How about defining the positions as follows.
A>B>C => A=1, B=2, C=3
A=B>C => A=1.5, B=1.5, C=3
A's position got worse in the second example. B's position got better.
Juho
On Nov 24, 2009, at 6:51 PM, Kristofer Munsterhjelm wrote:
It's fairly straightforward to define whether a candidate is helped
after a change of ballots if "helping" is limited to win/not win: if
the candidate wasn't in the set of winners (ranked first on the
social ordering), but is after the modification, the candidate was
helped. It is also not that difficult to define it for a social
ordering without ties: if the candidate moves from qth place to pth
place, p < q, then he was helped.
But how would one define this for an ordering with ties? The problem
with defining it in terms of candidates higher ranked is that if
A > B > C > D = E turns into A > B = E > C > D, C is "helped"
according to that metric, even though intuitively it seems like he's
not so. On the other hand, defining it in terms of ranks above the
set containing the candidate has problems when the possible number
of sets change. For instance, A > B > C > D turning into A = B = D >
C doesn't seem to have "helped" C, although now he's second, whereas
before the change, he was third.
Is there any consistent way of defning help and harm, in the context
of candidates, when the social ordering may contain ties?
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