In the United States elections will be held in November 2026 for all 435 seats in House of Representatives and  33 seats in the 100 seat Senate. Oversimplifying a trifle, seats are held by either members of the Democratic or Republican parties. Party primary elections sometimes occur to choose the candidate who will be on the ballot for the party. Many elections involve more than two "alternatives." Nearly all elections in America involve a ballot where the voter votes for exactly one candidate and the candidate with the most votes wins - plurality voting. Reformers urge that democracy would be strengthened by using more "expressive" ballots - ordinal ranked ballots, perhaps with indifference, yes/no ballots, approval ballots, etc.. Some communities have moved towards having more "expressive" ballots and in some places votes will occur to allow these more expressive ballets. When more expressive ballots are used and there are 3 or more candidates there are heated debates about what is the "best" method to decide the "winner."

Two widely discussed approaches are based on ideas of Condorcet and Borda. There are many scholarly treatments of these matters. For those who might need a place to start learning about the issues you can look at this web page of Donald Saari

https://www.math.uci.edu/~dsaari/Saari_preprints.htm

The issues are complicated and subtle.

Regards,

Joe

——————————————
Joseph Malkevitch

Email:
jmalkevitch@york.cuny.edu
Web page:
http://york.cuny.edu/~malk/

Hay Joseph,

Some corrections to https://web.york.cuny.edu/~malk/ranked-voting-math-club-2022.pdf :

On p 28, you said: "ii. preference, ordinal, or ranked ballot List choices in preference order without ties. Sometimes one is limited as to the total number of candidates one can rank (five for NYC voting)"

In fact, whether ties are allowed or not do not depend solely on whether is "preference" or "ordinal" or a ranked ballot.  It depends on the method.  Most Condorcet methods do not disallow equal rankings.  IRV, Borda, or Condorcet methods that are modifications of IRV may disallow equal ranking.  But straight-ahead Condorcet (with some appropriate completion method) or Ranked Pairs or Schulze or Minimax do not have a problem with equal rankings.

On p 39, the methods you shown are not "cardinal ballots".  They are ordinal ballots.  Only Score (incl. STAR) and Approval are cardinal ballots.

On p 41, there are 9 possible ways to mark your ballot with 3 candidates (or 10, if we count a blank ballots or "undervotes").  Not 6.

Since you were asking for feedback...

bestest,

--

r b-j . _ . _ . _ . _ rbj@audioimagination.com

"Imagination is more important than knowledge."

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.
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Dear Robert,

Thanks for your comments on the slides that I prepared for a 50 minute verbal presentation for an audience that had no background regarding mathematical insights into voting and election situations. To avoid lots of "detail" I said verbally that I would typically restrict my attention to voting situations where the voters ranked all the alternatives without indifference. In that setting if there are 3 alternatives there are 6 ballot types. The way I count there are 13 types of ballots with three choices, no truncation and indifference allowed.

In "informal" discussions often words will be interpreted by different people differently. Even in more scholarly environments sometimes there is "sloppy" terminology. A good example is the use of the term Borda or Borda Count. This account:

https://en.wikipedia.org/wiki/Borda_count

May not be the same as what one might see elsewhere using the terms Borda or Borda Count. Mathematics tries to be "precise" but you can find mathematics books where the word trapezoid defiines different collections of quadrilaterals.

Regards,

Joe

From: robert bristow-johnson rbj@audioimagination.com
Sent: Thursday, March 26, 2026 5:53 PM
To: Joseph Malkevitch jmalkevitch@york.cuny.edu; Joseph Malkevitch via Election-Methods election-methods@lists.electorama.com; Election Methods Mailing List election-methods@electorama.com
Subject: Re: [EM] Election methods scholarship

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Hay Joseph,

Some corrections to https://web.york.cuny.edu/~malk/ranked-voting-math-club-2022.pdf  :

On p 28, you said: "ii. preference, ordinal, or ranked ballot List choices in preference order without ties. Sometimes one is limited as to the total number of candidates one can rank (five for NYC voting)"

In fact, whether ties are allowed or not do not depend solely on whether is "preference" or "ordinal" or a ranked ballot.  It depends on the method.  Most Condorcet methods do not disallow equal rankings.  IRV, Borda, or Condorcet methods that are modifications of IRV may disallow equal ranking.  But straight-ahead Condorcet (with some appropriate completion method) or Ranked Pairs or Schulze or Minimax do not have a problem with equal rankings.

On p 39, the methods you shown are not "cardinal ballots".  They are ordinal ballots.  Only Score (incl. STAR) and Approval are cardinal ballots.

On p 41, there are 9 possible ways to mark your ballot with 3 candidates (or 10, if we count a blank ballots or "undervotes").  Not 6.

Since you were asking for feedback...

bestest,

--

r b-j . _ . _ . _ . _ rbj@audioimagination.com

"Imagination is more important than knowledge."

.
.
.