[EM] Best Single-Winner Method (Steve Bosworth)

steve bosworth stevebosworth at hotmail.com
Sun May 26 13:04:47 PDT 2019

Hi Ted,

Thank you both for your response to Cris’s IBIFA and for your discussion of your proposed Relevant Rating (RR) method.

Of course, both IBIRA and RR are great improvements on the methods that are actually being used to elect single-winners.  However, it still seems to me that both fall short of the advantages offered by MJ.  With regard to your description of RR,  please correct any of the self [clarifications which I’ve added within the square brackets].  At the same time, I still seem to be misunderstanding RR because, as yet, I do not see how you have arrived at your conclusions with regard your example election.

In the hope of assisting our dialogue, below I have added a table which records the way MJ would count that election.  That table uses Balinski’s recommended grading language. He argues that the following 6 grades provide a most discerning, meaningful, and informative way for voters to express their judgments about the suitability of candidates for office: Excellent, Very Good, Good, Acceptable, Poor, or Reject.  These six follow the empirical evidence and arguments offered by G.A. Miller (1956,”The magical number seven, plus or minus two: Some limits on our capacity for processing information”. Psychological Review 63: 89-97) and use by Balinski and Laraki Majority Judgment, 2010 MIT.

Correct me if I’m mistaken, but in contrast to MJ, it currently seem to me that your RR suffers from two major flaws:

  1.  RR does not guarantee that the winner is the one candidate who has received the highest available grade on at least 50% plus one of all the ballots containing at least one rating other than 0 (i.e. other than Poor or Reject in the language suggested by Balinski).  Provided an MJ citizen marks their ballot at least with one of the following grades for one of the candidates (Excellent, Very Good, Good, or Acceptable), this grade together with all their default “Rejects” for all the other candidates are counted.  Thus each and every voter equally and necessarily helps to determine each candidates median-grade.  The MJ candidate who has received the highest median-grade is the winner.  My previous post (copied below) explains the strong and relatively simple process by which any tie is broken.
  2.  RR does not treat each citizen’s vote equally.  All the relatively small number of Rejects given to the most popular candidates who gave at least Acceptable to a much less popular candidate are ignored when determining the median-grade of all the candidates (or such median-grades have no relevance for RR).  This is why RR seems both not to guarantee that the winner will be supported by an absolute majority of all those voting, and effectively wastes (ignores) all the votes of at least Acceptable given to unpopular candidates.  This inequality encourages voters to vote dishonestly (tactically).


I look forward to your clarifications and/or refutations.



From: Ted Stern <dodecatheon at gmail.com>
Sent: Friday, May 24, 2019 7:10 PM
To: cbenham at adam.com.au
Cc: election-methods at lists.electorama.com; steve bosworth
Subject: Re: [EM] Best Single-Winner Method

T:  Here's an attempt at a statement of Relevant Rating.  I would welcome any improvements in the explanation.

  1.  Voters rate each candidates a rating of max rating down to 0.  Blank ballots [regarding any particular candidate] will be counted as 0 / Disapproved  [i.e. Reject].  Any non-zero rating is counted as approved [i.e. Acceptable, Good, Very Good, or Excellent].  A ballot can contain any number of candidates at any rating level, but all equal-bottom rated candidates [Poor or Reject] will be counted as 0.  In other words, a ballot must disapprove at least one candidate [no, must grade at least one candidate as Acceptable, and by doing so will be counted as also graded every other candidate as Reject].
  2.  A candidate's total approval rating on a set of ballots is the total number of those ballots rating the candidate above zero.
  3.  For each candidate X, imagine dividing the ballots into MAXRATE+1 piles [i.e. ballots that give each candidate either Excellent or one of the other 5 Balinski grades ]. [I.e.] In each pile are [all] the ballots rating X at rating R [with any of Balinski’s grades] from MAXRATE down to zero [i.e. Reject].  For a particular rating R [i.e. median-rating], we can see the total vote for X at and above that rating by adding up the sizes of the piles from MAXRATE down to R.  Then, looking at the [number of] ballots in the remaining piles (R-1 down to 0) [i.e. the number lower than the median-rating grade], look for the candidate with highest total approval [i.e. the number higher than the median-grade plus the median-grade].
  4.  If [the number of] X's votes at and above a rating R [i.e. X’s median-grade] exceed the highest total [number of ballots which approve approval for any candidate on ballots that rate X below R [i.e. X’s median-grade], then X's relevant rating is R [?].
  5.  Starting at the top rating, see if any candidates have that relevant rating (i.e., they satisfy criterion 4).
  6.  If there is at least one such candidate, then see if there is among them a candidate Y for whom the total number of ballots rating Y above R is also greater [in number] than the highest total approval for any candidate on the ballots that rate Y below R.  If so, the candidate Y with highest total number of ballots rating Y above R is the winner.  Otherwise, the candidate X satisfying criterion 5 with highest total number of ballots rating X at R and above is the winner [I am still not clear whether this description might only be a more cumbersome way of expressing the count used by MJ and which I presented in the last paragraph of my earlier post as copied at the end of this post – please clarify?].
  7.  If there is no non-zero rating at which a candidate can be found who satisfies criterion 4, then the candidate with highest total approval is the winner.

S:  Stern’s example election :

Converted to Balinisk’s MJ vocabulary (Excellent (e), Very Good (vg), Good (g), Acceptable (a), Poor (p), or Reject (r).

Possible translation by Balinski’’s MJ:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

Candidates:     A         B          C          D          E          F          G          H

                        49:e     49:vg   03:a     10:e     38:e     38:vg   05:e     05:a

                        03:vg   03:e     10:a     05:vg   03:E

                        0          10:vg   38:a     0          0

                        0          0          0          0          0

Total High:       52        62        51        15        41        38        05        05

Median:           r           vg        r           r           r           r           r           r

Total Low:        53        43        54        90        67        67        100      100

                        0          0          0          0          0          05:r      0          0

                        0          0          0          0:         05:r      0          38:r      38:r

                        05:r      05:r      05:r      38:r      10:r      10:r      10:r      10:r

                        38:r      38:r      0          03:r      03:r      03:r      03:r      03:r

                        10:r      0          49:r      49:r      49:r      49:r      49:r      49:r

B is the winner with a median-grade of Very Good.  However, if the 5 citizens who voted for GDH were absent or entirely ignored as seems to be the case with RR, A would be the winner with a new median-grade of Excellent (e).  At the same time, since these 5 citizens did vote, but RR elects A, A is being elected by a large minority of all the votes cast.  How can this be justified if every citizens’ vote must be treated with equal respect in a democracy?
T:  Here's a somewhat contrived example of an election in which Relevant Rating and IBIFA get a different result.

49: A > B

03: B > A > C

10: D > B > C

38: E > F > C

05: G > D > H

Ratings of 3, 2, 1, 0.

At rating 3, we see that A has 49 vs total approval C:51 on the complementary ballots, so A's relevant rating must be below 3.

At rating 2, A has 51 [?52] ballots at and above rating 2, as opposed to C's approval of 48 [?51] on complementary ballots, so A's relevant rating is 2.  But we see that B has 61 [62] ballots at and above rating 2, meeting the same criterion.  If we are using IBIFA, B wins with 61 [?62] vs A's 48.  But using Relevant Rating, we see that A's 3-level total of 49 is greater than the total for C on ballots voting for A below 2, and B's total at rating 3 does not exceed the highest approved candidate on ballots that exclude B at 3, so A wins the RR tie-breaker.

S: Please clarify all the above reasoning.


Stern’s RESULTs (printed in red:???????????????????????):

Candidates:     A         B          C          D          E          F          G          H

                        49:3     49:2     03:1     10:e     38:e     38:vg   05:e     05:a

                        03:2     03:3     10:1     05:vg   03:e

                        0          10:2     38:1     0          0

                        0          0          0          0          0

Total High:       52        62        51        15        41        38        05        05

Median:           r           vg        approval

Total Low:        48        38        49        90        67        67        100      100

                        0          0          0          0          0          05:r      0          0

                        0          0          0          0:         05:r      0          38:r      38:r

                        05:r      05:r      05:r      38:r      10:r      10:r      10:r      10:r

                        38:r      38:r      0          03:r      03:r      03:r      03:r      03:r

                        10:r      0          49:r      49:r      49:r      49:r      49:r      49:r



On 21/05/2019 5:16 am, steve bosworth wrote:

Re: Best Single- Winner Method


Firstly, please correct me if I am mistaken but currently I am assuming that  we all would ideally want the Best Single-Winner Method:

  1.  To be simple enough so voters  can both use it and understand how it is counted;
  2.  To minimize the wasting of citizens’ votes (see below),  and
  3.  To guarantee that the winner among 3 or more candidates is the candidate most supported by at least 50% plus one (an absolute majority) of all the citizens voting, and
  4.  To offer as few incentives and opportunities as possible to vote tactical.

Given these desires, currently I see Majority Judgment (MJ) as superior to all of the above methods on each of these counts.  However, since the above discussions have not mentioned MJ, I assume that many contributors would reject this claim for MJ.  This is why I would very much appreciate receiving any of your clarifications or explanations of how my claim for MJ cannot be sustained.  What important flaws to you see in MJ?

To help you to marshal your criticisms of MJ, please let me explain more full my own understandings and reasons for favoring MJ.  Firstly, I see a citizen’s vote as being wasted quantitatively to the degree that it fails equally to help one of their most trusted candidates to win.  A citizen’s vote is wasted qualitatively to the degree that it instead helps to elect a candidate whom they judge less fit for office, rather than an available candidate judged to be more fit.

Other than in MJ, such waste is present in all the existing methods, whether they ask voters to rank, score, or approve as many of the candidates as they might wish.  Of course, most dramatic is the waste provided by plurality or First-Past-The-Post voting.

To counter qualitative waste, Balinski and Laraki (Majority Judgment, 2010 MIT) argue that our capacity for judging qualities of human behavior can be most meaningfully expressed in an election by each voter grading each candidate’s suitability for office as either Excellent (ideal), Very Good, Good, Acceptable, Poor, or “Reject” (entirely unsuitable).  These grades are more discerning, meaningful, and informative than merely expressing preferences or using numeric scores , X’s or ticks.  Such grading makes it more likely that the highest quality candidate will be elected in the eyes of the electorate.

Each candidate who is not explicitly graded is counted as ”Reject” by that voter.  As a result, all the candidates will receive the same number of evaluations, but a different set of grades from the voters.  The Majority Judgment (MJ) winner is the one who has received grades from an absolute majority of all the voters that are equal to, or higher than, the highest median-grade given to any candidate. This median-grade is found as follows:

  *   Place all the grades, high to low, top to bottom, in side-by-side columns, the name of each candidate at the top of each of these columns.
  *   The median-grade for each candidate is the grade located half way down each column, i.e. in the middle if there is an odd number of voters, the lower middle if the number is even.

If more than one candidate has the same highest median-grade, the MJ winner is discovered by removing (one-by-one) any grades equal in value to the current highest median grade from each tied candidate’s total until only one of the previously tied candidates currently has the highest remaining median-grade.

Also, in contrast to the alternatives, Balinski  explains how MJ reduces by almost half, both the incentives and opportunities for effective tactical voting.  Thus, each voter has every appropriate incentive, not only to vote but to reveal their honest evaluations of each candidate.

Thus, to me, using MJ should be simpler and more satisfying because grading many candidates is both easier and more meaningful than ranking or scoring them.  Also, finding and comparing the median-grades of all the candidate is quite simple.  Unlike MJ, IRV, Condorcet methods, and Scoring do not guarantee the election of the candidate most preferred by at least 50% plus one of all the citizens voting.  Unlike IRV but like Condorcet methods and Score, MJ does not eliminate any candidate until the winner is discovered.

Finally, I would favor the following Asset option to be added at the bottom of each MJ ballot:  Any citizen who currently feels that they do not yet know enough about any of the candidates to grade them, can instead give their proxy vote to the Register Elector who will do this for them.  They could do this  by WRITING-IN the published code of that Registered Elector.

I look forward to your comments.

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