A natural use case for geometric expected value is in situations where “compounding” occurs. In finance we deal phenomenon like comound interest. Hence the average rate of return is calculated using the geometric mean.
I think compunding occurs in lots of phenomon (population, economy). It may even applicable for modelling utility in some circumstances—perhaps the happiness of a community is boosted by the happiness of all the individuals.
Whether geometric expected utility itself should be maximised is subjective/ philosophical in my opinion. In our culture our concept of “total” is the addition. The total cost of a shopping list is the sum of the prices of all it’s individual items. And hence our concept of “total utility” is similarly the addition of the utility of each actor. However, I wonder if on an alien planet, their concept of “total” is multiplicative. As far as I can reason, there is no reason to think one concept of “total” is more natural than the other.
Aside from arithmetic (addditive) and geometric mean, there are other means like the harmonic mean. https://en.wikipedia.org/wiki/Pythagorean_means. Maybe we can also ask if we should maximise harmonic expected value too? Or even more exotic kinds of “means”:
I can’t see a good reason why any one of them is a more natural choice over any other. I genuinely think that our usage of the ordinary arithmetic expected value is purely cultural. And this forms part of my critique of leaning so heavily on expected value in the first place.
A natural use case for geometric expected value is in situations where “compounding” occurs. In finance we deal phenomenon like comound interest. Hence the average rate of return is calculated using the geometric mean.
I think compunding occurs in lots of phenomon (population, economy). It may even applicable for modelling utility in some circumstances—perhaps the happiness of a community is boosted by the happiness of all the individuals.
Whether geometric expected utility itself should be maximised is subjective/ philosophical in my opinion. In our culture our concept of “total” is the addition. The total cost of a shopping list is the sum of the prices of all it’s individual items. And hence our concept of “total utility” is similarly the addition of the utility of each actor. However, I wonder if on an alien planet, their concept of “total” is multiplicative. As far as I can reason, there is no reason to think one concept of “total” is more natural than the other.
Aside from arithmetic (addditive) and geometric mean, there are other means like the harmonic mean. https://en.wikipedia.org/wiki/Pythagorean_means. Maybe we can also ask if we should maximise harmonic expected value too? Or even more exotic kinds of “means”:
https://en.wikipedia.org/wiki/Logarithmic_mean
https://en.wikipedia.org/wiki/Heinz_mean
https://en.wikipedia.org/wiki/Identric_mean
I can’t see a good reason why any one of them is a more natural choice over any other. I genuinely think that our usage of the ordinary arithmetic expected value is purely cultural. And this forms part of my critique of leaning so heavily on expected value in the first place.