Utility and Equivalence

Credit: deviantART / sekeroglu

In a previous post, I outlined the four von Neuman-Morgenstern axioms for preferences (completeness, transitivity, continuity and independence), discussed my interpretation of the outcome consequentialism that underlies sensible interpretations of the axiom, and finished by talking about how they imply the existence of utility functions:

Theorem (von Neumann–Morgenstern utility theorem): Assuming the VNM axioms, there exists some linear (weighted-average preserving) function \(u: \mathcal{O} \rightarrow \mathbb{R}\) such that: \[o_1 \prec o_2 \Leftrightarrow u(o_1) < u(o_2)\]

These functions aren't unique: given any such function you can translate it or multiply it by a positive scalar to get another function that represents the same preference relation \(\prec\). In fact, for all intents and purposes, utility functions can be considered invariant under positive affine transformations.

Castigate or forgive...?

(N.B. This post's a little rough around the edges; I wrote most of it near the start of this year and haven't given it the level of polish it deserves.)

When I was in year seven, I used to torment a boy — who was several years above me and twice my size, mind you — by throwing my lunchbox at him whenever I saw him in the yard. Eventually, teachers had to intervene.

When I was in year seven, I was pressured into all kinds of embarrassing antics by a boy in my classes who was probably happy that they were all laughing at someone other than him. Nothing ever became of that.

Credit: deviantART / tawkwondonj

A few months back I stumbled across this short piece by Gint Aras, writing for The Good Men Project, titled Should we forgive apologetic bullies?. Take a look: it's short and bittersweet. In it, Aras poses the titular question, asking about reformed bullies, about past aggressors who really do seem to have seen the error of their ways.

The idea of retribution transcends cultures. One need look no further than our traditional stories — whether it's the hubris and (literal) downfall of Icarus, the creation mythos of Australian Aboriginal Dreamtime stories, or the countless deluge myths that exist across cultures, punishment has always gone hand in hand with crime.

It's not too much of a stretch, then, to say that we're born with a sense of tit-for-tat justice. (One imagines that creatures who let their selfish brethren off the hook did not do a particularly good job at staying in the gene pool.) It's not taught. Society didn't invent our moral compasses: it merely reinforced them.

So here's a question. Whence cometh forgiveness?

Thoughts on "Monsters University"

I didn't see Monsters University while it was in the cinemas, making it the second Pixar movie in a row that I've missed (after last year's Brave). But I finally got around to it on a plane trip a few days ago (ermahgerd i was on a plane), and was suitably impressed.

Of course, I have a soft spot for Pixar movies, but no wonder — their script writing has been solid for as long as I can remember... and of course, their visuals have always been artful as well as cutting-edge. (Maybe Cars 2 was crap? I've never seen it; I think it was a direct-to-DVD release.)

Still frame from Monsters University.

In any case, the movie was a lot of fun. Here are a few things I was thinking immediately after seeing it.