[EM] weighted pairwise method: clarification

[James Green-Armytage]

Dear election methods fans,

	I believe that I was overly terse when I stated the definition of the
methods I proposed yesterday. I think that I made it pretty easy to
misunderstand. So I'm trying to rewrite the central part where I actually
define the method. How about this...?

<beginning of definition>

Weighted pairwise comparison method

Ballots:
1. Ranked ballot. Equal rankings are allowed.
2. Ratings ballot. e.g. 0-100, whole numbers only. Equal ratings allowed.
Note: You can give two candidates equal ratings while still giving them
unequal rankings. However, if you give one candidate a higher rating than
another, then you must also give the higher-rated candidate a higher
ranking.
 
Tally:
1. Pairwise tally, using the ranked ballots only. Elect the Condorcet
winner if one exists.
	If no Condorcet winner exists:
2. Determine the direction of the defeats using the ranked ballots for a
pairwise comparison tally.
3. Determine the strength of the defeats by finding the weighted magnitude
as follows. We’ll say that the particular defeat we’re considering is
candidate A beating candidate B. For each voter who ranks A over B, and
*only* for voters who rank A over B, subtract their rating of B from their
rating of A, to get the marginal utility. The sum of these winning
marginal utilities is the total weighted magnitude of the defeat.
4. Now that the directions of the pairwise defeats have been determined
(in step 2) and the strength of the defeats have been determined (in step
3), you can choose from a variety of Condorcet completion methods to
determine the winner. Beatpath and ranked pairs are my preferred choices.

<end of definition>

	The main idea is that the *direction* of the pairwise defeats are
determined by the rankings information, and the *strength* of the pairwise
defeats are determined by the ratings information.
	Step 2, the part where you determine the direction of the defeats, is
nothing new. It's just an ordinary pairwise tally.
	Step 3 is a bit trickier. Let's say that we are dealing with a defeat of
A over B. We already know that A has won the pairwise comparison, and now
we are only concerned with the strength of the A-->B defeat. The tricky
thing about step 3 is that we're only looking at ballots which rank A over
B, that is the "winning votes". For each of those ballots that rank A over
B, we're finding the difference in ratings between A and B. For example,
if A is rated 85 and B is rated 80, the difference, or marginal utility
for that voter, is 5. 
	(Under the rules I've listed above, it's possible that a voter will rank
A over B, but give both A and B a score of, say, 70. In that case, the
vote would count towards the strength of the defeat, except for the fact
that 70-70=0. However, what is *not* possible is to have a negative number
among the winning marginal utilities. If a voter ranks A over B, then that
voter must give A a greater or equal rating to B. And if a voter doesn't
rank A over B, then they cannot contribute to the strength of the defeat.)
	Anyway, you sum up the marginal utility of A over B, for all of the
voters who rank A over B. This is the sum of the winning marginal utility,
which is exactly the number that we are using to determine the defeat
strength. Once we've found the sum of winning marginal utilities for each
of the pairwise beats, then we can move on to step four.
	However, I just want to emphasize one last time that, for A's beat over
B, we're not looking at the voters ballots who ranked B>A or A=B, only
those who ranked A>B. For the voters who ranked B>A, you could sum the
marginal utility of B over A, and call it the "losing marginal utility".
In fact the losing marginal utility may be greater than the winning
marginal utility, but it still doesn't matter; the direction of the A-->B
defeat remains unchanged. This method doesn't look at the losing marginal
utility. Actually, I think that if you compared the winning marginal
utility against the losing marginal utility and determined the direction
of the defeat by which one is larger, you'd basically be back to using
plain old cardinal ratings. Which is okay in itself, I guess, but that's
not what I'm going for with this method. This is supposed to be a hybrid
between Condorcet and cardinal ratings, which maintains the 'integrity' of
the pairwise comparisons themselves while using the ratings to determine
their relative strength.
	I am anxious to read your questions and comments on this method. Please
let me know if the proposal still seems unclear.


my best,
James Green-Armytage