[EM] Arrow's axioms

[Markus Schulze]

Hallo,

to demonstrate that a given election method violates a given
criterion it is sufficient to find a single example where this
election method violates this criterion. When in this very
special example each voter casts a complete ranking of all
candidates then this does not mean that you have to presume
that each voter always casts a complete ranking of all
candidates. 

In other words: To prove Arrow's Theorem it is not necessary
to presume that each voter always casts a complete ranking of
all candidates. Actually, it is not even necessary to presume
that each voter casts transitive (= non-cyclic) preferences.
It is sufficient to presume that each voter can cast a complete
ranking when he wants to do this.

[Of course, when you allow cyclic preferences then the Pareto
criterion has to be modified in such a manner that it says that
_when each voter casts transitive preferences_ and no voter strictly
prefers candidate B to candidate A and at least one voter strictly
prefers candidate A to candidate B then candidate B must be elected
with zero probability.]

*********

To prove Arrow's Theorem it is also not necessary to presume that
the used election method is deterministic. A very good paper is
"Distribution of Power Under Stochastic Social Choice Rules"
(Econometrica, vol. 54, p. 909-921, 1986) by Prasanta K. Pattanaik
and Bezalel Peleg. They prove that no paretian non-dictatorial
rank method can satisfy the following criterion ("regularity"):
Adding candidate Z must not increase the probability that
candidate A (with A <> Z) is elected.

Markus Schulze