Solution to which problem? I am not sure what is supposed to be problematic.
That if you use backward induction on acting rationally at each step, you will be worse off. You will predict later that you’ll change your mind, unless you can force your future self to honor a commitment (or plan) you’d no longer want to keep when it actually comes time to honor it.
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.
However, I do not think it can be infinite. The function f(x) = x can take an arbitrarily large value, but not an infinite value (its range is the set of real numbers).
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.
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.
That if you use backward induction on acting rationally at each step, you will be worse off. You will predict later that you’ll change your mind, unless you can force your future self to honor a commitment (or plan) you’d no longer want to keep when it actually comes time to honor it.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.
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.