[Election-Methods] independence of mutually irrelevant alternatives?

Steve Eppley SEppley at alumni.caltech.edu
Sat Apr 12 09:14:31 PDT 2008

```Hi,

I think there are at least two errors in Greg's message.  First,
consider this example:

35   10   10   45
---  ---  ---  ---
X    Y    Y    Z
Y    X    Z    Y
Z    Z    X    X

A majority of the voters (10 + 45) rank X strictly bottom, so X should
be irrelevant given a method that satisfies Greg's IMIA criterion, if I
correctly understand his definition.  Is X irrelevant given the IRV
voting method?  No, IRV elects Y if X does not compete but elects Z if X
does compete.

Second, Greg appears to be using an incorrect definition of Local IIA.
The real Local IIA was defined by H Peyton Young no later than 1993 in
his book Equity In Theory And Practice.  Irrelevance of the alternatives
outside the Smith set is not LIIA.  For more information, see the notes
claimed it satisfies LIIA.

Few voting methods satisfy the real LIIA.  One of them is Maximize
Affirmed Majorities (MAM). (MAM is sometimes called Ranked Pairs, but
Ranked Pairs is a related but different method defined by Nicolaus
Tideman in 1987.  Ranked Pairs also satisfies LIIA.)  Young promoted
another method that satisfies LIIA, but it fails clone independence.
(Young's method does also satisfy a criterion he called Reinforcement,
which seems unimportant.)

Regards,
Steve
--------------
Greg wrote:
> I'm attempting to classify different kinds of irrelevant alternatives.
> In particular, I'm trying to partition irrelevant alternatives into
> those that IRV eliminates and those it does not. It seems that IRV
> eliminates failures of IIA if and only if it represents a failure of
> the mutual majority criterion. That is, IRV is independent of the
> addition of any candidate who, when added, belongs to a subset of
> candidates S, such that more than half the voters strictly prefer
> every candidate outside S to every candidate in S. Could we call this
> Independence of Mutually Irrelevant Alternatives (IMIA) criterion? To
> make an analogy: the Smith criterion is to Local IIA as as the Mutual
> Majority criterion is to IMIA. Let me know if this is already a
> well-defined notion.
>
> Greg
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>

```