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

[Jameson Quinn]

IMC seems to me to be too narrow to be a general criterion, if only one
custom-built voting system passes it. WIMC is an interesting refinement of
Condorcet and Smith. But neither belongs on Wikipedia without a "reliable"
citation.

Jameson

2013/7/5 <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 voting (IRV) for mayor was repealed this week by a margin of less than 4%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
> ----
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