[EM] Which branch of mathematics does voting theory belong to?

[Kristofer Munsterhjelm]

On 12/2/21 6:05 PM, Colin Champion wrote:
> To judge from the literature I would suppose that voting theory was part 
> of first-order logic or of graph theory, but it seems clear that the 
> right answer is Bayesian decision theory.

I would agree with you when the methods are being judged on a how good 
it is according to e.g. how often it's susceptible to strategy, or how 
often it returns the ideal winner, or its VSE results.

But I don't think the properties approach is quite as much Bayes stats. 
Investigating what kind of properties are compatible or not is not 
really a statistical thing: either a method fails (say) monotonicity or 
it passes it. Trying to invent methods with new combinations of passing 
criteria is... I would say, closer to linear algebra than anything, 
although the exact linalg approach for doing so (what I almost did to 
show unmanipulable majority and Condorcet as incompatible, and what 
Craig Carey did to arrive at IFPP) doesn't really scale.

One could also argue that it's all economics, because public and social 
choice originate in economics (namely, a way to apply economics to 
political questions). Things like the median voter theorem and more 
generally Hotelling's law feel very "econ-ish".

> Unfortunately the constructive approach seems to be numerically 
> intractable in cases of interest. If the number of voters was small, the 
> observations would provide probabilistic information which could be 
> integrated under the prior in the normal way. But as the number of 
> voters increases, the information becomes increasingly deterministic - 
> it degenerates to a set of equations. And therefore two cases arise. 
> Either the equations fully determine the parameters of the voter 
> distribution, in which case the prior almost drops out of the 
> calculation; or the equations constrain the voter parameters to a curved 
> manifold in which only the prior remains to be integrated.

Let's say we use a spatial model. Then the parameters of that spatial 
model would still be free (e.g. how many candidates, how many 
dimensions, what kind of distribution do the candidates and the voters 
follow). Deciding the values of these parameters might be like setting a 
prior, because it doesn't necessarily arise from the empirical 
distribution no matter how many voters you have.

For instance, let's say you want to figure out the Condorcet efficiency 
of IRV. You use a spatial model based on current US politics, which has 
a very high probability of being two-party. So you find out that 
Plurality has near 100% Condorcet efficiency. But this is kind of 
misleading, because the political situation is the way it is in order to 
not make Plurality err to begin with.

I guess what I'm saying is that it doesn't seem like the hard parameters 
scale badly with the number of voters; the difficult bit is instead how 
to make sure the numbers are useful. The number of preferences does 
indeed scale pretty badly, but I'd imagine Monte Carlo helps with this.

In some cases it might be possible to integrate exactly. One idea I've 
had but never got around to implement is to make exact Yee maps. For a 
particular point being the center of voter opinion (distributed 
according to a Gaussian with some given variance), we can count the 
proportion of say, A>B>C votes by taking the intersection of the polygon 
of the Voronoi map designated closest to A, and the polygon of the 
second-closest Voronoi map assigned to B, and the polygon of the 
max-distance Voronoi map assigned to C. Then we can decompose the 
polygon into triangles and integrate over them. This approach scales 
badly in the number of candidates, but the number of voters are no 
longer part of the picture; the fractions should be in the limit of 
number of voters going to infinity (in practice some large finite number 
due to numerical precision limits). Is that what you mean by the 
equations fully determining the voter distribution?

-km