[EM] Election design

Mon Sep 12 07:49:27 PDT 2022

Dear Richard,

The example John Paulos uses in some of his writings (e.g. A Mathematician Reads the Newspaper, pgs. 104-106) he “got” from me. My example appeared in an article I wrote for Annal 607, NY Academy of Sciences, 1990, pgs. 89-97 but also in the early editions of a book for liberal arts mathematics student courses called “For All Practical Purposes,” of which I am a coauthor. The example was in part of the book by the late game theorist William Lucas. However, Bill does not explicitly mention he got the example from me.

Data from actual elections with ordinal ballots show that there are elections where different election decision methods yield different winners. When designing “better” systems one consideration is that whatever the system used is that it should not “regularly” give results different from “highly regarded” methods.  Borda and Condorcet (when there is a Condorcet winner) often agree but as the example I show indicates, not always.

Regards,

Joe

------------------------------------------------
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

My email is:

jmalkevitch at york.cuny.edu

web page:

http://york.cuny.edu/~malk/
________________________________________
From: Richard Lung [voting at ukscientists.com]
Sent: Sunday, September 11, 2022 5:46 PM
To: Forest Simmons
Cc: Joe Malkevitch; EM
Subject: Re: [EM] Election design

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Hello Joe,

Your example interested me, because of one like it, by von Paulos, in his Numeracy dictionary. It also produced 5 different results. But when the Condorcet pairings were arithmetically weighted, they agreed with the Borda count. The example allowed very little margin for error, but I could tell by inspection of this small example that the two most rational counts agreed. It's just a question of the methods using the most preference information.

I'm an old man, somewhat muzzy on a night, but managed to get Borda and Condorcet agree with sufficient information input. I make it Borda gives D the win with 136 points (2 more than E).
E has highest Condorcet pairing with 17 votes to D at 14. Likewise weighted Condorcet gives  E one vote advantage over D (28 to 27). But compare the differences between all the pairings (except with A which is the same difference from D and E) And D wins by 34, in all pairings and E wins by 29 in all pairings.

Conclusion. The different information inputs of different election methods allow different results especially for contest outcomes leaving very little margin for error. But the more rational the methods, the more likely they are to agree.

Cheers,
Richard.

On 11 Sep 2022, at 8:00 pm, Forest Simmons <forest.simmons21 at gmail.com<mailto:forest.simmons21 at gmail.com>> wrote:

Great example!

A geometric example of this type involving the four major cities of Tennessee can be found spread across several electowiki articles explaining various election methods.

The example deduces the preference ballot of each voter from the assumption that voter V would prefer (as Tennessee state Capitol, for example) city X over city Y, if voter V lives geographically closer to  X than to Y.

It could be the choice of an Olympic venue rather than capitol location, and the metric could be travel cost instead of distance as the crow flies.

This kind of geometrical preference deduction can be made for any election embedded in a metric space of any kind, concrete like this one, or abstract like a hypothetical Yee Disgram.

Warren Smith has several of them on his website.

The first time I saw such a geometrically derived ballot set forty years ago, I was very surprised that election methods like Borda, Bucklin, Copeland, Coombs, Hare, etc... all with reasonable heuristics, could give so many conflicting results. I was well aware that Arrow and others had given abstract examples showing the possibility of rock paper scissors cycles ... but the existence of a planar distribution of voters concentrated in four cities giving rise to a cycle of Euclidean distance preferences was a big surprise to me.

Anybody who doubts the existence of the concrete reality of cardinal ratings in some practical democratic decision making contexts should peruse Warren's examples on rangevoting.org<https://urldefense.proofpoint.com/v2/url?u=http-3A__rangevoting.org&d=DwMFaQ&c=yJ4UutiJRKf_XEsHUtOboHQQfLtKBfRC_OV2xJVGo18&r=7pxaWK0WuEo0HxsF_q5vQGhLPUOIFx_uY8ytjKVDVA4&m=2drCEZmsp3FjmPrhcQQpp8kEco_q4A--8cZPRLtw0ruNNRKkfXVRvqLNiuaKeHnG&s=l4TKNRwr1AYUTsqlmh7GHBCsz-cgLzaywcGa2R2s1ds&e=>

Note that Kendall-tau is an example of an abstract metric on permutations of candidates, i.e. possible finish orders. This Kendall-tau metric is the one used for rating/scoring the possible finish orders in the Kemeny-Young method.

How can we break the public imagination of democratic possibilities out of its current narrowly confined prison?

More teachers like Joe Malkevich are needed ... many, many more .... very, very sorely needed!

-Forest

On Sun, Sep 11, 2022, 10:01 AM Joe Malkevitch <jmalkevitch at york.cuny.edu<mailto:jmalkevitch at york.cuny.edu>> wrote:
Hi:

This post is a reaction to recent list discussions.

The  election below (highest rank at the left) shows the votes of 55 voters who produced ballots without ties or truncation, putting  to the side if ballot rules allowed indifference or truncation. I designed this example for students in various mathematics courses that included some attention to mathematical modeling to explore the notion of the will of the voters.
The method used to decide the election matters for the result.

18 votes ADECB
12 votes BEDCA
10 votes CBEDA
  9 votes DCEBA
  4 votes EBDCA
  2 votes ECDBA

If you use the ballots to choose a single candidate to win using:

Plurality
Run-off between two candidates with largest number of first place votes
Sequential run-off (IRV)
Borda
Condorcet (Select candidate who can beat all others in a 2-way race if there is one)

You discover the he 5 methods yield 5 different winners!

The backdrop for this example (and others in its spirit) are the theorems of Arrow, Satterthwaite and others that relate election methods to “desirable and fairness” properties.
It also relates to the issue of the skills real world voters can provide via “honest” ballots and how one should design elections which involves the choice of ballot type and the system used to count the ballots. There is also the issue of how the voters get information about the candidates and use this information to fill out there ballots. Polls whose accuracy is hard to be sure of often seem to be more important in how some voters vote rather than what the candidates stand for. What one does also depends on what “objective function” is being used.

Regards,

Joe

------------------------------------------------
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

My email is:

jmalkevitch at york.cuny.edu<mailto:jmalkevitch at york.cuny.edu>

web page:

http://york.cuny.edu/~malk/
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