[EM] Best Single-Winner Method

[Ted Stern]

For reference, here was my previous statement of EXACT:

http://lists.electorama.com/pipermail/election-methods-electorama.com/2017-December/001646.html

in my example, I misstated B's total at and above rating 2 as 61; it should
have been 62.

I should also note that without the irrelevant 5 G ballots, MJ chooses A
(like Relevant Ratings), but MJ's winner changes to B when irrelevant
ballots are added.

Ted

On Fri, May 24, 2019 at 12:10 PM Ted Stern <dodecatheon at gmail.com> wrote:

> 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_1116843553331214489_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_1116843553331214489_m_-2818294270155005541_m_-2981034959420349233__msoanchor_1>Numerical
>>> scores
>>>
>>>
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>>
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