# [EM] Deterministic Epsilon Consensus Idea stimulated by a question of Steve Bosworth (was Election-Methods Digest, Vol 207, Issue 9)

Forest Simmons forest.simmons21 at gmail.com
Wed Oct 6 19:38:57 PDT 2021

```Steve's query about Chiastic Approval included the following ....

Also, correct me if I'm mistaken that XA does not guarantee that its
winner will be elected with the support of a majority of all the votes
(ballots) cast.

The short answer is "no" ...  no method can guarantee majority voter
support for its winner, unless they can guarantee that at least one
candidate is ranked, rated, scored, or graded above bottom on more than
half of the ballots submitted.

The long answer is, "Why stop at half or two-thirds, as some methods
require ... why not go for full 100% consensus?"

But, you may object, full consensus is not always possible. Well, neither
is forty percent support always possible, but that doesn't stop the
Constitution from requiring two-thirds of the voters' support for certain
kinds of amendments, etc.

One expedient that has been suggested is the NOTA option for the case when
the quota is not met.  This option gives new meaning to the word "approval"
... as Mike Ossipoff used to say, you approve a candidate if you would
rather see her elected than have to come back next month to vote for
someone else.

I would like to suggest another option based on the standard MJ grade
ballot ...

Each candidate X gets a score that is given by the sum ..

S(X) = Sum (over j from zero to five) of the product

a(j)*epsilon^j,

where epsilon is a value to be determined by the voters ... and the
respective values of a(j), for j in {0, 1, 2, 3, 4} are the number of
ballots on which candidate X is graded strictly above reject, poor,
acceptable, good, or very good, respectively.

Also each voter has the option of voting for a value of epsilon in the set
{.01, .02, ... .99, 1.00}. The median of the distribution of these votes
determines the value of epsilon.

Elect the candidate X with the max value of S(X) (once the epsilon value
has been determined).

Note that if, for some j, the coefficient a(j) is the total number of
ballots, then we can say candidate X is a full consensus candidate at level
j.

If there are several full consensus level j candidates, then the higher
degree terms will determine the winner.

Thanks!

FWS

> FROM: Steve
> TO: Forest
> Re: Majority Judgment
>
> Thank you for telling me about  Andy Jennings' "Chiastic Approval  (XA)".
>
> However, please explain why do you say it is a "drawback" for MJ
> initionally to have a large number of "ties". You say this even when you
> also correctly say that the next steps in MJ count rationally and
> "cleaverly ... resolve such ties". Do you agree with me that this has the
> majoritarian advantage of guaranteeing that the winner is supported by at
> least 50% plus 1 of all the votes cast, and is unique in both have the
> highest median grade and the largest number of grades equal to or above the
> value of this highest median grade?
>
> Also, in what sense do you see XA as being "more robust" than MJ?
>
> At the same time, unfortunately, XA seems to allow votes to be express
> merey by numbers. I understand MJ's use of word grades (Excellent,Very
> Good,... Reject) to be democratically superior because grades are more
> meaningfully and informatively expressive of the qualitative judgments that
> can be made by voters.
>
> Also, correct me if I'm mistaken that XA does not guarantee that its
> winner will be elected with the support of a majority of all the votes
> (ballots) cast.
>
> For the above reasons, I do not yet see why you said that: "The voter
> instructions and other features of the voter interface
> environment for Jennings' method are identical to those of MJ." What do
> you think?
>
> I look forward to our next dialogue.
> Steve
>
>
>
> Great questions and suggestion!
>
> However, in some venues not every voter has the patience to grade
> potentially dozens if not hundreds of candidates as in past California
> governor recall elections which were plenty cumbersome with FPTP style
> ballots (which should have been voted and tallied under Approval rules). In
> that context many more voters will have enough patience to vote in the
> final round.
>
> In my opinion MJ is a great improvement on most Cardinal Ratings/Score
> based methods, but the certainty of tied median grades is a drawback
> despite its signature clever method for resolving such ties.
>
> A closely related but more robust method invented by Andy Jennings makes
> ties vanishingly rare while preserving all of the advantages of MJ
> including use of familiar, easy to understand grade style ballots with
> minimal strategic incentive for grade inflation/exaggeration.
>
> The voter instructions and other features of the voter interface
> environment for Jennings' method are identical to those of MJ.
>
> The method is called Chiastic Approval  (XA) because of an X shaped (i.e.
> Chi shaped) graphical interpretation showing how MJ, Approval, and XA are
> related to the candidates' grade distributions.
>
> [Those distributions are discontinuous graphs that stair step down through
> the six grade levels. A vertical line through the middle of such a graph
> crosses at the midrange approval cutoff point between acceptable and poor.
> An horizontal line through the middle of the graph crosses it at the median
> ... which illustrates the ambiguity of the median, since an horizontal line
> will either miss the stair stepping graph entirely, or will intersect it
> along an entire line segment. On the other hand, a diagonal line of unit
> slope will cross the graph at precisely one point .. at the instant it
> crosses from below to above the graph. The descending distribution graph
> and the ascending diagonal line (segment) form the shape of the Greek
> letter chi.]
>
> It is pleasant for election geeks to contemplate such schematic diagrams,
> but the beauty will be lost on anybody else ... best to save it for those
> who might appreciate it :-)
>
> The algebraic representation will be even more of an imposition ... best to
> shred any  reference to it except as documentation of the software:
>
> The distribution function F whose graph forms the descending staircase
> alluded to above is defined as follows:
>
> F(x) is the percentage of ballots that rate  candidate C greater than or
> equal to x ... as x increases F(x) decreases.
>
> So the chiastic approval of candidate C is the greatest value of x such
> F(x) is no greater than x.
>
> Putting it all together we have ...
>
> The chiastic approval for candidate C is the greatest value of x such that
> x is less than or equal to the percentage of ballots that rate (i.e. grade)
> candidate C at or above x.
>
> I warned you!
>
> How many of you understand the Huntingto-Hill method of apportionment? Yet
> that is the official method for determining how many representatives each
> state gets in Congress and the Electoral College. And people trust in it
> unquestioningly just as they implicitly trust the medical pharmaceutical
> complex.
>
> Chiastic Approval is arguably easier to explain than MJ except perhaps at a
> superficial level that avoids the tie breaking details of MJ.
>
> Once you understand the beauty and efficiency of both Chiastic Approval and
> Approval Sorted Margins it becomes apparent that no Condorcet compliant
> deterministic social order (single winner order of finish) can surpass
> XASM,  Chiastic Approval Sorted Margins, IMHO:-)
>
> Forest
>
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