Always impossible!

[Markus Schulze]

Dear Mike Saari,

you wrote (7 Jul 1998):
> In a message dated 98-07-03 16:42:43 EDT, you write:
>
> > In literature, there are many different proves of the
> > impossibility of a non-manipulatable election method.
> >
> > All these proves have the same structure:
> >
> > 1) Take some axioms, such that a tie is possible.
> >    That means: Take some criteria, such that it is possible
> >    that A > B > C > A or in other words such that
> >    a) Candidate A would have been elected, if candidate C
> >       hadn't run for office.
> >    b) Candidate B would have been elected, if candidate A
> >       hadn't run for office.
> >    c) Candidate C would have been elected, if candidate B
> >       hadn't run for office.
> >
> > 2) Assume, that there is a tie.
>
> In a rated (as opposed to ranked) voting system, each candidate
> receives "votes" according to some method which allows them to
> be tallied.  (Think of Olympic scoring as a simple example.)
> After the votes for each candidate is tallied, each candidate
> then receives a resulting "score".  One of several methods can
> then be used to determine a winner:
> -Under some methods, the candidate with the highest score is
> automatically deemed the winner
> -Under another possible method, any and all candidates (if any)
> who receive a score above some pre-set threshold is declared the
> winner.  Note that this method can possibly result in multiple
> winners, or possibly result on "no winner".  This might be more
> applicable for parliamentary situations where the set of choices
> is unbounded.
>
> In either case, note that a "circular result" e.g. A > B > C > A
> cannot occur. Therefore such a voting method is not subject to
> the same style of "impossibility proof" as are other voting
> systems.

The problem is, that even every rated election method (as long as
it meets Neutrality, Anonymity, Pareto, and Non-Dictatorship)
is vulnerable by exaggeration or leads occasionally to ties.
The reason: If there are only two candidates (A and B) and the
number of voters, who strictly prefer candidate A to candidate B,
is larger than the number of voters, who strictly prefer
candidate B to candidate A, then every election method (as long as
it meets Neutrality, Anonymity, Pareto, and Non-Dictatorship)
necessarily elects candidate A or is vulnerable by exaggeration.
Again: I don't want to say that it is a democratic principle, that
candidate A is elected. I only want to say, that the election
method elects candidate A or is vulnerable by exaggeration.

You wrote (24 Apr 1998):
> I disagree the very obvious-sounding second paragraph.  Here is
> my counterexample.  Consider the following hypothetical
> situation: 60% rate Coke as "Excellent" and rate 7-Up as
> "Very Good". 40% rate Coke as "Detestable and poisonous" and rate
> 7-Up as "Very Good". These are honest, true feelings - not
> exaggerated.

I agree with you, that there are situations, where an alternative,
which is strictly prefered to every other alternative by an
absolute majority of the voters, is not necessarily the "best"
alternative. But I see the following problems:

(1) It is the task of an election campaign to make those voters,
    who only have a weak preference of two alternatives, get a
    strong preference. Example: Suppose, that there are some
    voters, who think that Clinton is "good" and that Dole is
    "average." Then the task of Clinton's election campaign would
    be to make these voters think, that Clinton is "excellent"
    and that Dole is "disastrous."

    In other words: The election campaigns distort the absolute
    preferences of the voters while they usually don't change
    the relative preferences of the voters. Thus, it is sensefull
    to consider no absolute preferences but only relative
    preferences.

(2) In your example with Coke and 7-Up, there should be an
    interview among the voters about their opinion about Coke
    and 7-Up. After this interview, the voters should decide,
    whether Coke should win or they should respect the
    opinion of that minority who doesn't like Coke.

    But there should be no power of veto for every small
    minority.

(3) The task of a decision method is to "create" a decision,
    where there is no unanimity. If there is unanimity, then
    there is no need for a decision method.

    In other words: The task of a decision method is to
    determine, which voters or which alternatives should be
    ignored to "find" a decision. A decision method will always
    violate the opinion of some voters.

(4) It is not the task of an election method to protect a
    minority from an intolerant majority. An election method
    cannot do that. To protect a minority, this is the task
    of the constitution and of the jurisdiction.

By the way: As far as I remember correctly, you wanted to post
an election method, that uses absolute preferences and that is
not vulnerable by exaggeration. You haven't done it yet.
Could you -please- do that?

Markus Schulze