[EM] Best Single-Winner Method

Ted Stern dodecatheon at gmail.com
Fri May 24 12:10:25 PDT 2019

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 will be counted as 0 / Disapproved.  Any non-zero rating is counted
   as approved.  A ballot can contain any number of candidates at any rating
   level, but all equal-bottom rated candidates will be counted as 0.  In
   other words, a ballot must disapprove at least one candidate.
   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.  In each pile are the ballots rating X at rating R from MAXRATE down
   to zero.  For a particular rating R, 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 ballots in the remaining piles (R-1 down to 0),
   look for the candidate with highest total approval.
   4. If X's votes at and above a rating R exceed the highest total
   approval for any candidate on ballots that rate X below R, 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 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.
   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.

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 ballots at and above rating 2, as opposed to C's
approval of 48 on complementary ballots, so A's relevant rating is 2.  But
we see that B has 61 ballots at and above rating 2, meeting the same
criterion.  If we are using IBIFA, B wins with 61 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.

MJ and Median rating both choose B.

Condorcet winner is A.

Since I announced this method (as "EXACT") a year or so go, I've found a
fairly natural way to extend this to quota-based multiwinner, also.  If
interested, I will describe details.

On Fri, May 24, 2019 at 11:09 AM Ted Stern <dodecatheon at gmail.com> wrote:

> Hi Chris,
> You write (emphasis mine):
> On Wed, May 22, 2019 at 5:12 AM C.Benham <cbenham at adam.com.au> wrote:
>> Steve,
>> Earlier I responded briefly to your 4-point list: yes, yes, not
>> possible,yes.
>> To expand a bit, MJ is a Median Ratings method with a relatively
>> complicated tie-breaker, *justified by I've no idea what.*
>> Bucklin is likewise a Median Ratings method and will usually give the
>> same winner, but the "tie breaker" is simple.
> The justification for the Majority Judgment tie breaker is that when two
> candidates have the same median rating, the proper way to resolve the tie
> is to look at the minimal number of ballots that must be removed to change
> their median rating.  So it is similar to Minimax Pairwise Margins
> justification of minimal change.  This tie breaker leans towards those
> candidates with stronger support above the median rating.
> My privately proposed method of Relevant Ratings (formerly known as EXACT)
> uses a MJ-like tie breaker with your IBIFA method to get a similar effect.
>> An example I recently came up with to critique another Bucklin-like
>> proposal:
>> 46: A
>> 03: A>B
>> 25: C>B
>> 23: D>B
>> 97 ballots  (majority threshold = 48)
>> (If  you want MJ-style  multi-slot ratings ballots, assume that all the
>> voters have given their favourite the highest possible
>> rating and those that rated B above bottom all gave B the same middle
>> rating and that truncating here signifies giving the
>> lowest possible rating).
>> MJ and Bucklin  both rightly elect A.   IBIFA  and IRV also elect A.  A
>> is the Condorcet winner: A>B 49-48, A>C 49-25, A>D 49-23,
>> A>E 49>0.
>> A is the most Top-rated candidate:  A49,  C25,  D23,  B0, E0.
>> So suppose the votes are counted and it is announced that A has won, but
>> just before this is officially and irrevocably confirmed
>> someone pipes up, "Hang on a minute, we found a few more ballots!"
>> (Maybe they are late-arriving postal votes that had been
>> thought lost.)
>> These 3 new ballots are inspected and found that all they do is give the
>> highest possible rating to E, a candidate with no support
>> on any of the other 97 ballots. What do we do now?  Laugh and carry on
>> with confirming A as still the winner?  No.
>> 46: A
>> 03: A>B
>> 25: C>B
>> 23: D>B
>> 03: E
>> 100 ballots  (majority threshold = 51)
>> Now MJ  and Bucklin and any other Median Ratings method elects B.  All
>> methods that I find acceptable elect A both with
>> and without those 3E ballots.
>> To expand a bit my earlier response to your point 4, I think it's highly
>> desirable for a method to meet FBC (the Favorite Betrayal
>> Criterion), especially in the current US situation.
>> Potentially popular candidates the mainstream media and political
>> establishment doesn't like can be sunk by fake polls and
>> the voters' fear of some perceived Greater Evil.
>> They can say "Forget candidate X!  X is only polling at 1 or 2%. That
>> justifies us ignoring X. If you vote for X you'll just be helping
>> Greater Evil win!".  And their prophesy that X won't be a viable
>> candidate tends to be self-fulfilling.
>> But if the used voting method meets FBC, the voter can in response reply
>> (or at least think) "Ok, maybe I have to top-rate some
>> 'realistic' Compromise candidate to maximise  the chance of beating
>> Greater Evil, but I like X and I know that it can't possibly hurt
>> me to also top-rate X so I will."
>> If the method meets FBC no voter who knows and understands that can be
>> cowed into not voting their sincere favorite at least
>> equal-top.  FBC is met by Approval, MJ and the currently proposed version
>> of Bucklin, and IBIFA.
>> Of those I judge IBIFA to be by far the best.  IRV and all Condorcet
>> methods unfortunately fail FBC.
>> For a method that doesn't meet FBC, I consider meeting the Condorcet
>> criterion to be desirable. Unfortunately all Condorcet
>> methods are (to greatly varying degrees) vulnerable to Burial strategy,
>> i.e. insincerely down-ranking to make a higher (usually
>> top) ranked candidate win.
>> In my humble opinion the best of the methods that are invulnerable to
>> Burial is IRV.
>> Chris Benham
>> On 21/05/2019 5:16 am, steve bosworth wrote:
>> Re: Best Single- Winner Method
>> Sennet Williams,  Forest Simmons, Robert Bristow-Johnson, Abd dul Raman
>> Lomax, and Chris Benham have recently addressed each others’ claims
>> about IRV, 3-slot Methods, IBIFA, and Asset.  This discussion prompts me
>> to request some help later after I have clarified several issues.
>> 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 possibilities for voting 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[MOU1]
>> <#m_-2818294270155005541_m_-2981034959420349233__msocom_1> , 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 receiv 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.
>> Steve
>> ------------------------------
>>  [MOU1] <#m_-2818294270155005541_m_-2981034959420349233__msoanchor_1>Numerical
>> scores
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