Salandriagado wrote:Right, so Xero keeps talking about things being "better", and that brings us to something that's actually vaguely close to my area of expertise, and I've got some really boring paperwork to procrastinate from, so it's time for some maths.
Now, in order to talk about something being "better" (or "more X" for any X) than something else, you need to have what's called a
poset in mind: that's a "
partially
ordered
set" (nobody ever accused mathematicians of being good at naming things). That is: you need a set of things you'd like to be able to compare (in this case, the set of all allocations of economic resources), and you need a comparison rule between them that satisfies a few properties (for technical reasons, we'll define "equal to or better than", then just drop out the "equal to" bit later): we need:
- Everything is equal to or better than itself (obviously: it's equal)
- If A is equal to or better than B, and B is equal to or better than A, then A and B are equal (can't be better in both directions, otherwise nothing makes sense)
- If A is equal to or better than B, and B is equal to or better than C, then A is equal to or better than C (we need this to be able to compare things in parts)
For convenience, we'll write "A ⊵ B" for "B is equal to or better than A" and "A ⊳ B" for "A is better than B" (that is: A ⊵ B and A ≠ B) (and the obvious mirror image versions). With these, the above turn into:
- A ⊴ A
- If A ⊴ B, and B ⊴ A is equal to or better than A, then A = B
- If A ⊴ B, and B ⊴ C, then A ⊴ C (we need this to be able to compare things in parts)
Now, the important part here is the "p": this is a
partial order: there's no guarantee, given any random A and B, that either A ⊴ B or A ⊵ B: it's entirely possible for two things to just not be comparable at all. Adding a requirement that every two items are comparable gives a totally ordered set (which, for reasons I don't understand, absolutely nobody calls a "toset").
With this, Xero's core claim comes down to three statements, with D the allocation resulting from his silly donation-voting setup, B the allocation resulting from actual voting, and
A the set of all allocations of economic resources:
- There is an objective way to make A into a poset.
- This poset structure in fact makes A into a totally ordered set (not actually necessary to his conclusion, but it's what he's been claiming).
- Even if (2) is false, then at the very least B and D are comparable (this is the version of (2) that is actually necessary to his claim).
- Specifically, B ⊲ D.
Let's start with the good news: claim 1 is true! Not only is there a poset structure on
A, but this structure is commonly used by economists (in the sense that
people actually write papers about it), but it's also objective, and natural (I'm really not going to go into enough category theory to explain what that means precisely, but it's roughly that this is the "obvious" way to do it). This structure is the Pareto poset, where we say that A ⊴ B if moving from A to B is a Pareto improvement or Pareto neutral (that is: such a change makes nobody worse off). You can easily check that this satisfies the three properties above: (2) is the only non-obvious one, but it's not that hard to show.
And now, the bad news: this isn't a total order. That is: claim (2) is wrong. In fact, it's
really wrong: this is a very
coarse poset, in the sense that very few things are comparable. Particular things that are incomparable in this include all pairs from "the current system", "Salandriagado gets all the money", and "Gallo gets all the money". It's really, really coarse.
And finally, the really bad news: (3) (and therefore (4)) is provably false as well. Indeed, Xero has claimed this himself, a great many times: some people clearly will be made worse off by such a change, starting with elected politicians.
So, that's Xero's core thesis literally mathematically disproven. The only possible method by which it
might be saved is if Xero can give us a different objective poset structure on
A that is either natural or comes with a very good reason to use it over the natural option, and justify why he's using this rather than the one everybody else uses, and then show that this satisfies (3) and (4).
So, Xero, I ask again: what do you mean by "better"? What poset structure are you using, and how are you going to justify that choice?
Posted: Tue Jun 26, 2018 6:19 am
