Hi Steve, you wrote:

What I understood from Kristofer's Jan 4 explanation of page 15 is that this halving of the manipulabilityis not meant to be a comparison to any other methods. It's a comparison to a (rather strange) hypotheticalsituation. If so, this claim on its own could be true but is of unclear value.
Possibly this argument is used to build up to a larger argument. But when it gets stated on its own it feelsmisleading to me, because there's no way for the reader to understand what this "half" is half of.
Kevin

  De : steve bosworth <stevebosworth@hotmail.com>

À : "election-methods@lists.electorama.com" election-methods@lists.electorama.com; "rbj@audioimagination.com" rbj@audioimagination.com
Envoyé le : Dimanche 12 février 2017 5h48
Objet : Re: [EM] “goal of a better election method”

1. Level playing field
   for all voters ("One person, one vote")
2. No burden of tactical voting (therefore no punishment for voting sincerely) in a multi-candidate election. ?No voter regret.?
3. …. 10 ….”My suggested goal for a single-winner election (e.g. for a president, governor, or major) is Balinski and Laraki’s:“The purpose of … an election is to select, if possible, some candidate who shall, in the opinion of a majority of the electors, be most fit for the post…” (E. J. Nanson, quoted by Balinski and Laraki,Majority Judgment, MIT, 2011, p.209).Also, “to gather as precisely as possible, the true opinions and evaluations of individuals, and to determine as precisely as possible, the true aggregate wills of electorates ….” (Ibid, p. 388).Accordingly, I also seem to be coming to the conclusion that a refined improvement on Balinski and Laraki’s Majority Judgment (MJ) method called Highest Majority Judgment (HMJ) by me would be the most efficient way of electing this “most fit” candidate. As will be explained below, HMJ guarantees that the winner will be the candidate most valued for the job by at least an absolute majority of all the voters, i.e. the candidate believed on average by at least an absolute majority to be the most qualified for the office.  Thus, HMJ also seems to be the “easiest single-winner voting method to tolerate”. Please explain any mistakes I might be making if I see HMJ as superior to the other good methods, e.g. any known variant of Immediate Run-Off Voting (IRV), SCORE (Range), Maximized Affirmed Majorities (MAM), or APPROVAL.Firstly, I believe that Balinski and Laraki explain how MJ avoids both “Condorcet paradox” and the “Arrow paradox” (ibid, pp.182-3). Secondly, I believe that B&L “prove” (pp. 15, 19, 186-198) that MJ provides only about “half” the incentives or opportunities for anti-democratic “strategic” voting to be successful. If you disagree, please explain the flaw in their argument. Until I see such a flaw, MJ would seem to offer no reason for a savvy voter to expect to have a probable “strategic” advantage over less savvy voters, i.e. if and when they might choose to misuse MJ’s ballot, in effect, to “rank” rather than to “grade” the candidates. If B&L’s proof is valid (and it is also true for HMJ), it, more than the above methods, largely frees voters from the burden of perhaps having to dishonestly ‘grade’ some of the candidates. This is because with HMJ, it is probable that their honest ‘grades’ will do all they can to help elect the best candidate in their view. This is why Balinski and Laraki say that voting “honestly” with MJ is likely to be the “dominant strategy” (pp.190). Unfortunately, IRV, SCORE, MAM, and APPROVAL sometimes truly offer some very complicated reasons for very savvy citizens to vote dishonestly while hoping to make the election of their most preferred candidate more likely. By largely removing this burden, HMJ would seem to help to minimize such distortions to the democratic process, i.e. to make the election of the best candidate as likely as possible.I will now explain in more detail how HMJ works:
   • Uniquely, HMJ (and MJ) asks each voter simply to record their “evaluation” of each candidate by giving each candidate one of the following “grades” depending on how closely each candidate comes to fitting the citizen’s own image of an “excellent” candidate:EXCELLENT (5), VERY GOOD (4), GOOD (3), ACCEPTABLE (2), POOR (1), or REJECT (0) – each blank is interpreted as REJECT.   Such grades are likely to be meaningful at least to any person who has gone to school. It is easier to grade many candidates than to rank them. Every citizens’ “grade” for every candidate continues to be part of the count until the absolute majority winner is discovered.
   • Each candidate receives the same number of “grades”. To begin the count, all of each candidate’s “grades” are listed from highest to lowest, left to right. Next, each candidate’s “median-grade” is identified (i.e. the middle one on the right if there is an even number of voters).
   • The HMJ winner is the candidate discovered to have the highest average of all the grades she has receive to the left of her “median-grade” and including her “median-grade”. If there is more than one candidate with this same highest average grade, each is a potential winner (see Example 1 below). In this event, HMJ discovers the winner by comparing, one by one, the next grade that each has received immediately to the right of their respective medians. The winner is the first candidate in this sequential comparison discovered to have a next grade higher than any of the other previously tied potential winners(see Example 2 below).
     Example 1:Y wins with HMJ, but X wins with MJ & Bucklin MJX: VGGPP or 43311………..10/3 = 3.66Y: EEAAP or 55221 …………12/3 = 4Example 2:X wins with MJ & BMJ, but with HMJ, X & Y are initially tied. However, by comparing the grades to the right of the median, Y is discovered to be the HMJ winner.X: EVGPP or 54311………..12/3 = 4; 13/4 = 3.33Y: EEAAA or 55222…………12/3 = 4; 14/4 = 3.66In this way, HMJ guarantees that the winner will be the candidate most highly valued by at least an absolute majority of the electorate. While Balinski and Laraki clearly explain their own methods for breaking MJ ties, I see their methods to be much more laborious and less efficient than HMJ’s. Firstly, MJ identifies all the candidates who have received the “highest majority-grade”. If there is more than one such candidate, MJ’s simplest tie breaking process uses “majority-guages” (pp.9ff). This differs from HMJ’s by simply comparing each such candidate’s number of grades listed to the left and to the right of all the grades each such candidate has received which are the same as their common “highest median-grade”. If and when this comparison fails to discover a winner, then the “majority-values” (pp. 6ff) of the tied candidates are compared instead. This is done by sequentially discovery and listing each tied candidate’s new “median-grade” upon the removal of the current median-grade from the total list of all the grades each such candidate had initially received. This is repeated until one of these candidates is found to have the highest new median-grade. Thus, unlike HMJ, MJ does not average the variety of different degrees of “evaluation” listed to the left of each candidate’s “median-grade”, i.e. the grades that may be distributed differently among this group of grades received by each of the tied candidates.  Therefore, in contrast to the above rival methods, HMJ
   • offers the greatest encouragement for each citizen to vote;
   • allows and most strongly encourages (with MJ) each citizen most fully, exactly and honestly to express their different “evaluations” of each candidate; 
   • again, (with MJ) offersonly about “half” the incentives or opportunities for anti-democratic “strategic” voting to be successful. Thus, B&L argue that for most citizens, “sincere voting” will be the “dominant strategy” (pp. 190); and
   • has the virtue (with MJ) of not requiring any arbitrary procedure to discover the absolute majority winner among any initially tied candidates unless every voter has “graded” each candidate identically.
     Also, I note that in contrast to HMJ:
   • IRV can eliminate some candidates before the winner is discovered. Unfortunately, one of these eliminated candidates might be the one who is preferred by more voters than any other candidate. This is true even though IRV also guarantees that its winner will have been explicitly preferred by a majority over all the remaining candidates;
   • SCORE’s winner has not necessarily received ratings higher than 0 from a majority of the voters. Its number ratings of candidates are less meaningful than “grades”.  Also, especially if its highest rating is more than “7”, they would also be less “discerning” (Ibid, p.283) than the 6 different “grades” used by HMJ.  This is because empirical studies have discovered that most people cannot meaningfully distinguish between more than seven “grades” of valued human behavior;
   • MAM’s aggregation of all the voter’s preferences ignores some of the different ordinal preferences recorded on each voter’s ballot.  In contrast, all the different degrees of evaluation of all voters used by HMJ (i.e. “grades”) contribute to the discovery of its absolute majority winner. Also, many ordinary voters would find it much harder to understand how their MAM preferences are aggregated in an attempt to find the Condorcet winner (or the one produced by its tie breaking procedure). Finally, an MAM winner may not have been explicitly preferred by a majority of the voters;
   • APPROVAL does not allow citizens to express the full range of different degrees of “approval” that voters may feel with regard to the available candidates.  Consequently, a voter’s marking of the candidates she only weakly favors may help to defeat the candidate she most enthusiastically favors. Also, the APPROVAL winner may not have been “approved” by a majority of the voters.
     I look forward to your feedback.Steve

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On Sun, Feb 12, 2017 at 6:48 AM, steve bosworth stevebosworth-at-hotmail.com
|electorama electowiki/Example Allow| 9zz1sjkwvt@sneakemail.com wrote:

I definitely disagree with this quote, after learning of the difference
between majoritarian and utilitarian systems.  I'm solidly in the
utilitarian camp now:

The purpose of an election is to select the candidate who maximizes the
utility/happiness/satisfaction of the voters.  (All the voters; not half.)

Also, “... to determine as precisely as possible, the true aggregate wills

This sounds more like utilitarianism.  If the electorate's positions on
political issues are plotted in a multi-dimensional issue space, the winner
should be the candidate who is nearest to their centroid (= the aggregate
wills of the electorate).  This goal is incompatible with the previous goal
of majority rule.

http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html

On 02/12/2017 12:48 PM, steve bosworth wrote:

[...]

I suppose the concern that comes most clearly to mind is that MJ is not
meant to be a cardinal method, whereas using averages introduces a
cardinal (numeric) element.

To rephrase that, all that "plain" MJ knows about is that there's a
common standard among the people where

Excellent is better than Very Good
Very Good is better than Good
Good is better than Acceptable
Acceptable is better than Poor
and Poor is better than Reject.

It doesn't know what the common standard is, however.

To use a school grading metaphor, it knows that A is better than B, but
not that

A: 100%-97% of max attainable points
B: 97%-85%

(or what have you).

Suppose that we want to use MJ to determine the student with the best
performance. Then MJ is supposed to work equally well no matter whether
the grading scale is

A: 100%-97%
B: 97%-80%
(... etc)

of if the grading scale is

A: 100%-92%
B: 92%-77%
(... etc)

as long as all classes use the same grading scale (that's the "common
standard" qualification).

If you use averaging as part of an MJ method, then the ordinal
assumption is violated, because taking the mean doesn't just depend on
the relative order (i.e. A is better than B), but also on just how much
(A has a mean score of 98.5%, B has a mean score of 88.5% vs A has a
mean score of 96% and B has a mean score of 84.5%).

As a consequence, the method may produce better results in certain
scenarios (where the common standard is equally spaced), but it trades
that off by potentially producing worse results in other scenarios
(where the common standard is not equally spaced).

Or more simply put: making the scale cardinal, which you need to be able
to calculate means on the votes, adds more assumptions that may not be true.

For more on this particular objection, see B&L's "Election by Majority
Judgment: Experimental Evidence", p 45. and onwards ("Voting by Points
and Summing"). Quoting:

(My source is
https://1984f707-a-62cb3a1a-s-sites.googlegroups.com/site/ridalaraki/xfiles/ElectionByMajorityJudgment(ExperimentalEvidence)Final.pdf)

tiebreaking is surely better than using it everywhere (like in Range).
But the same theoretical arguments apply as soon as you're using a
numerical scale at all.

There's also a strategy argument, which could be generally argued like this:

• Either ties of the type where many candidates have the same median are
  common or they're not.
• If they're not common, then a mean tiebreak is not going to change the
  outcome often, so we can go with Bucklin or MJ's system.
• If they're common, then, since Range is more susceptible to strategy
  than MJ, using a mean tiebreak will make HMJ considerably more
  susceptible to strategy as well, and so should be avoided.

See this slide set by B&L for more on that:
http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf . In particular
they say:

"The unique [social grading function]s that are partially
strategy-proof-in-ranking are the order functions."

This means that the only grade-based voting methods that are partially
strategy-proof in ranking (which they define earlier) are order
functions, which are of the form "return the nth best grade" for some n.
(The median sets n=voters/2) Consequentially, to keep the SGF partially
strategy-proof-in-ranking, the tiebreak should also be an order function
with a different n, which it is in MJ and Bucklin, but not in HMJ.

B&L do not analyze the HMJ variant in itself, but they note:

"The unique aggregation functions that minimize the probability of
effective-manipulability are the middlemost. Point-summing-methods, f_1
and f_n maximize this probability."

which does suggest that using a point-summing method as a tiebreak will
weaken the method more than using another type of method.

In addition, averaging would make a method no longer
strategy-proof-in-grading. For the same reason that the median (being an
order function) makes MJ resistant to strategy, an order function tie
break makes MJ resistant to strategy in the tiebreak. B&L say:

"The unique strategy-proof-in-grading SGFs are the order functions.

If the mechanism is a point-summing method (the mean with respect to
some parametrization), for almost all profiles, all voters can manipulate."

(Last slide on "Strategy in Grading")

This is not to say that HMJ is a bad method. I would certainly choose
it if the alternative was, say, IRV or Borda. But using averaging does
undermine B&L's theoretical foundation of MJ (since it is no longer an
ordinal grade method), and if you accept their strategy-resistance
arguments, it also weakens MJ's resistance to strategy.

On 02/12/2017 12:48 PM, steve bosworth wrote:

[...]

I forgot to mention this strategy-in-grading problem.

Suppose we're using ordinary MJ, and candidate X's final grade is Good.
Someone who gave X a grade of "Very Good" has no reason to exaggerate to
"Excellent" because his vote is counted equally according to MJ's tie
breaker. This helps prevent the method from becoming Really Expensive
Approval where everybody just votes max or min.

However, if you use averages as a tiebreak, the voter might think: "I'm
reasonably sure X's final grade is going to be Good, but as there may be
other candidates with Good as a final grade as well, I should do my best
to make sure X's average gets as high as possible, which means that I
should vote Excellent instead of Very Good". If enough voters do that,
then the method slides into Approval.

And since we need all the help we can get to keep MJ from becoming
Approval (some Range advocates say that MJ would essentially become
Approval even in its current state), it's best to make this kind of
strategy ineffective. And B&L say that requirement narrows down the only
tiebreakers you can use into order functions -- for the same reason that
only an order function will do for the main scoring prior to any tiebreaker.