EDIT: my bad, the problem is that if you don’t use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesn’t leave you (or Sally) worse off and doesn’t require any commitment.
St Petersburg doesn’t require any state to have infinite value. Its value is (canonically) 2^n with probability 1/​2^n for each n at least 1. Always finite actual value, but infinite expected value.
The expected value of the St. Petersburg lottery is 1 + 1 + … = +inf. It involves finite terms, but infinitely many terms. I meant to relate f(x) = x in my comment to the expected value of the St. Petersburg lottery. If this involved an arbitrarily large number of terms, its expected value would be arbitrarily large, but not infinite.
I followed up here.
The expected value of the St. Petersburg lottery is 1 + 1 + … = +inf. It involves finite terms, but infinitely many terms. I meant to relate f(x) = x in my comment to the expected value of the St. Petersburg lottery. If this involved an arbitrarily large number of terms, its expected value would be arbitrarily large, but not infinite.