# [EM] Range voting fails IIA

David Cary dcarysysb at yahoo.com
Tue Nov 7 17:59:42 PST 2006

```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."

Wikipedia is nearly as informal but more restrictive in the scope of
its application:
"If A is preferred to B out of the choice set {A,B}, then
introducing a third, irrelevant, alternative X (thus expanding the
choice set to {A,B,X} ) should not make B preferred to A. In other
words, whether A or B is better should not be changed by the
availability of X."

But Range Voting doesn't satisfy IIA in either form.  Without loss of
generality, we'll look at the range voting that allows ballots to be
cast in the range from 0 to 100.  Then consider the following
example:

First, there are two candidates, A and B, and three voters, u, v, and
w.  Each of the three voters have a personal scoring system for the
candidates as follows:

u: A= 450,  B= 270
v: A=-320,  B=-200
w: A= 590,  B= 730

and the voters cast the following ballots:

u: A=100, B=  0
v: A=  0, B=100
w: A=  0, B=100

This produces totals of A=100, B=200, and B is the winner, beating A.

Now a third candidate, C, is introduced.  The personal scoring system
for each voter is extended to include C, but the scores for A and B
are unchanged:

u: A= 450, B= 270, C= 150
v: A=-320, B=-200, C=-800
w: A= 590, B= 730, C=  30

and the candidates cast votes as:

u:  A=100, B= 40, C=0     ( Vote = 1/3 * (Score - 150) )
v:  A= 80, B=100, C=0     ( Vote = 1/6 * (Score + 800) )
w:  A= 80, B=100, C=0     ( Vote = 1/7 * (Score -  30) )

Which produces totals of A=260, B=240, C=0, and A is the winner,
beating B and C. Hence a violation of IIA.

Note that in each case all voters cast what some have described as

-- David Cary

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