[EM] Three Tier, Dyadic via CR, etc.

Forest Simmons fsimmons at pcc.edu
Sat Aug 25 16:14:13 PDT 2001

Hi folks. I've been gone a couple of weeks, so it might take some time for
me to catch up.

Here's a few comments with regard to recent mail exchanged by Roy, Craig,
and Richard.

Richard's right. I have proposed half a dozen uses for Five Slot ballots,
including Dyadic Approval and other pairwise methods.

In my first posting I proposed that the five slots be viewed like this:

A > B >> D > E

similar to, but not quite the same as  Dyadic Approval.

In head-to-head comparison, an A or B counted one point over a D or F.
All other comparisons (when not equal) counted as 1/2 point.

In practice the five slots were to be alloted as follows:

A = favorite
B = preferred front runner
D = disliked front runner
F = devil incarnate
C = anyone between B and D

I consider it a lucky coincidence that the standard ABCDF grades provide
slots for all the main types of candidates.

The five slot ballot has many applications, but Five Slot Approval (FSA)
was Joe Weinstein's suggested use in public elections, because of the
simplicity of the ballot, and the simplicity of the scoring: the candidate
with the most passing grades wins the election.

FSA is Approval instrumentally, but with intermediate resolution CR

Craig's Three Tier ballot could be used in many ways, and could be thought
of as collapsing the outer categories of the five slot ballot to the next
inner categories.  I think that three slots are plenty adequate, although
the public might prefer the familiar ABCDF grades.

The following forwarded message is a recent one in which I discussed some
related issues, including using CR ballots as a basis for various pairwise

The zero to fifteen CR ballot that I proposed below is at the upper limit
of practical resolution as far as I'm concerned.  Any apparent further
resolution provided for on a fancier ballot is probably an illusion. Other
sources of imprecision dwarf its resolution deficiency.  [Compare the
student who says a circle whose circumference is 20 cm has a diameter of
exactly 6.36363636364 cm, by using 22/7 as an approximation to pi on a
fancy twelve digit calculator.]

Standard Cardinal Ratings (in which the winner is the candidate with the
highest average rating) is in no way inferior to standard Approval except
for the requirement of a more cumbersome ballot. If the Five Slot ballot
is used, the psychological advantages of CR may compensate for this slight

Which would you rather use the Five Slot ballot for?

1. A pairwise method with all comparisons of equal strength.
2. A pairwise method with some comparisons stronger than others.
3. Five slot CR (CR with resolution 5).
4. Five Slot Approval.
5. Other.

With regard to choices 2 and 3. They will both yield the ordinary Approval
winner, but the additional information from FSA will be more sincere, and
therefore more valuable for determining mandates, future strategy, etc.

I think Roy's runoff ideas have improved as he has started doing the
runoff's on the basis of pairwise matrices rather than trying to keep
track of the n factorial changes in the preference ballots like IRV.

Any pairwise method has potential application to Dyadic and Universal
Approval, since neither Martin Harper nor I is (or am, respectively)
totally committed to one particular completion method. So all of these
investigations are interesting to me.

We were launched on this particular search of elimination methods
satisfying the Condorcet Criterion by a desire to fix IRV in a way that
would be palatable to the same folks that now support IRV. 

We're going to have to simplify eventually if we want this journey to end
up there.

Of course, nobody can stop us from changing our destination whenever we
want to :-)

Meanwhile, we're all learning a lot, and having lots of fun!

Here's a brief offering in the IRV fixit category:

At each stage of the runoff eliminate either the candidate with the
greatest number of pairwise losses or the candidate whose maximimum
victory is minimal, whichever loses in a head-to-head comparison.


Both candidates for elimination reduce to the Condorcet Loser if there is
one, otherwise they are two different kinds of approximations of what it
means to be a Condorcet Loser.

Basically, we're pitting the Copeland loser against the minmax loser at
each stage of the runoff. [Here I'm using method loser to mean method
winner when the pairwise matrix is transposed, i.e. when preferences are

Obviously, the CW is never eliminated.

In three way contests, the Ranked Pairs winner is not eliminated.

In the examples that I have worked out, clones do not upset the results.


If there are two or more candidates with the same number of losses, then
the one nearest to having another loss is the one pitted head-to-head with
the minmax loser. 

One way to interpret "nearest to having another loss" is having the
smallest smallest winning margin.


Minmax doesn't satisfy reverse symmetry. A minmax winner can also be the
minmax loser (i.e. minmax winner when preferences are reversed).

That's one reason why we have to pit the minmax loser against the Copeland
loser: i.e. to make sure we don't eliminate the CW if there is one.

Unfortunately Copeland elimination by itself is not good enough to
guarantee the Ranked Pairs winner in a three way contest. 


---------- Forwarded message ----------
Date: Mon, 6 Aug 2001 15:34:13 -0700 (PDT)
From: Forest Simmons <fsimmons at pcc.edu>
To: election-methods-list at eskimo.com
Subject: Re: [EM] Introduction (cont.)

One more comment on the strategic equivalence of Range Voting and Approval

There are ways of using range ballots that do not yield the strategic
collapse to the extremes of the range. 

However, Smith's use of range ballots is of the type that do yield the
strategic collapse.

A zero to ten "Olympic" range ballot can be the basis for head-to-head
comparisons similar to the head-to-head comparisons of Condorcet methods.

The advantage of the zero to ten scale over the usual ranked preference
ballots are three fold:

You can rank two candidates equal if you feel like it.

If there is a big gap between adjacently ranked candidates, you can
express it.

The ballot itself is simpler in the case when there are more than ten

The price you pay is the inability to completely rank more than eleven
candidates. That's not so bad when you consider that a standard Approval
ballot can only completely rank two candidates. 

Here's a way of doing a zero to fifteen range ballot in such a way that it
would be hard to mark the ballot in a meaningless way:

Jose Blaze    (8)  (4)  (2)  (1)
Sheila M.     (8)  (4)  (2)  (1)
Jana P. Q.    (8)  (4)  (2)  (1)
A. Ron Bla    (8)  (4)  (2)  (1) 
J. Q. Sten    (8)  (4)  (2)  (1)
Sajh Amlkj    (8)  (4)  (2)  (1) 

Each candidate's score on this ballot is the sum of the digits shaded
(with a number two pencil) to the right of the candidate's name (or

Any combination of shaded and unshaded digits produces a validly marked

If none of the digits are shaded, then the score is zero for that

If voters know their addition tables up to seven plus eight or else
remember to bring a simple calculator, then they can vote this kind of

Furthermore, this kind of ballot is a Dyadic ballot in disguise, so it can
be easily used for Universal Approval, Various forms of Approval runoff,
and other methods that depend on Dyadic ballots.

[If you cannot easily find these in the archives, I can provide some
info on these Dyadic based methods.]

These are all good uses of range ballots that do not waste the expressive
power of range ballots through strategic collapse.

The simplest method that doesn't discourage full expression is Approval
itself. All scores above the midrange score (7.5 in the zero to fifteen
case) are considered approval votes. All other votes are considered lack
of approval.  

In this method, the extra expression doesn't influence the outcome of the
current election, but it does provide psychological benefits.

It gives the same election results as Smith's Range Voting method (in the
strategic case), but without requiring the voter to sacrifice expressivity
for the sake of strategy.

This form of Approval is inferior to some of the other alternative uses of
Range ballots mentioned above, but it is hard to beat when expressivity
and simplicity are both considered important. 

Joe Weinstein and I have a method called Five Slot Approval (FSA) based on
Grade Ballots (F, D, C, B, A, equivalent to a range of zero through four)
which incorporates this simplest expressive use of range ballots. 

It takes advantage of the voters' familiarity of being graded all through
school on the basis of those letters.

The candidate with the greatest number of passing grades (C or above) is
the FSA winner.


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