[EM] James--CP, AERLO, ATLO, defensive strategy definiltion

[Markus.Schulze at alumni.TU-Berlin.DE]

Dear Mike,

you wrote (27 March 2005):
> I told Markus that I was going to define majority
> rule soon. My definition of majority wishes is
> similar, and I guess that I'd better state that
> definition now, instead of being vague about what
> I mean by majority wishes and majority rule.
>
> If a majority prefer X to Y, that's a majority
> pairwise preference (MPP). The strength of that
> MPP is measured by the number of voters who prefer
> X to Y.
>
> An MPP for X over Y is outdone if there is a sequence
> of MPPs from Y to X, consisting of MPPs that are all
> at least as strong as the MPP of X over Y.
>
> To violate majority wishes means to elect someone who
> has an MPP against him that isn't outdone.
>
> Protecting majority wishes means avoiding a violation
> of majority wishes.
>
> [end of definition of protecting majority wishes]
>
> Majority rule:
>
> X has a majority pairwise vote against Y if a majority
> vote X over Y.
>
> Substituting majority pairwise vote for majority pairwise
> preference in the definitions above leads to a definiltion
> of majority rule instead of majority wishes.
>
> Well, it's better to say it explicitly:
>
> X has a majority pairwise vote (MPV) against Y if a
> majority vote X over Y.
>
> An MPV's strength is measured by the number of people
> who vote X over Y.
>
> An MPV for X over Y is outdone if there's a sequence of
> MPVs from Y to X consisting of MPVs that are all at least
> as strong as the one for X over Y.
>
> Violating majority rule means electing someone who has
> an MPV against him that isn't outdone.
>
> [end of definition of violating majority rule]

Such criteria have already been proposed in the past.

Suppose V is the number of voters.

Suppose d[X,Y] is the number of voters who
strictly prefer candidate X to candidate Y.

Suppose p(z)[X,Y] is the strength of the strongest
path from candidate X to candidate Y when the strength
of a pairwise defeat is measured by "z" (e.g. "z" =
"margins", "z" = "winning votes", "z" = "votes against").

Then I proposed the following criterion in 1997:

   If p(wv)[A,B] > V/2 and p(wv)[B,A] < V/2,
   then candidate B must be elected with zero
   probability.

Steve Eppley proposed the following criterion in 2000:

   If d[A,B] > V/2 and p(wv)[B,A] < V/2,
   then candidate B must be elected with zero
   probability.

Markus Schulze