The diverging series seems to be a version of the St Petersburg paradox, which has fooled me before. In the original version, you have a 2^-k chance of winning 2^k for every positive integer k, which leads to infinite expected payoff. One way in which it’s brittle is that, as you say, the payoff is quite limited if we have some upper bound on the size of the population. Two other mathematical ways are 1) if the payoff is just 1.99^k or 2) if it is 2^0.99k.
Interesting point!
The diverging series seems to be a version of the St Petersburg paradox, which has fooled me before. In the original version, you have a 2^-k chance of winning 2^k for every positive integer k, which leads to infinite expected payoff. One way in which it’s brittle is that, as you say, the payoff is quite limited if we have some upper bound on the size of the population. Two other mathematical ways are 1) if the payoff is just 1.99^k or 2) if it is 2^0.99k.