[EM] Exact spatial model probabilities?

[Daniel Carrera]

On Wed, Jan 26, 2022 at 6:53 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> So when figuring out the proportion who'd vote A>B>C in the continuous
> (V->infty) case, you could first create a Voronoi map for all the
> candidates. The volume of this region gives you how many voters would
> vote A first. Then you'd take the region for A and calculate the Voronoi
> map within it for every candidate but A excluded. The subregion that's
> now closest to B gives you the number of voters who vote A>B>...; and so
> on.
>


You've convinced me that there's a lot of value in doing the exact
geometric solution. I would be interested to see simulations with very
large numbers of voters. Real elections can have 10^7 voters. I haven't
figured out how to compute the volume of a Voronoi cell, but I did have a
thought: What if we simplify the problem by reducing the number of
candidates and issues?

For real elections, we don't really care about an arbitrary number of
characters or an arbitrary number of issues. Political views are often
described in just a two-dimensional space, with one "economic" axis (left
<-> right) and one "social" axis (libertarian <-> authoritarian). That
alone is enough to describe most of politics. The number of parties can be
larger, but honestly, the countries that have >3 prominent parties are few
and they probably use PR anyway. If all we do is find exact solutions for 3
parties (e.g. two major parties and a challenger) or 4 parties (e.g. two
major parties + Green + Libertarian) and a 2-dimensional issue space, that
would be a very relevant case that would tell us a lot about the real world
elections we care about.

Finding the area of a Voronoi cell in a 2D space with just 3-4 cells
doesn't sound difficult. I haven't done the math, but it feels like high
school math problem.


However, that doesn't solve the integral. My idea was to get the (as
> close to exact as possible) probability that some random chosen voter
> will vote A>B>C. Say that this is p(A>B>C).
>
> Since the voters are independent, this then gives a probability that
> some random election of V voters will be of a particular type; e.g. you
> can figure out the probability that a three-candidate election will be
> e_s = (A>B>C, A>B>C, B>C>A). Let the probability that we will encounter
> election e be p(e). Then the probability of the example election is just
> p(e_s) = p(A>B>C) * p(A>B>C) * p(B>C>A).
>


I'm a bit confused by the notation. If p(A>B>C) is the probability that a
random voter picks A>B>C then what is p(A>B>C) * p(A>B>C) * p(B>C>A)?

I also don't get what you mean by "e_s = (A>B>C, A>B>C, B>C>A)" being an
election.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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