Ideal (?) Instant Consensus voting method

[Saari at aol.com]

(method recap at end of this message)
I note that this method will not reliably find the most central (highest)
point in the case of a simple Gaussian distribution.

For example, suppose a group was voting on the "best" height to hang some
pictures.  
(We are describing a simple idealized situation here to allow a rigorous
determination of the "best" outcome.)
Assume that the voters' heights form a standard Gaussian (bell-curve)
distribution - the expected distribution for a large group.  Assume all
voters vote their honest preferences.  Assume that looking up and looking
down are equally uncomfortable, the further from ideal the worse, and that
all voters are identical in all other ways except on the issue in question
(which is solely a function of their height).

This is just about the purest, simplest possible situation.  (Many other
possible situations are too complex to permit establishing an
unambiguously-correct solution, but this one is not.)  Perhaps it is not
realistic, nevertheless we should expect a proposed voting system to product
the correct result for such an extremely simple case, right?

For a simple Gaussian distribution, clearly the "best" result is at the
middle of the central peak, i.e. at the 50-percentile point.  Surely we can
all agree on this, right?  The most central point will definitely maximize
the number of contented members and minimize the overall level of discontent
- better than any solution to one side or another.

What result will INSTANT CONSENSUS (or Instant Runoff) produce?  It depends
on the particular distribution of candidates.  For example, suppose that
candidate A is located at the 40-percentile point (i.e. well left of center).
 Candidate B is located at the 50-percentile point (i.e. dead center).
 Candidate C is located at the 65-percentile point (i.e. well right of
center).

Under INSTANT CONSENSUS, the vote results will look like:
45% vote: A > B > C
42% vote: C > B > A
13% vote: B > A > C  -or-  B > C > A
The most central candidate (B) will be eliminated, leaving the win to
candidate A.

INSTANT CONSENSUS (or Instant Runoff) based on preferential voting thus will
not reliably choose the most central candidate in the case of a simple
Gaussian distribution.

Mike Saari


>SteveE writes:
>Happy April,
>
>This seems like a good time to explain the ideal single-winner
>voting method, known as INSTANT CONSENSUS.  It's nearly
>identical to the Instant Runoff method, being an iterative
>elimination & transfer method, except that Instant Consensus
>iterates completely instead of stopping short.
>
>
>The Voting
>----------
>Each voter ranks the candidates in order of preference, from 
>most preferred to least preferred.
>
>The Tally
>---------
>For each candidate,       
>   Count the ballots which rank that candidate most preferred.
>
>Repeat the following...
>
>   Eliminate the candidate which has the smallest count.
>
>   For each ballot which was counted for the eliminated candidate,      
>      Transfer the ballot to the most preferred    
>      uneliminated candidate on that ballot.   
>
>...until all but one candidate has been eliminated.