harfe comments on Should we Maximize the Geometric Expectation?

[harfe]

Here is another argument against Geometric utility: It does not work if negative utilities are involved: $u^p$ is undefined if $u$ is negative. And I think some real-world experiences that involve suffering have negative utility.

[Eevee🔹]

Well, if we allow complex numbers, a lottery over all negative utilities would result in a real geometric mean, but for a mixture of positive and negative utilities, we’d get imaginary numbers.

For example, consider lottery $A$ with Pr(-5) = 0.5, Pr(-3) = 0.3, and Pr(-2) = 0.2. Then

$G\left(A\right)=(-5{)}^{0.5}\cdot (-3{)}^{0.3}\cdot (-2{)}^{0.2}$.

The (-1)’s factor out, giving us

$G\left(A\right)=(5^{0.5}\cdot 3^{0.3}\cdot 2^{0.2})\cdot (-1{)}^{0.5+0.3+0.2}=(5^{0.5}\cdot 3^{0.3}\cdot 2^{0.2})\cdot -1$,

which is a negative number.

Now consider lottery $B$ where one of the utilities is positive—e.g. we have Pr(-5) = 0.5, Pr(3) = 0.3, and Pr(-2) = 0.2. Then we’d get

$G\left(B\right)=(-5{)}^{0.5}\cdot 3^{0.3}\cdot (-2{)}^{0.2}$

$=(5^{0.5}\cdot 3^{0.3}\cdot 2^{0.2})\cdot (-1{)}^{0.5+0.2}$

$=(5^{0.5}\cdot 3^{0.3}\cdot 2^{0.2})\cdot (-1{)}^{0.7}$

$=(5^{0.5}\cdot 3^{0.3}\cdot 2^{0.2})\left(\cos \right(0.7\pi )+i\sin (0.7\pi \left)\right)$,

which is an imaginary number. The magnitude is equal to the weighted product of the magnitudes of the individual utilities, but the argument (the angle it makes with the positive real axis on the complex plane) is $\pi$ times the total probability mass of any negative utilities. This makes comparisons impossible because the complex plane is an unordered set.

[Radical Empath Ismam]

A utility between 0 and 1 effectively plays the same role as “negative utilities” do under regular arithmetic expected value. So a geometric perspective can work if you map what you would ordinarily consider to be negative utilities to the interval 0 to 1, and restrict utility to $R^{+}$.

But why would you? It doesn’t offer any benefit over regular artihmetic expected value calculations.