[EM] Discussion about majority complaints

[Forest Simmons]

>
> 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.)

What about a unanimous complaint of this type?  i.e. when the Condorcet 
winner W is not approved by anybody.


> 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.

... relative to the strongest complaint against W?


Suppose that

  candidate j's complaint against candidate k has strength c(j,k),

and that

  candidate k's rebuttal against candidate j has strength r(k,j).

Then it seems like we should find the k that minimizes the maximum (over 
j) of the value  c(j,k)/r(k,j) .

The greater the complaint, the stronger the rebuttal required to address 
it.

So for each system of measuring complaint strength and rebuttal strength, 
we get a minmax election method.


>  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
>


Thanks for asking excellent questions, as well as your insights towards 
answering them.

Forest