine St Petersburg style games without really large utilities.

One way is with tiny utilities. If it took you n tosses to get to heads, you get a utility of 2ⁿα, where α > 0 is a fixed infinitesimal. The expected payoff won’t be infinite, but the mathematical structure is the same, and so the paradoxes should all adapt.

Another way is with tiny probabilities. Let G(n) be this game: a real number is uniformly randomly chosen between zero and one, and if the number is one of 1, 1/2, 1/3, ..., 1/n, then you get a dollar. Intuitively, the utility of getting to play G(n) is proportional to n. Now our St Petersburg style game is this: you toss a coin until you get heads, and if you got heads on toss n, you get to play G(2ⁿ).