BrownHairedEevee comments on Should we Maximize the Geometric Expectation?

[BrownHairedEevee]

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.