Range Voting Redux

[kevin_standlee]

Y'all might want to go have another look at the discussion on Range Voting from a few weeks ago. Advocates have (somewhat belatedly by LJ standards) chimed in on the subject here and here.

Update, 23 Jan 2007, 00:30: And now many other places. Whew!

Comments

New proof that Range Voting > IRV, Condorcet, etc.

[(Anonymous)]

We shall show a sense range voting is superior to any
pure-rank-ballot voting method.

Theorem:
These criteria, for a single-winner voting system based on
pure-rank-order-ballots, are incompatible:
1. AFB = avoids favorite betrayal
2. ICC = immune to candidate cloning
3. reduces to simple majority vote in 2-candidate case
4. symmetry under candidate renaming
5. tiebreaks (if any) are random equally likely.
6. adding a new candidate to the election whom all voters unanimously
rank unique-bottom,
does not change the winner.

Proof:
Consider these 3 votes:
A>B>C
C>A>B
B>C>A
by symmetry axiom 4 this is a perfect 3-way tie.
However we shall argue under axioms 1-3 that A must win, which is a
contradiction that establishes the proof.
If A does not win, then B or C does.

If B wins, then the C-voter would betray C to vote A>C>B getting
A>B>C
A>C>B
B>C>A
and then {B,C} is a clone set and hence by axioms 2 and 3 then A must win
and hence the betrayal worked and hence we get a contradiction with
axiom 1.
[The alternate dishonest vote, which is not a C-betrayal, C>B>A,
would not work since B still would win:
A>B>C
C>B>A
B>C>A
now {B,C} is a clone set, so by axioms 2 & 3 B or C must win;
but winner here must be B and not C (and not BC tie) because if it were C
or BC tie then the A-voter could betray: B>C>A causing
B>C>A
C>B>A
B>C>A
in which case B must win by axiom 6.]

If C wins, or if BC tie, then the A-voter can betray A to vote B>A>C
getting
B>A>C
C>A>B
B>C>A
whereupon {A,C} is a clone set and hence by axioms 2 and 3 then B must win
and hence the betrayal worked and hence we get a contradiction with
axiom 1.
[The alternate dishonest vote which is not an A-betrayal, A>C>B would
not work since
C still would win when
A>C>B
C>A>B
B>C>A
because {A,C} is a winning clone set
and if A wins (or AC tie) then B>C>A voter betrays: C>A>B to
make C win
A>C>B
C>A>B
C>A>B
by axiom 6.]
Q.E.D.

Remark 1.
I do not presently know if this theorem can be extended to permit
equalities in vote-rankings.

Remark 2.
Antiplurality voting obeys all 6 axioms except for #2.

Remark 3.
Schulze beatpaths voting obeys all 6 axioms except for #1.

Remark 4.
Range voting obeys all 6 axioms if all range votes are
"normalized" so voters (obeying the recommendations for voting on the
http://rangevoting.org front page) always gove the best candidate the top
score and the worst the bottom score in a 2-candidate election
[i.e. in practice with voters who are not idiots].
But with possible-idiot voters, range fails axiom #3.

So we have proven a sense in which range is superior to EVERY
pure-rank-ballot voting
method, and using two of the most important voting criteria AFB and ICC.

Remaining Open question: what happens if we permit rank order votes to
have EQUALITIES in them?