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 <[email protected]> > 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 voting (IRV) for mayor was repealed this week by a margin of less than 4% 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 > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
---- Election-Methods mailing list - see http://electorama.com/em for list info
