[EM] The voter median candidate generalized to multidimensional issue space

[Forest Simmons]

It occurred to me that some readers might need a key to the geometric
language in my recent posting about the Achilles heel of the Voter Median
Order.

Think of the preference orders ACB, CAB, CBA, BCA, BAC, and ABC positioned
on the face of a clock at the respective even hour marks 2, 4, 6, 8, 10,
and 12.

If you wanted, you could put A at one O'Clock, C at five O'Clock, and B at
nine O'Clock, to show their respective positions relative to the full
rankings.

This is enough to get the general picture.  For the computational details
(for eigenvalues, eigenvectors, etc.) the clock is mapped rigidly into a
three dimensional coordinate system:

ABC --> [1,0,-1]
ACB --> [1,-1,0]
CAB --> [0,-1,1]
CBA --> [-1,0,1]
BCA --> [-1,0,1]
BAC --> [0,1,-1]


Forest


On Mon, 13 Jan 2003, Forest Simmons wrote:

> A simple example illustrates the Achilles heel of the VMO (Voter Median
> Order).
>
> 3 ABC
> 2 BCA
> 2 CAB
>
> The axis of symmetry is the line from ABC through the midpoint X of the
> line joining BCA and CAB.
>
> This midpoint X is closer to CBA than to ABC on the line joining these two
> opposite orders, so the VMO is BCA.
>
> MinMax, Ranked Pairs, Kemeny, and Borda (hence Black) all yield the
> opposite order ABC, which makes sense because this example is obtained
> from a perfectly symmetrical set of ballots by adding one ABC ballot.
>
> Why does the VMO give the exact opposite of the "right" answer?
>
> Because the symmetry breaker ballot ABC reveals an issue axis that was
> hidden before that ballot was included.
>
> Is that issue axis real? Or is it just the result of random fluctuation?
>
> It is impossible to answer this question with any certainty in such a
> statistically small sample.
>
> For this reason, it appears that the VMO is not reliable for small
> samples.
>
> Before leaving this example, consider the symmetrical subset of ballots
>
> 2 ABC
> 2 BCA
> 2 CAB,
>
> and suppose that you had a way of knowing (independent of the ballots)
> that the main issue axis was parallel to the perpendicular bisector of the
> segment joining BCA and CAB.
>
> Since the BCA and CAB ballots are closer to the CBA end of issue space
> than they are to the ABC end, it is clear that a 4:2 majority prefers the
> CAB outcome to the ABC outcome on the main issue, in spite of the symmetry
> of the ballot distribution.
>
> In general, there is no reliable way to determine the issue space axes
> in the case of symmetry or near symmetry of ballot distribution, so the
> VMO is not reliable except when the ballot distribution itself reveals
> statistically significant issue dimensions.
>
> A statistical test of the hypothesis that the eigenvalue estimators are
> distinct would show us the level of confidence that we could have in the
> VMO.
>
> Here's a similar example where the computations are easy:
>
> 2 ABC
> 1 BCA
> 1 CBA
> 1 CAB
>
> Any method based on the pairwise matrix rightly yields a precise three way
> tie.  This includes Borda and all of the popular Condorcet Methods.
>
> However, the VMO order is CBA. In fact, the unique median position X is
> exactly the same as in our first example above.
>
> The positive eigenvalues corresponding to eigenvectors
>
>               [1 0 -1] and [-1 2 -1], respectively,
>
> are seven and three.
>
> The ratio 7/3 appears to be significantly greater than unity, but with
> only a few of degrees of freedom to be shared between the numerator
> and denominator, we cannot have much confidence in this conclusion.
>
> Note that the top VMO candidate is also the IRV winner.  That's not too
> comforting.


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