# [EM] More Kemeny-Young Thoughts

Forest Simmons forest.simmons21 at gmail.com
Wed Mar 2 14:34:30 PST 2022

```Kemeny-Young is a geometric method based on the Kendall-tau distance
between rankings.

If we combine this idea with the Universal Domain axiom that the only
relevant voter information for determining the outcome of an election is
the ordinal information contained in the ballot rankings of the candidates,
then for all intents and purposes we can identify the space of voters with
the space of candidate rankings.

Let's see where this leads when we adopt the Kendall-tau distance as a
metric on the ballot space.

For example, consider the ballot profile

40 A>B>C
35 B>C>A
25 C>A>B

The Kendall-tau distance between any pair of the three faction rankings is
exactly two units of distance; it takes a minimum of two swaps to turn a
permutation of three symbols into another such permutation of the same
cyclical parity.

So geometrically the three distinct ballot orders of the three respective
factions represent voters at the three respective vertices of an
equilateral triangle of side two unts.

Note that the candidates themselves are supposed to be voters. Which of the
three ballot orders gives A's position on the equilateral triangle? And
which represent B and C?

It's not hard to see by a process of elimination (and assumption of sincere
ballot preferences) that A has to be at the A>B>C ballot position, B at the
B>C>A position, and C at the C>A>B vertex of the equilateral triangle.

If that is the case, how can we take seriously the ballot order of the 40
members of the A>B>C faction that says that they unanimously, strictly
prefer candidate B to candidate C?

All forty of them are equally close (a distance of 2 units) to candidates B
and C. How is it that not even one of them prefers C to B?

Evidently the Kendall-tau metric is not consistent with all geometric based
affinities.

So what do we do when our randomly generated factions give us a profile
that is inconsistent with the Kendall-tau metric?

Suppose that it was generated by random points in Euclidean space with a
Euclidean L_2 metric or some other vector space metric. Then what do we say
when it is not consistent with the Kendall-tau metric?

Which do we trust more ... L_p metrics or Kendall-tau?

The best answer to that depends on how seriously we take the Universal
Domain axiom. Kendall-tau is topologically compatible with Universal
Domain, while the vector space metrics are not.

But, you object, Kemeny-Young, based on Kendall-tau is clone dependent.

Two replies:(1) so are the L_p metrics in most ballot profile generation
contexts. (2) Kendall-tau is easily de-cloned while completely retaining
its UD compatible topology

For clarity let me reiterate: no vector space metric is topologically
compatible with Universal Domain.

Personally, I'm not afraid of coloring outside of the lines ... most of my
favorite methods are hybrids that violate UD ... but UD purists need to
know the inherit UD inconsistency in geometrically generating ranked choice
ballot profiles with UD incompatible topology ... namely flat vector space
topology as opposed to the curved space topology of permutation/ranking
space, which resides on the boundary of an (n-1) dimensional ball when
there are n candidates.

I hope this "abstract nonsense" (as Professor Klaus Bichtler used to
glowingly call it) has helped to connect certain conceptual categories in
relation to each other. Our understanding grows as our internal network of
such connections becomes more complete ... especially as the degrees of
separation from familiar intuitive concepts decrease.

El mié., 16 de feb. de 2022 4:47 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

>
>
> El mié., 16 de feb. de 2022 4:33 a. m., Kristofer Munsterhjelm <
> km_elmet at t-online.de> escribió:
>
>> On 16.02.2022 08:58, Forest Simmons wrote:
>> > Suppose you had a nice metric on a candidate space for measuring the
>> > disparity between members of the electorate, and you wanted to use the
>> > metric to find the most representative member of the electorate. How
>> > would you use that metric?
>> >
>> > There are many possibilities, but let's consider three of them that are
>> > fairly natural and easy to understand:
>> >
>> > Let V be the set of voters, and let K be the subset of V consisting of
>> > the candidates who managed to get their names on the ballot.
>> >
>> > 1. Elect the candidate k that minimizes the total distance to the  other
>> > voters, i.e. elect argmin{TotDist(k,V)|k in K} where TotDist(k,V) is the
>> > total distance from k to the other members of V.
>> >
>> > 2. Elect the candidate k closest to the most representative voter,
>> > namely argmin{TotDist(v,V)|v in V}. This is analogous to the Condorcet
>> > dictum, "elect the candidate closest to the voter median."
>>
>> These approaches make use of the ballots that the candidates cast,
>> right?
>
>
> That's a possibility, but not what I had in mind. For example, in method
> two, once you have found the secret anonymous ballot B that minimizes the
> total distance to the other ballots, just assume that ballot B's favorite
> is the candidate closest to ballot B. This ballot B could well be the
> ballot of one of the candidates, but there is no way of knowing that. This
> is the same reasoning that Kemeny-Young uses to identify the candidate
> closest to its idealized ballot.
>
> Another possibility is (for each candidate k) find the ballot B(k) that
> minimizes the total distance to the ballots that rank k as favorite. Then
> consider B(k) to be the position of candidate k in ballot space. That might
> be the best approach for multi-winner K-Y.
>
> Arguably either one of these inferences of the location of the winning
> candidate is just as good as the (computationally expensive) idealized
> ballot inference of Kemeny-Young.
>
> That would seem to be more in the spirit of Asset than anh
>> ordinary voting method. If there's a secret ballot, then the candidates
>> are incentivized to lie on their ballots; and if the candidates' ballots
>> are not secret, then the candidates have to predict the election
>> accurately while still making their ballots pull the outcome in their
>> direction.
>>
>> There's nothing wrong as such with the second - it fits with the Asset
>> themselves - but one should be aware that it's not an ordinary voting
>> method and couldn't be used for choosing what movie to watch, budget
>> item to fund, or similar.
>>
>> ...
>>
>> That makes me think, though. Could one say that the distinguishing
>> feature of proxy and Asset-type elections is that they violate
>> anonymity? This because the proxies are privileged in the sense that
>> their votes are amplified by however many other voters decide to back
>> them.
>>
>> Perhaps not so much for traditional Asset, where you first choose a
>> "parliament" that then negotiates until some winner crosses a threshold.
>> That method has multiple rounds: first voting for assets, then the
>> parliamentary formation, then the final winner is decided; and it
>> doesn't violate anonymity in either round.
>>
>> But a one-shot Asset-like where you either designate a proxy (whose vote
>> you'll copy) or vote directly... would. I think.
>>
>> -km
>>
>
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