[EM] Burlington dumps IRV; Immunity from Majority Complaints (IMC) criterion

[seppley at alumni.caltech.edu]

FairVote wrote (elsewhere, cited in EM): "... the use of instant runoff
voting (IRV) for mayor was repealed this week by a margin of less than 4%
in Vermont's largest city of Burlington. ..."

That looks like a case where a voting method's failure of the Immunity
from Majority Complaints criterion (IMC) led to the voters dumping the
voting method.

IMC is a criterion I wrote about in the EM maillist many years ago.  It's
motivation is this:  Suppose a majority rank x over y but x does not
finish ahead of y (in the election's order of finish).  They may complain
that x should have finished ahead of y, using "majority rule" as their
argument.  If they are not rebutted, the voting method is on the chopping
block since a majority have considerable power to enact change.  In the
most dangerous case, where y is the winner, y's mandate is undermined and
the complaining majority would be especially motivated to replace the
voting method in order to elect x.  It would be problematic to try to
rebut (and placate) them by arguing the merits of criteria (reinforcement,
participation, monotonicity, etc.) for which there is no consensus
regarding importance since the majority might not consider those criteria
important, or might not understand them.  So it is desirable to be able to
turn their own "majority rule" argument against them.  Therefore, the
voting method should satisfy the following criterion:

Immunity from Majority Complaints (IMC)
---------------------------------------
Let V(a,b) denote the number of voters who rank a over b, for all
alternatives a & b.
For all x & y, if V(x,y) > V(y,x) and the order of finish does not place x
ahead of y, there must exist an arrangement a1, a2, ..., ak of a subset of
the alternatives such that a1 = y and ak = x and all three of the
following conditions hold for each ai in {a1, a2, ..., ak‑1}:

     (IMC‑1)  A majority rank ai over ai+1.
     (IMC‑2)  The number of voters who rank ai over ai+1
                is at least as large as V(x,y).
     (IMC‑3)  ai is ahead of ai+1 in the order of finish.

IMC-2 means the majority who rank ai over ai+1 is at least as large as the
complaining majority for every ai in {a1, a2, ..., ak‑1}. (When there are
many voters, as in a public election, two pairwise majorities will rarely
be exactly the same size.  So the majority who rank a1 over a2, the
majority who rank a2 over a3, etc., will all usually be larger than the
complaining majority.)

Satisfaction of IMC allows the complaining majority to be rebutted using
their own argument:  By IMC‑1 & IMC‑2, majorities at least as large as the
complaining majority said x should finish behind ak‑1, ak‑1 should finish
behind ak‑2, ..., and a2 should finish behind y.  And they do finish that
way, by IMC‑3.

Condition IMC‑3 matters because if some ai does not finish ahead of ai+1,
the complaining majority can point out a flaw in the rebuttal: the voting
method thwarted the majority who rank ai over ai+1 because it found
sufficient evidence that they are wrong about ai & ai+1; therefore those
voters do not contribute evidence that x should finish behind y. 
This would be especially problematic if ak‑1 does not finish ahead of x,
since in that case no evidence remains that x should finish behind any
alternative.

Only one voting method satisfies IMC: Maximize Affirmed Majorities (MAM).

Satisfaction of IMC implies satisfaction of many other desirable criteria:
top cycle (also known as the Smith set criterion), Condorcet, independence
from clones, minimal defense (also known as Ossipoff's strong defensive
strategy criterion), etc.

Most voting methods not only fail IMC, they also fail a criterion weaker
than IMC: Weak Immunity from Majority Complaints (WIMC): If more than half
of the voters prefer some x over the winner w, there must exist an
alternative z such that both of the following hold:
     (WIMC-1) The number of voters who rank z over x is
               at least as large as the number of voters
               who rank x over w.
     (WIMC-2) z is ahead of x in the order of finish.

WIMC is weaker than IMC in three ways:
(1) WIMC covers only the most dangerous case in which a majority prefer a
loser over the winner.
(2) The complaining majority in WIMC is an absolute majority, more than
half the voters.
(3) Perhaps a less comprehensive rebuttal could suffice:  By the
complainers' own "majority rule" argument, x should finish behind z (and
does).  Thus x shouldn't be the winner (and isn't).

WIMC is stronger than the Smith set criterion (which is stronger than the
Condorcet criterion) because satisfaction of WIMC implies the winner is in
the Smith set (also known as the top cycle, defined as the smallest
non-empty subset such that every alternative in the subset is ranked by
more than half the voters over every alternative not in the subset).
(Proof: Suppose the winner is not in Smith; we must show WIMC is violated.
 Since Smith isn't empty and an order of finish is acyclic, we can pick x
in Smith such that no alternative in Smith finishes ahead of x.  Thus all
alternatives ahead of x are not in Smith, so no alternative ahead of x is
ranked over x by a majority.)  So it is easy to show that every voting
method that fails the Condorcet criterion also fails WIMC and IMC.  These
include Hare (a.k.a. Instant Runoff and the Alternative Vote) and Borda. 
They also include Approval voting, which fails in spirit since polling can
establish the existence of a majority who prefer a loser over the winner,
in the cases where the restrictive ballot format does not elicit that
information.

Should IMC and WIMC be added to Wikipedia?

Regards,
Steve