[EM] Discussion about majority complaints

[Jobst Heitzig]

Hello all!

I'd like to ask you about your opinion concerning the following aspect
of voting:

After the election of a candidate W, some majority of the voters may
complain that they all find some other candidate X better than W. Long
ago I have called this a "binary argument", but let us call this a
MAJORITY COMPLAINT X>W in accordance with Steve (although one could
imagine more general complaints that this one, for example a majority
could complain that they all have W below their approval cutoff etc.) By
definition, W faces at least one potential majority complaint whenever
it is not the (sincere) CW.


There are some QUESTIONS here which I feel we have not discussed
sufficiently:

1. How "serious" are majority complaints? Are there more and less
serious ones? Can we measure the "seriousness" of a complaint
quantitatively? Is the complaint more serious when X is the "second
place" winner (i.e. the candidate which would win when W was dropped)?

2. What possibilities to rebut such a majority complaint should we
expect our election method to provide us with? In particular, when we
use a method which provides us with a beatpath from W to X (a "rebutting
beatpath"), should we design our method so that the rebutting beatpaths
are as strong as possible or as short as possible?

3. When rebutting a majority complaint X>W by pointing at a beatpath
W>A>...>B>X, should we take into account that these people constituting
these "rebutting defeats" may belong to the complaining group? In
particular, when we argue that the rebutting beatpath is "stronger" than
the complaint, shouldn't we then try to measure "strength" in a way
which avoids counting the same voter towards both the strength of the
complaint and to the strengths of the defeats we use to rebut the complaint?

4. For methods which construct a complete social ordering of all
candidates, should we even be prepared to rebut *all* complaints of the
type "Y is placed higher than X but we prefer X to Y", or should we only
 care if Y is the winner?


Here's a bit of what I personally tend to think about these questions:

ad 1. I think all majority complaints are so serious that we should be
prepared to rebut them in some way.
  I also think that some types of complaints are so serious that they
should be avoided completely. If, for example, X covers Y or dominates Y
in some other strong sense, Y should not win (the corresponding criteria
are usually called "efficiencies").

ad 2. The rebuttal may be of different type:
  a) We may just point out that also the proposed exchange X would be
subject to a similar complaint, and we may try to design the method so
that this rebutting defeat is as "strong" as possible. For example, we
could elect the candidate W for which the minimum strength of all
strongest defeats against those candidates defeating W is as large as
possible.
  b) We may point at some beatpath W>...>X and design our method so that
this rebutting beatpath is as strong or as short as possible, or optimal
in some other sense.
  c) We may point out that W is "better" than X as measured by some
score like the Copeland score, or as indicated by some binary relation
like the Covering relation or any other type of "domination".
  Personally, I think we should use type b and look for short rebutting
beatpaths. More precisely, I think we should require (since it is
possible!) that there always be a beatpath of the form W>A>X whenever
X>W. Only as a second criterion we should try to have these rebutting
defeats as "strong" (see below) as possible .

ad 3. Assume that we argue that the two defeats W>A>X are both
"stronger" than the complaint X>W, and that we define "strength" as the
number of voters having the corresponding preference. Then those members
of the complaining majority which have X>W>A or A>X>W may not accept our
rebuttal but may point out that their other preferences shall not be
counted against their main preference X>W. Hence I think we should try
to measure defeat strength in a more sophisticated way, perhaps using
cardinal weights as James suggests. Or perhaps only by using an approval
cutoff and counting how often X is above and W is below the cutoff like
I suggested in the proposal named "grand compromise". When using the
latter measure of strength, it is never possible for some voter to be
counted towards both the strength of W>X and that of X>A.


Most probably all of you will have some opinion on this topic, and I
hope you will post them so that we can have a fruitful discussion here.

Yours, Jobst