[EM] Voting by selecting a published ordering

Wed May 10 15:18:34 PDT 2006

I've been exploring certain questions in relation to this topic of Candidate Published Orderings.

First of all, which combinations of candidate orderings make sense geometrically?

This question is not hard to answer in the case of three candidates, A, B, and C.

Suppose that candidates A and C are the furthest apart from each other as measured by some symmetric metric (Euclidean or not).

Then geometric consistency requires that the A and C orderings are ABC and CBA, respectively.  

The B ordering could be either BAC or BCA. 

Either of these can be obtained from the other by a permutation of the letters.

In summary, there is essentially only one case (up to permutations of the letters) that is geometrically consistent, when there are three candidates:

x:ABC,  y:BAC, z:CBA ,

and no matter the sizes of the factions x, y, and z, there will always be a Condorcet Candidate.

There are many more cases to consider when there are four candidates A, B, C, and D.

However, it turns out that (up to permutations of the letters) there are only thirty (30) cases that are geometrically consistent, and 29 (all but one) of these are like the above case in that there will always be a Condorcet Candidate, no matter how the faction sizes w, x, y, and z are distributed.

The geometrically consistent case that does not always have a Condorcet Winner is the following:

w:ACDB, x:BCDA, y:CBDA, z:DBCA.

Even this case has a Condorcet Winner for precisely 87.5 percent of the distributions of faction sizes w, x, y, and z.  The exceptions occur when all of the following relations hold simultaneously:

x+y<50%, z<50%,  x+z>50%

which describes precisely one eighth of the volume of the tetrahedron given by

x+y+z < 100% 

in the first octant of the x, y, z, coordinate system. (Note that w is determined from x, y, and z by  w+x+y+z=100% .)

To see that this case is geometrically consistent, locate candidates A, B, C, and D respectively, at the points
(4,2), (0,0), (1,0), and (0,2) in a Cartesian Coordinate Plane. 

The important thing about these positions is that

d(A,B)>d(A,D)>d(A,C)>d(C,D)>d(B,D)>d(B,C) .

I'll stop here for now, to give you all a chance to digest these interesting facts.

To me they are rather amazing facts, and potentially useful,  IF we can figure out how to incorporate them appropriately into this candidate published ordering setting, or more generally, into a one faction per candidate setting.