Six criteria

[Markus Schulze]

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

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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.

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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
      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.

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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