[EM] Which branch of mathematics does voting theory belong to?
Forest Simmons
forest.simmons21 at gmail.com
Thu Dec 2 20:44:00 PST 2021
It's like the parable of the blind men and the elephant ... you see it from
the point(s) of view that are most familiar to you.
El jue., 2 de dic. de 2021 3:48 p. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:
> 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
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