[EM] thoughts on weighted pairwise

[Jobst Heitzig]

Dear James!

I hope your not angry on me since I argue so fiercly against ratings. I
still agree with you that preferences can have different "strengths"
which we could try to take into account. I just think that we should do
it somewhat differently. In the last days I tried to find a way to use
the nice mechanism of weighted pairwise without using ratings and
without comparing the "strengh" of preferences of *different* voters,
since I think two preferences A>B and B>C can only be compared by
strength in a meaningful way when they belong to the preferences of the
*same* voter.

So let me suggest that we try to do it like this:
	1. Let the voters express binary preferences, and let them express that
some of these preferences are more important than others if they want to
express this. The explanation of "A>B is more important than C>D" would
be that in case that these two preferences become competitive in a
situation where the method tries to resolve a majority cycle, then the
more important preference should have precedence. I will use the obvious
notation (A>B)>(C>D) and (A>B)=(C>D) here.
	2. Now when it comes to resolving a majority cycle, say A>B>C>D>A,
determine not the "absolute" strength of each defeat by using either the
margin or the winning votes or the difference in utility, but determine
the "relative" strengths of each pair of defeats. More precisely, let us
say that the defeat A>B "beats" the defeat C>D when...
	the number of voters who express A>B and
		either don't express C>D
		or express C>D as a weaker preference than A>B
	is larger than
	the number of voters who express C>D and
		either don't express A>B
		or express A>B as a weaker preference than C>D.
>From these "beats" determine which defeat should be dropped!

In your example:

26: B >> D  > K
22: B >> K  > D
26: D  > K >> B
 1: D  > B  > K
21: K  > D >> B
 4: K  > B  > D

would give majorities B>D>K>B
and the following "relative strengths of defeats":

B>D vs. D>K: 52 vs. 27 since the first 26 have (B>D)>(D>K),
	hence B>D "beats" D>K.
D>K vs. K>B: 27 vs. 51 since the other 26 have (K>B)>(D>K),
	hence K>B "beats" D>K.
K>B vs. B>D: 51 vs. 52 since the 4 who express both preferences
	did not express different strengths,
	hence B>D "beats" K>B.

So D>K is the unique weakest defeat here just as in weighted pairwise.
But in the above way, the strength of voter i's preference K>B does
never get mangled with the strength of voter j's preference B>K but is
only compared to the strength of its other preference D>K!

I sincerely hope we can perhaps develop such an approach a bit further
by combining our efforts!

Unfortunately, the "beats" between defeats may give cycles, so there
must be some means to resolve these cycles of "higher order" to
determine which defeat should be dropped... For example, we could define
the strength of each beat as the number of people supporting the beating
defeat, then apply Tideman to the beats using these strengths in order
to determine a linear ordering among the defeats, and finally apply
whatever base method (beat path, river, or again Tideman) to the defeats
using this order!

What do you think, could this be a compromise?

Yours,
Jobst