[EM] There's nothing wrong with Average Rating.

[Markus Schulze]

Dear Ken,

I wrote (29 Feb 2004):
> My favorite formulation of Arrow's Theorem is Pattanaik and Peleg's
> formulation (Prasanta K. Pattanaik, Bezalel Peleg, "Distribution of
> Power Under Stochastic Social Choice Rules," Econometrica, vol. 54,
> p. 909-921, 1986). In their formulation, this theorem says that
> there is no rank method that is non-dictatorial and satisfies Pareto
> and regularity. "Regularity" says that adding candidate Z should
> not increase the probability that candidate A (with A <> Z) is
> elected.

You wrote (29 Feb 2004):
> But is there any such non-rank method? (I presume "rank"
> means a ranked-preference method, which CR is not.)

Arrow proved that there is no single-winner election method with
the following four properties:

   1) It is a rank method (= a ranked-preference method).
   2) It satisfies Pareto.
   3) It is non-dictatorial.
   4) It satisfies IIA.

All four properties are needed to get an incompatibility.
For example, RandomDictatorship is a paretian rank method that
satisfies IIA, RandomCandidate is a non-dictatorial rank method
that satisfies IIA, Approval Voting is a paretian non-dictatorial
method that satisfies IIA, my beatpath method is a paretian
non-dictatorial rank method.

******

You wrote (29 Feb 2004):
> So is it correct to say that Arrow did not prove that "there
> is no perfect voting system"; he only proved that the methods
> he deems to be acceptable are imperfect?

Even though the presumption that the used single-winner
election method is a rank method is necessary to prove
Arrow's Theorem, this presumption is not necessary to prove
the Gibbard-Satterthwaite Theorem. The Gibbard-Satterthwaite
Theorem says that there is no paretian non-dictatorial
method that isn't vulnerable to strategical voting.

Markus Schulze