Condorcet and Irrelevance of Alternatives

[Blake Cretney]

--

On Fri, 28 Aug 1998 12:47:50   David Catchpole wrote:
>On Thu, 27 Aug 1998, Norman Petry wrote:
>
>> David,
>> 
>> Welcome to the list.  You wrote:
>> >for the following system (and therefore that IA is not satisfied for
>> 
>> The point you may be missing is that we usually use criteria to evaluate
>> voting _methods_, not particular "systems" of ballot configurations.  You
>
>I apologise for my misuse of the word "system". I do know I should have
>used the word such as "case" (such is misnomenclature...). However I do
>say that a criteria has worth even if it is a certainty that it cannot be
>satisfied in some non-tied cases.

I think the word you are misusing is "Criteria".  When we say that
a method satisfies a criterion, we mean for all possible cases.
Otherwise we say that it fails the criterion.  If a method sometimes
passes a criteria, and sometimes fails it, then technically, it
fails it.  If you want to differentiate a method that passes a
criterion in certain cases, you should define a new criterion to
do this.  For example,

A method satisfies criterion A if it always selects the Condorcet 
winner if one exists, and when one exists, the election would have
been unaffected by the elimination of a candidate.

>
>> Of course, IIAC can always be satisfied by some *unreasonable* methods.
>> According to Lorrie Cranor,
>> 
>> "A voting system is independent from irrelevant alternatives if it always
>> produces the same results given the same profile of ordinal preferences.
>> This criterion has consequences in two areas: it prohibits voting systems
>> that take into account cardinal preferences or ordinal preferences beyond
>> each voter's first choice, and it prohibits voting systems that include
>> elements of randomness." -- Lorrie Cranor, "Vote Aggregation Methods"
>> 
>> Therefore, unless we're prepared to live with plurality voting, we're going
>> to have to forget about trying to use IIAC as a means of assessing voting
>> methods.
>
>PLURALITY GIVES INDEPENDENCE OF ALTERNATIVES? Hello!; IIA is precisely the
>concept of an independence to splitting. Guess what the *BIG* problem with
>plurality is?
>
>Sorry for being a little rude, but I get a bit riled up when I see a
>completely illogical statement. I'm your typical lefty, I guess...
>

Actually, this isn't a completely illogical statement.  It is in fact
true, at least the way the author intends it.  You see, mathematicians
tend to consider criteria in terms of expressed preferences.  They do 
this because otherwise it would be difficult to prove anything, since
you would be constantly guessing peoples actual preference.

As a result, because no one has an expressed second preference, the
election is unaffected by the removal of a candidate, based on
expressed preferences.  In reality, we know that if a candidate
was removed, their supporters would vote a second preference, so
the election result could be affected.  However, technically, this
doesn't matter to IIAC.

Another example is Approval Voting.  It too passes IIAC, but for a
similar reason.  In reality, whether one marks a candidate approved
or not is bound to be affected by what other candidates are running.
But because only expressed preferences are used, Approval can pass
IIAC.

So, I was wrong when I said that no voting system would pass IIAC.
I was thinking of those actual preferences instead of the expressed
ones.

The point is that IIAC only really makes sense with regard to systems
that allow a complete preference order to be expressed.  And they
don't satisfy it.



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