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.<div>
<br></div><div>Jameson<br><br><div class="gmail_quote">2013/7/5 <span dir="ltr"><<a href="mailto:seppley@alumni.caltech.edu" target="_blank">seppley@alumni.caltech.edu</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
FairVote wrote (elsewhere, cited in EM): "... the use of instant runoff<br>
voting (IRV) for mayor was repealed this week by a margin of less than 4%<br>
in Vermont's largest city of Burlington. ..."<br>
<br>
That looks like a case where a voting method's failure of the Immunity<br>
from Majority Complaints criterion (IMC) led to the voters dumping the<br>
voting method.<br>
<br>
IMC is a criterion I wrote about in the EM maillist many years ago. It's<br>
motivation is this: Suppose a majority rank x over y but x does not<br>
finish ahead of y (in the election's order of finish). They may complain<br>
that x should have finished ahead of y, using "majority rule" as their<br>
argument. If they are not rebutted, the voting method is on the chopping<br>
block since a majority have considerable power to enact change. In the<br>
most dangerous case, where y is the winner, y's mandate is undermined and<br>
the complaining majority would be especially motivated to replace the<br>
voting method in order to elect x. It would be problematic to try to<br>
rebut (and placate) them by arguing the merits of criteria (reinforcement,<br>
participation, monotonicity, etc.) for which there is no consensus<br>
regarding importance since the majority might not consider those criteria<br>
important, or might not understand them. So it is desirable to be able to<br>
turn their own "majority rule" argument against them. Therefore, the<br>
voting method should satisfy the following criterion:<br>
<br>
Immunity from Majority Complaints (IMC)<br>
---------------------------------------<br>
Let V(a,b) denote the number of voters who rank a over b, for all<br>
alternatives a & b.<br>
For all x & y, if V(x,y) > V(y,x) and the order of finish does not place x<br>
ahead of y, there must exist an arrangement a1, a2, ..., ak of a subset of<br>
the alternatives such that a1 = y and ak = x and all three of the<br>
following conditions hold for each ai in {a1, a2, ..., ak‑1}:<br>
<br>
(IMC‑1) A majority rank ai over ai+1.<br>
(IMC‑2) The number of voters who rank ai over ai+1<br>
is at least as large as V(x,y).<br>
(IMC‑3) ai is ahead of ai+1 in the order of finish.<br>
<br>
IMC-2 means the majority who rank ai over ai+1 is at least as large as the<br>
complaining majority for every ai in {a1, a2, ..., ak‑1}. (When there are<br>
many voters, as in a public election, two pairwise majorities will rarely<br>
be exactly the same size. So the majority who rank a1 over a2, the<br>
majority who rank a2 over a3, etc., will all usually be larger than the<br>
complaining majority.)<br>
<br>
Satisfaction of IMC allows the complaining majority to be rebutted using<br>
their own argument: By IMC‑1 & IMC‑2, majorities at least as large as the<br>
complaining majority said x should finish behind ak‑1, ak‑1 should finish<br>
behind ak‑2, ..., and a2 should finish behind y. And they do finish that<br>
way, by IMC‑3.<br>
<br>
Condition IMC‑3 matters because if some ai does not finish ahead of ai+1,<br>
the complaining majority can point out a flaw in the rebuttal: the voting<br>
method thwarted the majority who rank ai over ai+1 because it found<br>
sufficient evidence that they are wrong about ai & ai+1; therefore those<br>
voters do not contribute evidence that x should finish behind y.<br>
This would be especially problematic if ak‑1 does not finish ahead of x,<br>
since in that case no evidence remains that x should finish behind any<br>
alternative.<br>
<br>
Only one voting method satisfies IMC: Maximize Affirmed Majorities (MAM).<br>
<br>
Satisfaction of IMC implies satisfaction of many other desirable criteria:<br>
top cycle (also known as the Smith set criterion), Condorcet, independence<br>
from clones, minimal defense (also known as Ossipoff's strong defensive<br>
strategy criterion), etc.<br>
<br>
Most voting methods not only fail IMC, they also fail a criterion weaker<br>
than IMC: Weak Immunity from Majority Complaints (WIMC): If more than half<br>
of the voters prefer some x over the winner w, there must exist an<br>
alternative z such that both of the following hold:<br>
(WIMC-1) The number of voters who rank z over x is<br>
at least as large as the number of voters<br>
who rank x over w.<br>
(WIMC-2) z is ahead of x in the order of finish.<br>
<br>
WIMC is weaker than IMC in three ways:<br>
(1) WIMC covers only the most dangerous case in which a majority prefer a<br>
loser over the winner.<br>
(2) The complaining majority in WIMC is an absolute majority, more than<br>
half the voters.<br>
(3) Perhaps a less comprehensive rebuttal could suffice: By the<br>
complainers' own "majority rule" argument, x should finish behind z (and<br>
does). Thus x shouldn't be the winner (and isn't).<br>
<br>
WIMC is stronger than the Smith set criterion (which is stronger than the<br>
Condorcet criterion) because satisfaction of WIMC implies the winner is in<br>
the Smith set (also known as the top cycle, defined as the smallest<br>
non-empty subset such that every alternative in the subset is ranked by<br>
more than half the voters over every alternative not in the subset).<br>
(Proof: Suppose the winner is not in Smith; we must show WIMC is violated.<br>
Since Smith isn't empty and an order of finish is acyclic, we can pick x<br>
in Smith such that no alternative in Smith finishes ahead of x. Thus all<br>
alternatives ahead of x are not in Smith, so no alternative ahead of x is<br>
ranked over x by a majority.) So it is easy to show that every voting<br>
method that fails the Condorcet criterion also fails WIMC and IMC. These<br>
include Hare (a.k.a. Instant Runoff and the Alternative Vote) and Borda.<br>
They also include Approval voting, which fails in spirit since polling can<br>
establish the existence of a majority who prefer a loser over the winner,<br>
in the cases where the restrictive ballot format does not elicit that<br>
information.<br>
<br>
Should IMC and WIMC be added to Wikipedia?<br>
<br>
Regards,<br>
Steve<br>
----<br>
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</blockquote></div><br></div>