In Scott Garrabrant's excellent Geometric Rationality sequence, he points out an equivalence between modelling an agent as

Maximizing the expected logarithm of some quantity $V$ , $E [l n (V)]$
Maximizing the geometric expectation of $V$ , $G [V]$

And as we'll show in this post, not only can we prove a geometric version of the VNM utility theorem:

An agent is VNM-rational if and only if there exists a function $V$ that:
- Represents the agent's preferences over lotteries
  - $L ≺ M$ if and only if $V (L) < V (M)$
- Agrees with the geometric expectation of $V$
  - $V = G [V]$

Which in and of itself is a cool equivalence result, that $E [U]$ maximization $⟺$ VNM rationality $⟺$ $G [V]$ maximization. But it turns out these are just two out of a huge family of expectations we can use, like the harmonic expectation $H$ , which each have their own version of the VNM utility theorem. We can model agents as maximizing expected utility, geometric utility, harmonic utility, or whatever representation is most natural for the problem at hand.

Expected Utility Functions

The VNM utility theorem is that an agent satisfies the VNM axioms if and only if there exists a utility function $U : Δ Ω \to R$ which:

Represents that agent's preferences over all lotteries
- $L ≺ M$ if and only if $U (L) < U (M)$
Agrees with its expected value
- $U = E [U]$

Where $Δ Ω$ is a probability distribution over outcomes $ω \in Ω$ .

The first property is easy to preserve. Given any strictly increasing function $f$ ,

$f (U (L)) < f (U (M))$ if and only if $U (L) < U (M)$

So $f \circ U$ also represents our agent's preferences. But it's only affine transformations $f (U) = a U + b$ that preserve the second property that $f (U) = E [f (U)]$ . And it's only increasing affine transformations that preserve both properties at once.

f-Utility Functions

But what if we were interested in other ways of aggregating utilities? One of the central points of Scott's Geometric Rationality sequence is that in many cases, the geometric expectation $G$ is a more natural way to aggregate utility values $V$ into a single representative number $G [V]$ .

We can represent the same preferences using the expected logarithm $E [l n (V)]$ , but this can feel arbitrary, and having to take a logarithm is a hint that these quantities $V$ are most naturally combined by multiplying them t...

The Math of Geometric Utilitarianism

The Math of Geometric Utilitarianism