[EM] Approval-Condorcet hybrid encouraging truncation

[Forest Simmons]

Thanks for the clarifications.  I should have been able to get your drift,
but I was too tired or lazy at the time.

You just gave me another use for dyadic ballots:

One version of dyadic ballot looks like

Nader 	(8)	(4)	(2)	(1)

Gore  	(8)	(4)	(2)	(1)

Bush	(8)	(4)	(2)	(1)

etc.


The voter may shade as many numerals as desired.  The result will be a CR
ballot with each candidate's cardinal rating given by the sum of the
shaded numerals to the right of the candidate's name.

The shading pattern is a binary representation of the CR value.

These dyadic CR ballots are used to create four pairwise matrices, from
crude to fine depending on how many of the binary places are ignored.

The crudest is the matrix based only on the 8 column.  This is the
"approval" matrix.  In other words, on the CR scale from zero to fifteen,
the approval cutoff is between seven and eight.

The finest matrix is based on all four columns.

How could we do something like Ranked Pairs with these four matrices?

First set in place all of the wins common to all four matrices.

Then set into place all of the wins common to the three finest matrices,
strongest to weakest, as long as they don't contradict previously frozen
wins.

Then do the same with the two finest matrices, and finally with the finest
only.

The resulting order establishes the winner.



The idea is that these matrices represent an hierarchy of strengths of
preference.




Forest

On Tue, 11 Mar 2003, Steve Eppley wrote:

> On 12 Mar 2003 at 0:33, Kevin Venzke wrote:
> > Forest Simmons wrote:
> > >I must be missing something. Could you give an
> > >example in which the approval winner is not the
> > >winner of the method?
>
> Here's an example:  Suppose 100 voters vote on 2 candidates A & B as
> follows:
>
>         60    40
>         A      B
>         B      --
>         --      A
>
> Clearly B is the "approval" winner, if you interpret being ranked
> over the line as being "approved." (An interpretation I think should
> be avoided, since "approval" is a misleading absolutist concept and
> *relative* preferences are what really matter.  Elections are
> primarily for making social choices from the actual candidate, not
> sending vague messages about candidates being better or worse than
> unspecified non-alternatives like "hold a new election" or "leave the
> office vacant" or some "mediocre fictitious composite candidate.")
>
> Although more votes rank "B over the line over A" than vice versa,
> the number that rank B over A does not exceed the number that rank A
> over B.  Therefore the line has no force in this example.  Thus,
> since the family includes methods that are Condorcet-consistent and
> methods that satisfy the Top Cycle criterion, it's not hard to find
> methods in the family that elect A. (See below.)
>
> > I don't think it's itself a method.  It's a "family of
> > voting methods" of them, like he said.  The two rules
> > specify what the ballot must look like, and who cannot
> > win.  There is a lot of freedom permitted as to who
> > actually wins.
>
> I did mention a particular method in that family, by describing a
> variation of MAM that gives utmost precedence to majorities that
> "straddle" the dividing line.  That method would elect A in the
> example above.
>
> > To give an example for your question, it would be
> > permissible for a method in the family to elect a
> > candidate who beats pairwise the approval winner.
> >
> > Steve Eppley wrote:
> > >2. For all pairs of candidates, say x & y, y is
> > >socially ordered over x if the number of votes that
> > >rank "y over x" exceeds the number of votes that
> > >rank "x over y" and the number of votes that rank "y
> > >over the dividing line over x" exceeds the number of
> > >votes that rank "x over the dividing line over y."
> >
> > Isn't it superfluous to say "y over the dividing line
> > over x" and vice versa?  Won't you get the same
> > results if you say "y over the dividing line" and vice
> > versa?
>
> By adding to both counts the number of votes that rank both x and y
> over the line...  That's briefer, and equivalent unless the voter may
> also rank the dividing line *equal* to some candidates.  In that case
> they can give different results.
>
> Actually, I'd need to reword the condition anyway to ensure it's
> acyclic when voters can put the dividing line equal to candidates.
>
> > >Assuming only one winner is to be elected, it's
> > >always possible to satisfy condition 2 (or the
> > >revised wording) since the subset of pairings that
> > >meet condition 2 is acyclic. (I have a proof of
> > >acyclicity, but it's tedious so I won't post it
> > >without a request.)
> >
> > I think this is intuitive.  Isn't it just a
> > consequence of it being impossible to have a cycle
> > with an approval ballot?
>
> I believe so, but I prefer proofs over intuition.  Sometimes my
> intuition is wrong, as when I mistakenly believed that MAM and
> BeatpathWinner satisfy Independence from Pareto-dominated
> Alternatives.
>
> > >I'm concerned that some voters wouldn't use the
> > >dividing line strategically as intended, and instead
> > >treat it as some sort of "sincere approval" dividing
> > >line. In that case, the dividing line may
> >
> -snip-
> > I also think my method could alleviate some of your
> > concern about the voters using the divider
> > efficiently.
>
> I didn't use the word "efficiently" and am uncertain what Kevin means
> by that here.
>
> > This is because the voter cannot order
> > among candidates whom they don't approve.  Thus if
> > they want to put the "dividing line" (which exists
> > only effectively) right after their first preference,
> > they cannot name compromises.
>
> That doesn't alleviate my concern.  I want them to rank some
> "compromise" candidate(s) over the line (at least in the scenarios
> where no candidate is the top choice of a majority) and neither your
> method nor the family I described force them to do so.  For example,
> some Nader voters might rank only Nader over the line.
>
> I think the voter should also be allowed to order the candidates
> below the line, so we can use those preferences to help find the best
> compromise.
>
> -snip-
> > >So even though many voting methods could be tweaked
> > >to be in the family that satisfies Sincere Defense,
> > >only the best methods should be considered.
> >
> > I wonder what you think would approach a "best
> > method."  Do you think tinkering with the ballot
> > format is completely out?
>
> MAM is the best method, of course.  :-)
>
> MAM satisfies Minimal Defense (and Mike Ossipoff's similar SDSC) so
> it provides a second line of defense when Sincere Defense doesn't
> have bite due to some of the majority failing to rank the compromise
> over the dividing line over the greater evil. (The underlying goal is
> to minimize the coordination costs for a "good" majority to be able
> to ensure the defeat of "evil" candidates.  The assumption is that
> sincere ordering is cheapest, downranking evil candidates is next
> cheapest, ranking compromises equal to favorites is next, and ranking
> compromises over favorites is most expensive to coordinate.)
>
> I can't begin to answer that last question until Kevin indicates
> which ballot formats he considers "untinkered."
>
> There may be some relevant format research done by software companies
> like Xerox, Apple and Microsoft.  If we assume the "drag & drop"
> metaphor has thrived for good reasons, maybe a lot of voters will
> intuitively understand how to drag and drop candidates into the
> desired ordering (top to bottom).  Try rearranging the icons on your
> computer's desktop, to get an idea how easy that can be, and imagine
> doing so using a touchscreen if you don't actually have one.
>
> -- Steve Eppley
>
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