[EM] Yee diagrams and Condorcet

[fsimmons at pcc.edu]

If we abandon the Euclidean metric, then we also abandon Voronoi Polygons; the corresponding idea for 
more general metrics is that of a Dirichlet region.

It would be amusing to see Yee diagrams based on L_1 and L_infinity metrics

Of course, Yee uses the  L_2 metric to make his pictures rotationally invariant to fit with his rotationally 
invariant bi-variate normal distributions.

If an L_2 metric is not used, then the pairwise distances between the candidates do not determine the 
diagram; their orientation must also be taken into account.

The whole idea of Yee diagrams was to show that certain methods (like IRV) could botch elections 
under the most ideal conditions.

Bad methods can yield good Yee diagrams, but methods that yield bad Yee diagrams flunk the test.

If a method looks good under the Yee-scope, and it is also monotone and clone independent, then it 
needs only one other quality for my support; it must be simple to vote.

By "simple to vote," I mean two things:  (1) The ballot is easy to understand and complete, and (2) 
strategic voters have little or no advantage over non-strategic voters.

I don't care as much whether the method is easy to count or easy to understand the details of the 
counting procedure.  That would be nice, but not as necessayr as the other features.

Think of the Huntington Hill method of apportionment that is used in this country after every census.  
How many voters understand its details?  Less than one in a thousand, but that doesn't matter; they 
understand the proportionality goal of apportionment, and they are willing to let the experts take care of 
the details.

How many voters in Australia understand or care about the Droop quota?  If they've heard about it, then 
they might know that it helps distribute the representation proportionately.  In any case they don't fret 
about it.

I haven't seen any Yee diagrams for SODA, but so far I think it has the best chance of fulfilling all of 
these requirements.

----- Original Message -----
From: Jameson Quinn 
Date: Wednesday, July 13, 2011 6:33 pm
Subject: Re: [EM] Yee diagrams and Condorcet
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> That proof assumes a euclidean distance metric. With a non-
> Euclidean one,
> the "planes" could have kinks in them. I believe I have heard 
> that the
> result still holds with, for instance, a city-block metric, but 
> I cannot
> intuitively demonstrate it to myself by imagining volumes and 
> planes as in
> this proof.
> 
> JQ
> 
> 2011/7/13 
> 
> > Actually, any centrally symmetric distribution will do, no 
> matter how many
> > dimensions.
> >
> > The property that we need about central symmetry is this: any 
> plane (or
> > hyper-plane in higher
> > dimensions) that contains the center of symmetry C will have 
> equal numbers
> > of voters on each side of
> > the plane..
> >
> > To see how this guarantees a Condorcet winner, let A and B be 
> candidates> such that A is nearer to the
> > center C than B is. Let pi be the plane (or hyper-plane in 
> dimensions> greater than three) through C that
> > is perpendicular to the line segment AB.
> >
> > By the symmetry assumption there are just as many voters on 
> one side of the
> > plane pi as on the other
> > side.
> >
> > Now move pi parallel to itself until it bisects the line 
> segment AB.
> >
> > All of the voters that passed through the plane pi during this 
> move went
> > from the B side to the A side of
> > the plane. So A beats B pairwise.
> >
> > Therefore, if there is a unique candidate that is closer to C 
> than any of
> > the rest , that candidate will beat
> > each of the others pairwise. Otherwise, all of the candidates 
> sharing the
> > minimum distance to C will be
> > perfectly tied for CW.
> >
> >
> >
> > > From: Bob Richard
> > > After looking up some old email threads, it now seems to me that
> > > I made
> > > a significant mistake in the post below. It is true that the
> > > model
> > > underlying Yee diagrams guarantees that there will always be a
> > > Condorcet
> > > winner. But apparently that has nothing to do with the two
> > > dimensions
> > > being orthogonal. It results from the fact that voters are
> > > normally
> > > distributed on both dimensions.
> > >
> > > --Bob Richard
> > ----
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> >
>