[EM] Approval meets IIA ?

[Markus Schulze]

Dear Chris,

I wrote (1 March 2004):
> 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 (3 March 2004):
> To my mind, Approval does NOT satisfy Independence of Irrelevant
> Alternatives (IIA), or even the much weaker Independence of Clones.
> For my demonstration, I am assuming that the voters know nothing but
> their own sincere ratings of all the candidates on the ballot, and
> that in that situation they all use the best "strategy"of approving
> all the candidates they rate above average, and no others.

I presume that adding a candidate doesn't change which voter approves
which of the already running candidates. This presumption makes sense
since Arrow's Theorem is only about strategic nomination and not about
strategic voting.

******

You wrote (3 March 2004):
> A wins 100 to 99. So adding a clone of A, which ALL the voters
> ranked last, changed A from a 1/100 loser to the winner.

Here is a very similar desideratum (Steve Eppley's "independence from
Pareto-dominated alternatives"; IPDA):

   Candidate x is weakly Pareto-dominated if and only if there exists
   a candidate y such that at least one voter ranks y over x and no
   voters rank x over y.

   IPDA says that the election outcome must not change if a weakly
   Pareto-dominated alternative is deleted from the set of nominees
   and from the votes.

Your example demonstrates that Approval Voting violates IPDA when the
voters vote in the manner presumed by you. However, the fact that also
Tideman's ranked pairs method and my beatpath method violate IPDA
demonstrates that this criterion is stronger than one might think
on first view.

******

You wrote (3 March 2004):
> How is Random Candidate a "rank method"?

A "rank method" is a mapping from a set of partial rankings each
of the same set of candidates to a probability distribution on this
set of candidates. Therefore, to be a "rank method" it is sufficient
that the result of the used method doesn't depend on information that
is not contained in this set of partial rankings.

Markus Schulze