[EM] definitions of immunity from majority complaints (was: attempt of a grand compromise)
Jobst Heitzig
heitzig-j at web.de
Thu Oct 21 14:20:15 PDT 2004
Dear Steve!
On the entrance to your site, you define
> immunity from majority complaints: If a majority rank an alternative
> x over the elected alternative w, then there must be a sequence of
> alternatives a1,a2,...,an such that a1 = w and an = x and the
> following conditions hold: (1) Majorities at least as large as the
> majority who rank x over w rank a1 over a2, a2 over a3, etc., and
> an-1 over an. (2) The election procedure generates a social ordering,
> and it ranks a1 over a2, a2 over a3, etc., and an-1 over an.
which seems to correspond to the following definition, found somewhere
deeper down on your site:
> Immunity from majority complaints #2 (IMC2): The voting procedure
> must let each voter vote any ordering of the alternatives, must elect
> a single alternative (denoted w), must socially order the
> alternatives in a "natural" way (i.e., consistent with the manner in
> which w is elected), and for all x in A such that V(x,w) > V(w,x)
> there must exist a sequence of alternatives a1,a2,...,ak in A such
> that a1 = w and ak = x and all three of the following conditions
> hold:
>
> 1. (aj,a(j+1)) is at least as large as (x,w) for all j in {1,2,...,k-1}.
> 2. V(aj,a(j+1)) > V(a(j+1),aj) for all j in {1,2,...,k-1}.
> 3. aj is socially ordered over a(j+1) for all j in {1,2,...,k-1}.
where "natural" is explained as:
> For instance, the following six techniques are fairly obvious:
[...]
> 3. For procedures which compute a transitive binary relation on the
> alternatives, it is natural to use that relation (or possibly its
> reverse) as the social ordering.
>
> 4. For procedures which distinguish an acyclic subset of the pairwise
> majorities, it is natural to construct a social ordering consistent
> with the acyclic subset.
Now, you claim that River cannot be considered to fulfil this by
extending it to construct a social ordering. But I only suggested to do
exactly what you call technique 3 above: Take as the social ordering the
river diagram, which is an acyclic subset of the pairwise majorities and
depicts an antisymmetric and transitive relation, i.e., an order
relation, and it perfectly fulfils your criterion! Just as in MAM, and
that's no surprise since River is very similar to Ranked Pairs.
However, you may respond that there is also a stronger definition on
your site which River does not fulfil since it refers to defeats against
candidates other than the winner. However, I neither understand its
motivation nor did I refer to that stronger version but to the one on
the starting page of your site.
As to "immunity from second-place complaints", River does *not* fulfil
that, as the following example shows (taken from the 104 examples): the
defeat strengths (B>C)>(A>C)>(C>D)>(D>A)>(B>D)>(A>B) lead to the
affirmation of B>C>D>A, when B is removed we affirm A>C>D, but A defeats B.
Yours, Jobst
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