[EM] unintended changes in pairwise preferences

[Stephen Turner]

This example shows something which seems undesirable,
though is probably rare.  It works both with ranked
pairs and beatpath.  

Two slightly different sets of ballots give very
different social orderings: a change in the pairwise
X-Y preference on a couple of ballots also changes
unrelated pairwise preferences in the social ordering.
 

("Unrelated" means that the two pairs involve 4
different candidates.)  

Ballots are ranked, and all preferences are strict. 
Candidates are W,X,Y,Z.
************ Profile 1 **************************
18:Z>X>Y>W
 9:W>Y>Z>X
 9:Y>W>X>Z
 5:W>Z>X>Y
 3:X>Y>Z>W
 2:Y>X>W>Z

Pairwise defeats:
Z>X 18
Y>W 18
X>Y  6
W>Z  4
The pairs Z,Y and X,W tie.

The ranked pairs and beatpath outcomes are both:
Z>X>Y>W (which happens to agree with the largest
faction above).  By the beatpath outcome, I mean that
each candidate disqualifies all those to its right.

************ Profile 2 **************************
(The only difference is that 2 of the 18 have reversed
X>Y to give Y>X.)

16:Z>X>Y>W
 2:Z>Y>X>W
 9:W>Y>Z>X
 9:Y>W>X>Z
 5:W>Z>X>Y
 3:X>Y>Z>W
 2:Y>X>W>Z

Pairwise defeats (only X>Y is different):
Z>X 18
Y>W 18
X>Y  2
W>Z  4
The pairs Z,Y and X,W tie.

The ranked pairs and beatpath outcomes are now both:
Y>W>X>Z.  So the 2 who reversed their X-Y preference
have achieved this, but at the cost of messing up the
much more important Z>W pair.  ("More important" as Z
is first and W last for all the original 18.)  The
remaining 16 members still form by far the largest
faction but the result is now nothing like their
desired one.
****************************************************
Of course, one of the causes is the near match between
the W-Z and X-Y pairwise defeats, which is why I say
that it is likely to be rare.  They differ by the
minimum for which they are not the same (this is 2
when all preferences are strict).

This has raised two questions for me.
(1) Is it known whether there can exist a procedure in
which a pair (P1,P2) can never be reversed in the
social ordering by changing only the unrelated (P3,P4)
pairwise preferences in one or more ballots?  
Dictatorship and anti-dictatorship obviously satisfy
this, but what about anything else?

(2) Among the criteria we usually discuss on this
list, we do not have one on "stability", which should
mean something like: "a small change in the ballots
should change the outcome as rarely as possible". 
This seems desirable.  Has it already been discussed
somewhere?
-- 
Stephen



		
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