[EM] second-order pairwise?

[James Green-Armytage]

>
>I hope your not angry on me since I argue so fiercly against ratings. 

	No, I'm not angry. Disagreement does not necessitate anger for me. I'm
glad that you are still interested in cooperation.

>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:
>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!
>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!

	This is a very interesting idea. I'm not positive that it can hold
together, but so far I have not been able to find a weak point. If it
holds together, it could be an excellent method.
>
	When I started work on strength-of-preference Condorcet, back in March,
my first idea was very close to this. But I didn't follow through on it. I
thought of second-order comparisons, and recognized the possibility of
second-order cycles (I can show you a couple if you like), but I didn't
get the idea of creating a transitive ordering of the defeats and then
plugging the values back into the first-order matrix. It's really quite
ingenious.
	The original name I had for the strength-of-preference Condorcet project
was "prioritized Condorcet" because it involved voters giving some
preference pairs a priority over others. But after a series of frustrating
and complex methods based on prioritized pairs (looking at my notes from
that time I find awful terms like "gradient equalizations", "static logic
gates", and "autostrategy"), I adopted a more cardinal approach,
eventually leading to my June 8 proposal.

>I sincerely hope we can perhaps develop such an approach a bit further
>by combining our efforts!
>
	Okay. I have spent several, several hours today looking at this new
method, and all I can really say so far is that I think it makes better
sense to look at the margins of defeat rather than just the winning side,
when in the SECOND-order matrix. I found one example where this change
prevents the success of a pretty severe strategic incursion. The incursion
could have also been prevented by enforcing a sort of second-order
transitivity, but I assume that you would be resistant to such a
restriction, so I won't insist on it unless I can find a situation where
strategy cannot be avoided otherwise.
	I'll do what I can to keep developing it, but I should say that this is a
very difficult method to evaluate because it is so complex to tally. Doing
just a four candidate cycle means looking at six beats, which means doing
15 beat-to-beat comparisons. Oy. (Yes, I did that once today... and guess
what? It was a tie.) I have found a couple situations where this method
gives a different result from weighted pairwise, but so far, such
situations have been in a distinct minority to situations where they
produce the same result. Which is in my eyes a good sign for this new
method, if you don't mind my saying so.
	Anyway, you should give this new method a name. Something like
second-order pairwise might sound sort of cool. 2OP for short...
>
>What do you think, could this be a compromise?
>
	It's possible. I don't intend to stop advocating weighted pairwise as my
most-preferred method, but I will keep an open mind to this new one as
well. Perhaps there will come a time when I will promote both of them.

my best,
James Green-Armytage