[EM] Re: New criterion: Independence Of Weakly Defeating Alternatives (IWDA)
Jobst Heitzig
heitzig-j at web.de
Mon Nov 1 14:21:38 PST 2004
Some hours ago I wrote:
> INDEPENDENCE OF WEAKLY DEFEATING ALTERNATIVES (IWDA):
> The winner must not change when an alternative X is added but for each
> option Y which is defeated by X, already some stronger defeat against Y
> is affirmed/undropped by the method.
>
> In the current wording, IWDA only applies to methods which affirm
> defeats (like Ranked Pairs, River, Kemeny) or drop defeats (like PC, SD,
> SSD).
>
> Ranked Pairs and Kemeny fail IWDA, as can be seen from the example I
> posted recently. In that example, C is the "weakly defeating alternative".
>
> Does SSD (Beatpath) fulfil IWDA?
The following example shows that also PC, SD, and SSD (Beatpath) fail IWDA:
Sincere prefs:
1 A>B>C>D
33 A>D>B>C
33 B>D>C>A
17 C>A>B>D
16 C>D>A>B
Defeats and winning votes: A>B>C>A>D, B>D>C
...>... A B C D
A 67 (34) 51
B (33) 67 51
C 66 (33) (34)
D (49)(49) 66
In {A,B,C}, all the above methods elect A by affirming A>B>C and
dropping C>A. When we add D to {A,B,C}, D is "weakly defeating" in the
sense of IWDA since D>C is weaker than the affirmed B>C. However, in
{A,B,C,D}, all the above methods elect D, hence they violate IWDA.
Proof that River does fulfil IWDA: Assume W is the River winner before X
is added. If X is "weakly defeating" in the sense of IWDA, it does not
defeat W since no defeat against W is affirmed. That is, W defeats X,
hence X is defeated by at least one candidate. Now it is easy to verify
that when we add the strongest defeat against X to the old River
diagram, we get the new River diagram. Thus the diagram changes only by
one arrow and the winner is still W.
So far, River seems to be the only method which fulfils IWDA.
Note that since IWDA refers to "strength" of defeats, so that its
meaning changes with the meaning of defeat strength. In particular,
Cardinal Weighted Pairwise will only fulfil IWDA when it uses River, not
when it uses Beatpath or Ranked Pairs.
Jobst
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