# Six criteria

Markus Schulze schulze at sol.physik.tu-berlin.de
Fri Jul 3 13:35:08 PDT 1998

```Correction:
In my last e-mail, it must mean "Anonymity" instead of
"Autonomy."

*******

Dear Mike Saari,

you wrote (3 Jul 1998):
> If these are the famous "Arrow" criteria, then one of
> them (I'm not sure which name) states that the outcome can
> depend ONLY on the "preferences" of the voters.

Arrow used the following criteria:

1) Unrestricted Domain Criterion

This criterion means:
1a) The election method is defined for every possible
set of rankings of the individual voters.
1b) The input of the election methods consists only
of the rankings of the individual voters.

2) Independence from Irrelevant Alternatives

Suppose, candidate A would have been elected, if candidate B
hadn't run for office. Then: If candidate B runs for office,
either candidate A or candidate B is elected.

3) Non-Dictatorship

The election method must not be a function of
only one voter.

4) Pareto

Suppose, every voter strictly prefers candidate X to
candidate Y, then candidate Y must not be elected.

*******

Arrow used neither Neutrality nor Anonymity.

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
b) Candidate B would have been elected, if candidate A
c) Candidate C would have been elected, if candidate B

2) Assume, that there is a tie.

*******

In my last e-mail, I only mentioned those criteria, that
are typically used to prove the impossibility of a
non-manipulatable election method. It is very simple to see,
that only those criteria are needed.

Markus Schulze

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