## Tuesday, February 22, 2022

### 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).