# [EM] (3) Best Single-Winner Method

steve bosworth stevebosworth at hotmail.com
Mon Jun 17 16:28:44 PDT 2019

```Hi Ted,

Thank you also for discussing MJ.  I will respond inline below but firstly I want to make it clear that in the end, I would like answers or explanations for the following two questions:

1.  Does your preferred voting method guarantee that its single-winner is supported by a majority of all the votes cast?
2.  If not, why is this method still favored by you?

________________________________

From: Ted Stern <dodecatheon at gmail.com>
Sent: Friday, June 14, 2019 12:33 AM
To: steve bosworth
Cc: cbenham at adam.com.au; EM list
Subject: Re: [EM] (2) Best Single-Winner Method-IBIFA vs. MJ

Hi Steve,

T:  I have developed a slight modification of Chris Benham's IBIFA that is more similar to Majority Judgment.  I've started editing a page describing the method at

Balinski and Laraki have an alternate form of Majority Judgment in which each candidate receives what they call a Majority Grade.

The Majority Grade has several parts.  The primary portion is the candidate's median rating.  That is, the rating MR at which the number of ballots approving candidate X at rating MR exceeds the total number of ballots rating X below MR.

S:  Yes.

S: It is not particularly important for our purposes, but my recollection is that B&L’s term for this three part summary of each candidates majority-grade is its “majority-gauge”.

T:  The secondary portion of the Majority Grade is a measure of what median rating the candidate would get if median ballots are removed until the rating changes.  But that rating is simply the rating that is "closer" to the median ballot.

S:  More exactly, the median-gauge starts with the verbal expression which contains all the grades that are equal in value to that candidate’s median-grade(e.g. Good).  This grade is in the middles of a complete list of all the grades received by a given candidate, e.g. in the middle of a list of all the grades received, listed highest to lowest, top to bottom (i.e. the middle grade if the number of voters is odd, the lower middle if the number is even).

T:  In other words, if the total number of ballots rating X below the MR is less than the total number of ballots rating X above the MR, then the secondary rating is MR+1, followed by the above MR total.  Otherwise, the secondary portion of the Majority Grade is MR minus one, followed by the at-and-above-MR total.

S:  Instead, according to my recollection of B&L, an example of a majority-gauge about which you are referring above is the follow: 42% Good+31%.

This would mean that 25% of each of all the grades given to this candidate are of a value equal to this candidate’s median-grade (i.e. Good).  Perhaps this way of reporting the grades received by each candidate helps some readers better to understand how MJ works.  However, I see the steps described earlier on pp. 5&17 of Majority Judgment as providing the simplest way to find the winner when there is a tie, i.e. one-by-one, remove one grade from the list of each tied candidate equal to the value of their shared highest median-grade.  Do this repeatedly until only one has the highest median-grade.

T:  So B&L are choosing to see the tie breaking as removing median ballots, but an alternative viewpoint would be to interpret the tie-breaking as comparing the above-median-rating strength to the below-median-rating-preference.

S: Yes.

T:  If you're talking about an election, I think the latter viewpoint is more meaningful.

S:  No, again I see the removal of ballots one-by-one currently reporting the same highest median-grade from each of the tied candidates as the simplest and most exact way of discovering the one candidate who continues to have the highest median grade.

T:  So what Chris and I are saying is that the meaningful part of that comparison is not simply the total number of ballots expressing a preference other than X, but whether there is a relevant alternative candidate Y whose total approval is comparable to that of X at that rating level. ….

S:  In some vague sense there could be some other “comparable” candidates but none of these would have this highest median grade at the end of he above tie breaking process.

T: If additional ballots don't contribute to the approval of a meaningful alternative candidate, then we experience a spoiler effect, a tyranny of the majority.

S:  Any election by any voting method might be changed by adding more ballots containing certain or preferences.  MJ is no exception.  As I see it, the great advantage that MJ offers is firstly that it allows each voters to express themselves fully and most meaningfully when grading the suitability of each candidate for office.  This means that the public will know exactly how many of each of the grades from Excellent to Reject each candidate received from all the voter after the count of the election, i.e. the public will be more informed by MJ’s results than by any alternative election.

Secondly, it guarantees that each of all the votes will be counted equally to help determine the median-grade of each candidate.  This allows the most discerning and meaningful discovery of the one candidate who has finally received the highest median-grade (i.e. received at least 50% plus one of all the grades which have a value at least as high as their highest median-grade).  This second benefit makes it clear that no citizens vote can be discarded as “irrelevant”.

T:  I mean, isn't the whole point of using a median rating in the first place an attempt to avoid the sensitivity to outliers inherent in the mean? ….

S:  Yes, it tends to moderate the effect of outliers but most importantly, it guarantees that the winner is explicitly supported by a majority.

T: …. If your answer experiences large changes in response to small changes in input, those of us in the numerical analysis community would call that ill-conditioned, something to be avoided.

S:  No mater what counting rules might be used, small changes in input can produce large changes.

T:  Personally, what I want to achieve with a voting method is to find the candidate closest to the center of mass of the population, but that's hard to do when each individual is making their own best guess to how near they are in preference space to each candidates.

S:  Since B&L show how MJ is the method most likely to prompt the honest grading of the candidates, doesn’t the fact that MJ also elects the candidate most highly supported by a majority of the citizens make it most probable that both this majority and the winner will be “closest to the center of the mass of the population”?

Also, recall that MJ allows each citizen to guarantee that they will have up to four sets of opportunities for one of their highly graded candidates to be the one elected.  Firstly, a candidate whom they give an Excellent will be elected by an absolute majority, provided a sufficient number of other citizens see that candidate as at least Acceptable.  If not, they have a second opportunity for any of the candidates they grade as Very Good to receive such a instead.  Similarly, they have a third and fourth opportunity for candidates they might have graded as Good or Acceptable.

T:  For all practical purposes, my relevant ratings method would give the same answer as MJ anyway.

S:  Perhaps, but to the extent that this is not guaranteed, RR would be flawed.  In fact, according to my current understanding of RR (as presented in your recently edited ElecoWiki article), it is flawed. RR prevents some approvals for a potential rival candidate from being counted for that rival, i.e. any approval for the rival which is on the same ballot that has already been added to the candidate who currently has the largest number of Top-ratings must not be added to the rival’s potential total.

Is my interpretation correct?  What do you think?

I look forward to the next step in our dialogue.

Steve

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