Condorcet and Irrelevance of Alternatives

[Norman Petry]

David,

Welcome to the list.  You wrote:

>My point exactly- you fail to acknowledge that no Condorcet winner exists
>for the following system (and therefore that IA is not satisfied for
>any outcome from that case). To say that it does not exist for all cases
>is a bit silly.
>
>I actually stated that where Condorcet winners do not
>exist, IA cannot be satisfied; but that where they do, it does where they
>are elected.

The point you may be missing is that we usually use criteria to evaluate
voting _methods_, not particular "systems" of ballot configurations.  You
made the point that where a Condorcet winner exists, the outcome satisfies
IIAC (Independence from Irrelevant Alternatives Criterion).  Therefore,
unless we could design a voting method which _always_ produces results
satisfying IIAC, then it's useless as a means of distinguishing between
voting methods.

Your own argument suggests that IIAC compliance only happens in rank
balloting when there is a Condorcet Winner.  When there are voting cycles,
IIAC cannot be satisfied by any reasonable method, therefore the appropriate
test to apply to single-winner methods is not IIAC, but the Condorcet Winner
Criterion (CWC?), which says that when there _is_ a Condorcet winner, the
voting method should select that candidate.  Methods which satisfy CWC are
better than those that don't, partly because they provide a degree of
independence from irrelevant alternatives, as you pointed out.  We can go
further, though, and say that even if there are cycles, it is often possible
to identify at least _some_ candidates who are completely irrelevant -- this
motivates the use of MIIAC.

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.


Norm Petry


-----Original Message-----
From: David Catchpole <s349436 at student.uq.edu.au>
To: election-methods-list at eskimo.com <election-methods-list at eskimo.com>
Date: August 26, 1998 7:11 PM
Subject: Re: Normative principles of elections- Condorcet and Irrelevance of
Alternatives (plus a brief hello and an STV clarification)


>My point exactly- you fail to acknowledge that no Condorcet winner exists
>for the following system (and therefore that IA is not satisfied for
>any outcome from that case). To say that it does not exist for all cases
>is a bit silly.
>
>I actually stated that where Condorcet winners do not
>exist, IA cannot be satisfied; but that where they do, it does where they
>are elected.
>
>See below for the Condorcet bit.
>
>(PS from now on I'll stay on with convention and call this IIA. I already
>stuffed up calling monotonicity homogeneity or something like that)
>
>
>On Wed, 26 Aug 1998, Blake Cretney wrote:
>
>>
>> --
>>
>> On Thu, 27 Aug 1998 09:17:34   David Catchpole wrote:
>> >On Wed, 26 Aug 1998, Mike Ositoff wrote:
>> >
>> >I guess it's hard to always have everyone agreeing, but I believe the
>> >criterion of irrelevant alternatives actually does allow for the
criteria
>> >below.
>> >
>> >The fact is that if a Condorcet winner exists, that winner
>> >satisfies IA, and vice versa. This is a good point about Condorcet,
>> >in case people are just being defensive without provocation.
>> >
>> >I personally disagree with the vagueness of just plain "majority rule",
>> >believing that more focused principles of majority rule need to be
>> >developed. As for the concept of strategy- IA has a large-scale
>> >positive effect on the behaviour of actors in an election.
>> >w
>> >I would appreciate it if you gave me examples of IA contradicting other
>> >ubiquitous criteria, as I am a relative newcomer to the "field";
however, I
>> >believe that the concept of IA itself is a particularly dominant one-
that
>> >the outcome of the election reflects the values of the electorate
>> >independent of the presence of any losing candidates.
>> >
>> IA, which I will call IIAC, is not possible.
>> Consider this example.
>>
>> Ballots cast
>> 45 A B C
>> 35 B C A
>> 20 C A B
>
> A to B- A
> A to C- C
> B to C- B
>
>A paradox of voting exists- No Condorcet winner exists in this case and
>therefore IA is not satisfied: QED. My point exactly.
>
>That one can say from this one anecdote that IA NEVER exists for any
>other elections is fanciful.
>
>> Let's assume your favorite method chooses A as the winner.
>> This means that B should be irrelevant.  So let's take it
>> out and see what the result is.
>> 45 A C
>> 35 C A
>> 20 C A
>>
>> That is 45 A>C 55 C>A.  Based only on these votes, it would
>> be absurd to choose A as the winner over C.  So we have to
>> conclude that any method that would choose A, does not
>> satisfy IIAC.
>>
>> But what about systems that would choose B.  That means C is
>> irrelevant.
>>
>> 45 A B
>> 35 B A
>> 20 A B
>>
>> So 65 A>B 35 B>A.  A is the obvious winner.  So we have to
>> conclude that any method that would choose C, does not
>> satisfy IIAC.
>>
>> But what about systems that would choose C.  That means A is
>> irrelevant.
>>
>> 45 B C
>> 35 B C
>> 20 C B
>>
>> So 80 B>C 20 C>B.  B is the obvious winner.  So we have to
>> conclude that any method that would choose C, does not
>> satisfy IIAC.
>>
>> So in fact no method will satisfy IIAC.
>> By the way, the criterion I was relying on which
>> contradicted IIAC was that if there are only two candidates,
>> the one with the most votes should win.  I think that's
>> called Paretto.  In fact, I suspect assuming Paretto is not
>> really necessary, and that it would be possible to construct
>> examples that showed that for a method that gives a victory
>> to certain levels of minority, it would have to give a
>> different answer to the same level of minority in a
>> different case.  The only real assumption is that you're
>> doing something that can be reasonably called voting.
>>
>> I think the best you can do is MIIAC, the Modified
>> Independence from Irrelevant Alternatives Criterion, which
>> says that a candidate is irrelevant if it is not in the
>> Smith set.  This will likely make the bulk of candidates,
>> especially unpopular ones, irrelevant.  As a result, your
>> chances of losing out because you vote sincerely are low,
>> and even if you wanted to vote insincerely it would be hard
>> to predict how to do this effectively.
>>
>>
>>
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>
>