[EM] Range voting fails IIA

[David Cary]

--- Abd ul-Rahman Lomax <abd at lomaxdesign.com> wrote:

> At 08:59 PM 11/7/2006, David Cary wrote:
> >Both CRV and Wikipedia claim that range voting meets the Independence
> >of Irrelevant Alternatives (IIA) criterion, but neither gives much
> >justification for such claims.
> >
> >Electorama succinctly but informally describes IIA as:
> >     "if one option (X) wins the election, and a new alternative (Y)
> >is added, only X or Y will win the election."
> 
> Election criteria are supposed to be objective. One of the aspect of 
> this, and I think it is -- or should be -- standard in applying the 
> criteria, is that the mental state of the voters is not relevant. 
> What is relevant is what is expressed on the ballots. (It was a 
> failure to consider this that resulted in the error of considering 
> Approval to have failed the Majority Criterion.)

The comparison of individual voter's mental preferences to the group
choice is at the core of many criteria, IIA and Majority Criterion among
them.  In many cases, these criteria have roots in the broader theory of
social choice that allows consideration of a wide variety of
possibilities, each associating some specified individual mental
preferences to a collective decision or set of preferences without having
to consider the mechanism, such as voting, for achieving that association.

In an important sense, the involvement of voter mental preferences does
not make the criteria any less objective.  The criteria are not required
to function as empirical criteria.  They can work quite well as
theoretical criteria.  In this context, the voter mental preferences are
not discovered but given, as in the example I gave, or even treated as
mathematical variables.

More precisely, in this context, human mental preferences are modeled by
some mathematical object, for example some ordering of the candidates, a
scoring function, etc.  This approach facilitates careful thinking about
voting systems by separating various issues and allowing consideration of
a broad range of possibilities.

A criterion expressed in terms of individual mental preferences is often
uninteresting when applied to an election method.  The application becomes
interesting when there are restrictions on how the individual mental
preferences are converted into votes, for example voting "sincerely", but
other restrictions are possible.  The restrictions convert the election
method into a non-trivial social choice function.  It possible to embed
the restrictions into the criterion, but since the restrictions may not
apply to all election methods, it can make more sense to invoke them as
needed. 

> One can certainly claim that an election method does not allow voters 
> to express something, such as more than two ranks, or preference 
> strengths. But that's not relevant here.

Formulation of a criterion often achieves precision and clarity at the
expense of generality, for example by making it particular to a specific
model of individual preferences.  This is done even though the principle
of the criterion could easily apply to other situations, or because there
are different ways of generalizing the criterion.

> 
> What does it mean to "add" a new alternative? In methods that are 
> purely ranked, we presumably assume that the new candidate fits 
> somewhere into the ranking system, and then the votes would be 
> changed accordingly, with no other complications. We would assume 
> that all relative ranks remain the same, for the candidates already 
> in the election, and that the newcomer is simply assigned one of 
> these ranks. If ranks must be exclusive, it is reasonable to allow 
> that a new rank is inserted at some position for the candidate, but, 
> again, except for that additional rank, nothing else changes.
> 
> The Range equivalent could only mean that the new alternative does 
> not disturb the existing range ratings. And if this is the meaning, 
> Range satisfies IIA, quite clearly.

For IIA there is clearly an implication that some things, but not
necessarily others, are held constant when an additional alternative is
added.  The informal definitions fail to address exactly what is held
constant.  It is certainly reasonable to specify that the mental
preferences and any rules or restrictions on how those preferences are
converted into votes are held constant. Call that IIA-P.  An alternative
is to specify that the actual votes remain constant.  Call that IIA-V.

It is pointless to argue that one is right and the other is wrong.  It is
important to acknowledge the two possibilities and be specific about which
one is being considered.  Each version of IIA has something different to
say.  Wikipedia too often is deficient by ignoring those distinctions.

> What Mr. Cary did was to assume that the voters were following a 
> strategy where one considers and adjusts Range votes based on the 
> "candidate space." I.e., with this strategy, one ranks the best at 
> max and the worst at zero, and then assigns values in between for the 
> others, based on expectation of value.

I gave an example that is consistent with such a restriction, but does not
require such a restriction.  I did assume IIA-P and that the voter's
preferences could be specified by a scoring function that was not bounded
by the voting range.

> However, this is internal process, and there are other reasonable 
> strategies for the voter to follow. Moving toward Approval style 
> voting is one of them. Approval satisfies IIA. Only if we think that 
> the voter is following some strategy which causes the voter to change 
> the approval votes already cast based on this newcomer could we say 
> that it does not satisfy it.
>
> But this is a ridiculous interpretation of IIA. Or at least it is one 
> that becomes very hard to satisfy. I do not find IIA to be crisply 
> defined. It seems to me that it is a criterion designed for ranked 
> methods, because one may make ready assumptions about the effect of 
> an added candidate on voting, by making presumptions about the effect of
> rank.

That the voters could have used other rules to convert their preferences
into votes is not directly relevant to the validity of the counterexample.

Making IIA general enough to apply to cardinal preferences as well as
ordinal preferences is not really the problem either.  The issue with
getting Approval Voting with "sincere" voting to satisfy IIA-P is the
issue of whether a "sincere" approval vote exists, is unique, or can
change as alternatives are added.  But those are recurring issues for
Approval voting regardless of whether IIA-P is being considered. 

> The nomination of Genghis Khan would probably not cause me to adjust 
> my Range Votes for any candidate.... I'd simply rank him at zero. The 
> nomination of a candidate that was obviously better to me than anyone 
> else on the list, however, might indeed cause me to change my range 
> ratings, and such adjustments might indeed, if other voters did not 
> agree with me, cause a slippage for one of these such that he loses 
> and the newcomer also loses. But this is hardly an "irrelevant 
> alternative." I'd think of an irrelevant alternative as someone 
> outside the reasonable candidate space, or, alternatively, a clone of 
> a candidate already in that space. These would not ordinarily affect 
> the winner, except possibly to substitute the newcomer for the 
> previous winner, thus satisfying IIA.
> 

There are two ways to make Range Voting satisfy IIA.  One is to say that
it satisfies IIA-V.  IIA-V essentially only considers how a voting method
tabulates votes.  It ignores any consideration of how the voting method
otherwise responds to voter preferences.  For example, IIA-V restricts
FPTP voting so that when a new alternative gets added only previous
abstaining votes are allowed to vote for it; no voter is allowed to switch
votes from another alternative.

Range Voting can also satisfy IIA-P to a limited degree if the the model
of voter preferences is limited to a scoring function that has a bounded
range and the conversion of votes is restricted the positive linear
function from the overall scoring range to the voting range, independent
of the alternatives being considered.  The ruminations about Ghengis Khan
illustrate the limitations of both approaches to getting Range Voting to
satisfy IIA.

While there is value in recognizing preference strength in modeling
people's preferences, in not being limited by an ordinal preference model,
it is also important to not gloss over the issues in defining what
preference strength is and how it might be represented, nor to avoid a
careful analysis of its involvement in an election method.

-- David Cary




 
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