So there’s been a push in the last few years — principally by libertarians — to try and break away from the traditional single-dimensional political model. (That’s left
vs. right
.) Their proposal involves a two-dimensional space, separating economic and social freedoms into independent axes. They then put together quick little quizzes which allegedly plot your position in their policy space. (A commentator has described these as those little tests which tell you you’re a libertarian
.)
There’s several questions you can raise regarding this proposal. One obvious one is: why should two dimensions be enough? While this description may give more explanatory room than the standard 1D model, why should it be superior to (say) a tribal
description of Canadian or American politics, where people are classified/stereotyped based on demographic concerns? Or if two dimensions are better than one, why should we stop there? Why not put together large questionaires on many, many issues and then do some clustering analysis to work out how many independent axes there really seem to be?
As interesting as those might be to pursue, they’re not what I’m working on right now. Rather, I’m curious about the geometric aspects of the problem. See, there’s a result by Black that states that, as long as voters’ preferences are single-peaked
, then one can find reasonable social choice procedures. (This isn’t true if preferences are allowed to vary arbitrarily; that’s Arrow’s Impossibility Theorem.) What single-peaked
means in this context is more or less that you can plot the various candidates along a straight line, and a voter’s preferences are determined strictly by their position on this line and how close they are to the different candidates. In other words, if the left/right model is actually accurate, then reasonable voting results are possible.
So what happens in this 2D space? Assuming that the libertarians’ model is accurate, does it give us guarantees of reasonable social choice? Or are there now sufficient degrees of freedom that Arrow’s Paradox comes into play? And if the latter is true, then under what circumstances do we need to worry?
I’ll post pictures of what I’m talking about later; for now, I just thought that some of my Faithful Readers might be interested in what I’m thinking about these days.