As I remarked a couple of weeks ago, I’ve been thinking about the two-dimensional policy space model that certain libertarians have come to favour. Specifically, I was wondering whether there’s an easy two-dimensional analogue of Black’s Theorem, which guarantees a transitive group preference for a set of individual preferences along a (one-dimensional) continuum.
In case you’re curious, the answer appears to be no. Black’s Theorem ends up choosing the preferences of the median voter along the continuum to be the group preferences; the reasoning goes that when it comes to a specific choice between two candidates, there will always be a majority that agrees with the median. (That’s a consequence of the geometry of the situation, and it’s not hard to work out why if you sit down with pen and paper for a few minutes.)
The problem in 2D is that it’s no longer clear what you mean by median. Unlike an average, what point you get for your median depends on your co-ordinatisation of the space, since the obvious generalisation of the concept is to just take the median along each axis. Thus, one can have the same set of points but get different medians by choosing different sets of axes to work from. I suppose that this is fine if one wants to posit that the libertarians have happened upon exactly the right model, but I think there’s reason to suggest that that’s not the case.
(Of course, then there’s the problem that it just doesn’t work; it’s easy to come up with a situation where any selection you come up with doesn’t have majority support.)
On a similar topic, I saw a suggestion in the comments of a blog lately to have a policy space based along three axes, corresponding roughly to the virtues
of the French Revolution: liberty (which in its extreme form would become anarchocapitalism), fraternity (fascism), and equality (communism). The cleanest way to do this mathematically would be to take convex combinations of the unit vectors in 3-space; the upshot of that model is that you can’t follow all three ideals equally well, and that there’s always going to be a trade-off somewhere.