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

Matthew Killebrew matkil4354 at ncpschools.org
Thu Dec 2 09:07:04 PST 2021

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On Thu, Dec 2, 2021 at 12:06 PM Colin Champion
<colin.champion at routemaster.app> 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.
>
> This is easiest to see under a spatial model, but I think it’s perfectly
> general. We have vague prior information about the attributes of voters
> and candidates (eg. their positions in space) and about voter behaviour
> (how they will cast their ballots in the light of these attributes). We
> can condition this prior information on the observations contained in a
> set of ballots and thereby compute the posterior probability of any
> desired function of the voters' and candidates' attributes. Our aim is
> to identify the most representative candidate, where the degree of
> representativeness can be expressed through a loss function, and where,
> therefore, we will seek to identify the candidate whose posterior
> expected loss is the least. The prior knowledge of voters' positions can
> be thought of as a distribution of distributions, eg. the voters come
> from a mixture of three identical and equally weighted circular
> Gaussians whose means come from a further Gaussian.
>
> This is a constructive approach which in principle might be used to
> determine the winner of an election. We'd just need to integrate out all
> the unknowns to find the expected losses of the candidates. This was in
> essence the approach adopted by Good and Tideman in 1971 but not pursued
> further. The same view of voting theory underlies the empirical
> evaluations which have taken place subsequently: elections are sampled
> under a vague prior, and the results are assessed under an appropriate
> loss function. The only feature which conceals the decision-theoretic
> basis is the persistent use of the term 'utility' where 'loss' would be
> more correct.
>
> 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.
>
> The former case was encountered by Good and Tideman, which is why their
> paper ended up as Bayesianism without the prior. Unfortunately their
> model (a single Gaussian) is too simple to be of interest, given the
> optimality of Condorcet methods under it.
>
> I say that the prior 'almost' drops out of the calculation because Good
> and Tideman's parameters have a degree of freedom which is independent
> of the information in the ballots. This lies in the radial distance of
> the three candidates from the centre of the circle whose circumference
> distance, and it may affect the candidates' losses, so it seems a
> suitable case for Bayesian treatment. But Good and Tideman adopt a
> squared-distance loss function, and under this loss function (and this
> function alone, I suspect) the radial distance is immaterial to the
> identity of the optimal candidate. (The authors claim that the same
> result applies to any loss function which depends solely on distance,
> but I believe this to be an error.) Thus Good - of all people - made the
> prior drop out completely.
>
> It would be more interesting to adopt a Gaussian mixture prior as
> sketched above. Even then we could perform nothing more than a
> computational thought experiment. A truly realistic model would have to
> allow for an arbitrary number of Gaussians in a space of any dimension,
> and incorporate valence and random effects, and would need to allow for
> tactical voting. It's hard to imagine any useful solution being obtainable.
>
> But if voting theory is an insoluble problem in Bayesian decision
> theory, then any voting method we encounter must be essentially ad hoc,
> even if it draws on bomb-proof reasoning from another branch of
> mathematics. At least we have the comfort of knowing that there is a
> rigorous method of evaluating the solutions which are proposed to us.
>
> CJC
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