[EM] > 2. Re: Does the 'Independence of Irrelevant Alternatives Criterion'

[Richard Moore]

 Re: Does the 'Independence of Irrelevant Alternatives Criterion' Imply a
 Condorcet Winner ?
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Forest Simmons wrote:

 >Yes, Arrow did use the IIA criterion, and yes, most folks here agree 
 >that it is the criterion that is too strict, and therefore should be 
 >relaxed in one way or another.

One way to interpret Arrow's Theorem is to say "you can't have all of 
these criteria, so you must give up at least one of them". If you 
insist on a number of criteria that cannot all be met by the same 
method, you are being too picky.

I'm one of those who think IIA is useless in sorting good methods from 
bad. The options are to ignore IIA altogether, or to replace it with a 
weaker (but more useful) criterion. I've been thinking about one 
particular way of "relaxing" IIA.

Suppose that candidate A beats candidate C by 30 votes, and candidate 
B beats candidate C by 20 votes. Might these margins tell us anything 
about the suitability of candidate A compared to candidate B, even if 
nothing else is known? I think this limited information provides 
support for the hypothesis that A is a better candidate than B. 
Removing candidate C from the race removes this information from 
consideration. The loss of information might conceivably justify a 
shift from a win by A to a win by B, but it would not justify a shift 
from a win by B to a win by A. Yet the latter shift could certainly 
occur in many methods.

The replacement for IIA that I have in mind is this: "If X wins and Y 
loses, and margin(X,Z) <= margin(Y,Z), then removing candidate Z from 
the election shall not cause Y to win and X to lose." Here, 
margin(A,B) represents the number of voters favoring A to B minus the 
number of voters favoring B to A. This value can be positive, 
negative, or zero.

I call this criterion "Independence of Non-supporting Information" 
(INI). A very simple but inexact paraphrase of INI is: "A given result 
should not depend on evidence that does not support that result."

A clear example would be the case where margin(X,Z) < 0, and 
margin(Y,Z) > 0. If X also beats Y in a pairwise comparison, then 
there is a Y>Z>X>Y cycle. If a single-winner method picks X as the 
winner, then removing Z breaks the cycle in a way that upholds that 
result. If INI is met, we would not see the winner change from X to Y.

Maybe it is possible for INI to be met while meeting all the Arrow 
criteria except IIA. Of course, INI might still be too strong, but 
there are many methods that meet this criterion but fail IIA.

The methods I have found that pass INI are:

- Lone-mark plurality (which passes in the most trivial way, because 
all ballots that were cast for a candidate that is removed simply 
don't count for any candidate)

- Approval

- Borda, if ballots with tied rankings are not allowed

- Copeland

Methods that fail INI:

- FPTP (based on ranked ballots, as opposed to lone-mark plurality)

- IRV

- Borda, if ballots with tied rankings are allowed

- Bucklin

Shulze and Ranked Pairs, I expect, also fail INI. I don't have proof 
of this but I think I have an idea of how to construct a 
counterexample. If I can complete this soon, I will post it.

I'm a little concerned that all the ranked methods that I've found to 
pass INI so far exhibit clone sensitivity. Maybe there is another 
impossibility theorem lurking there. If anyone can find a ranked 
method that does not exhibit clone sensitivity, and passes INI, I'd be 
very interested. If anyone can suggest a criterion in the same spirit 
as INI, but that is less exclusive of clone-insensitive ranked 
methods, that would also be interesting.

  -- Richard