Exaggerated opinions
Markus Schulze
schulze at sol.physik.tu-berlin.de
Sat Apr 25 09:12:17 PDT 1998
Dear Mike,
to my opinion, candidate X is the obviously best candidate only
if every voter strictly prefers candidate X to every other
candidate. In every other situation, it is questionable, whether
a given candidate should win.
You wrote (24 Apr 1998):
> 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 personally think that 7-Up may be the best outcome here, but this
> is only an opinion. I cannot "prove" this, and I do not need to.
> The "best" outcome for such a situation can and must depend on the
> decision method already chosen by
> the group.
>
> My point is NOT an argument as to which outcome is actually best.
> MY POINT IS that this counterexample calls into question the assertion
> that "if an absolute majority prefer A to B, then EVERY sensefull
> democratic voting method must necessarily choose A."
>
> The fact that I can define a simple example where at least a good case
> can be made that B is a better overall choice, even though more than
> half of the voters prefer A to B, means to me that the assertion that
> every sensible voting method MUST choose A is highly suspect.
That is not the point. I didn't say, that, if there are only two
candidates (A and B) and if an absolute majority of the voters prefers
candidate A to candidate B, then every sensefull democratic voting method
necessarily elects candidate A.
I said, that, if there are only two candidates (A and B) and if an absolute
majority of the voters prefers candidate A to candidate B, then every
voting method, that is sensefull and democratic and that is invulnerable
by exaggeration, necessarily elects candidate A.
To my opinion, a voting method is sensefull and democratic if & only if
it meets the following criteria:
1) Weak Pareto Criterion:
Candidate B cannot be elected, if there is a candidate A, such that
every voter strictly prefers candidate A to candidate B.
2) Non-Dictatorship Criterion:
The voting method must not be a function of only one voter.
3) Neutrality Criterion:
Every candidate is treated equally.
4) Anonymity Criterion:
Every voter is treated equally.
5) Weak Monotonicity Criterion:
If there are only two candidates (A and B) and candidate A is elected,
then he is elected also if a voter changes his opinion about candidate
A to the benefit of candidate A.
If there are only two candidates (A and B) and candidate A is elected,
then he is elected also if a voter changes his opinion about candidate
B not to the benefit of candidate B.
6) Determinism Criterion:
The voting method must not be unnecessarily random. That means, that
in every situation, at least one of the following two statements is
valid:
a) There is a unique winner.
b) There is a (not necessarily unique) voter, such that he can change
his opinion about the candidates, so that there is a unique winner.
In my last e-mail, I demonstrated, that
a) if there are only two candidates (A and B) and if an absolute majority
of the voters prefers candidate A to candidate B,
b) the used voting method is "sensefull and democratic" (i.e., it meets
the above mentioned six criteria), and
c) the used voting method is invulnerable by exaggeration,
it must necessarily elect candidate A.
****
By the way, in one of my last e-mails, I introduced the
following definition (12 Aug 1997):
> Pairwise Majority Criterion:
>
> Suppose, there are only two candidates X and Y.
> Suppose, an absolute majority of the voters prefers
> candidate X to candidate Y.
> Then: A voting method meets the "Pairwise Majority
> Criterion" (PMC) if & only if X wins.
I demonstrated in that e-mail, that every voting method,
that meets the "Pairwise Majority Criterion" is manipulatable
by irrelevant candidates and by order-reversal. That means:
Arrow's theorem and Gibbard's and Satterthwaite's theorem
are valid.
Today's e-mail and my old e-mail (12 Aug 1997) can be
summarized as follows:
Every sensefull and democratic voting method is
a) manipulatable by exaggeration or
b) manipulatable by irrelevant candidates and by order-reversal.
[Remark: An "irrelevant candidate" is a candidate, who changes the
result of the elections without being elected. "Order-reversal"
means, that a voter ranks the candidates in a different order than
his real preferences are.]
To my opinion, a voting method, that is manipulatable by
irrelevant candidates and by order-reversal, is better than a
voting method, that is manipulatable by exaggeration.
The reason: It is very difficult and risky to manipulate
with irrelevant candidates or with order-reversal, because
a) there must be a suitable candidate, who can change the
result of the elections without being elected,
b) you have to know very exactly how the other voters will vote,
c) it can happen, that a less prefered candidate wins the
elections if you vote tactically without knowing exactly
enough how the other voters vote.
To manipulate in an election, where the used voting method
is manipulatable by exaggeration, you usually only have to give
an "absolutely perfect" to all those candidates, you
prefer to the expected winner, and an "absolutely unacceptable"
to all those candidates, you don't prefer to the expected
winner.
Markus
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