On Jun 24, 2011, at 3:13 AM, Juho Laatu wrote:
On 24.6.2011, at 1.56, [email protected] wrote:
In sports round robin tournaments where burial strategy is
meaningless, what is the best measure of
defeat strength? In other words, if you were going to use, say,
beatpath to find a total order of the teams
from best to worst after the conclusion of a tournament, would you
use wv, margins, or something else
to measure defeat strength?
Let's think about basketball for example. Let W be the winning
points and L be the losing points from a
match between two teams. The margins defeat strength would be W-
L. The wv defeat strength would
be just W. Another possibility would be the point ratio W/L or
equivalently log(W) - log(L). Between
margins and the point ratio would be sqrt(W) - sqrt(L), which
seems better to me from an intuitive point
of view.
What rational basis could we use to decide between these measures?
We should understand how the results of the winner selection
algorithm relate to our real world needs. A good explanation and
justification is what we need.
The basic explanation behind margins is "number of required extra
votes to win everyone else pairwise". For the proportion / ratio
approach the corresponding explanation would be "proportion/ratio of
required extra votes to win everyone else pairwise". If people think
that margins is too much on one side and ratio is too much on the
other side, then they might be happy with the sqrt approach. I don't
have a good natural explanation available for it. Maybe there is one
(a bit more complex than for the other two) (maybe below in your
mail). The explanation for winning votes is also a bit more complex
and a bit one-sided (since it does not count the defending votes). I
think winning votes have some strategic defence flavour and implicit
approval flavour in it (but others may have different viewpoints on
what is the best explanation for winning votes).
my spin on why Margins makes the most sense is:
1. the margins of logs is sorta the same as the percentage defeat
strength. in an election with 100 voters, a 55/45 defeat is precisely
as strong (as indicative of the voter sentiment) as a 550/450 defeat
in an election of 1000 voters. both represent the same degree of
rejection by voters of the 45% candidate.
2. the measure of importance of an election is proportional the
number of voters participating in it. if very few people weigh in on
an election, it must not be very important.
3. so, just like in the beginning of the Dead Poets Society, we
multiply these two measures of import together to get a composite
measure of import. if in a Condorcet cycle you have one pairwise
election with a 10% margin and 100 voters (a margin of 10 votes) and
another that is 5% of 400 voters (a margin of 20 votes), while the
later displays a smaller degree of rejection of voters it involves so
many more voters that it speaks more loudly than the result of the
former.
4. simplicity has its attraction.
5. Winning Votes communicates something regarding the number of
voters participating, but says nothing about how close the election
was. an election with a razor-thin result, even with a lot of people
voting, does not measure well the will of the people. if it's 99,999
voters and 50,000 said Candidate A and 49,999 said Candidate B, that
does *not* say that Candidate A has such a great mandate to lead.
Margins says his/her mandate to lead is 1 and Winning Votes says it's
50,000. how can that make any sense?
if there are N candidates, the number of pairs is N*(N-1)/2, there are
two vote totals per pair, so the set of vote totals is N*(N-1). the
entire election is decided solely on those N*(N-1) numbers. if there
is a CW, the entire election is decided on the set of margins (again,
back to half: N*(N-1)/2), in fact with a CW it's based on the signum
function of those margins: N*(N-1)/2 bits of information (ignoring
ties).
again, y'all heard me bitch about the stupid rectangular "Defeat
Matrix". i think the layout should be a triangle arranged in such a
way that the UR half of the defeat matrix is folded down onto the LL
half. the winning candidate of a pair is listed above the loser for
each pair in one cell of the "Defeat Triangle". the election can be
survey with a glance (whereas with the stupid defeat matrix, you have
to look across the matrix to see the other salient number and then you
have to remember which number goes with which candidate). if margins
are used, each pair of numbers can be replaced with a single number,
the margin, and the entire election is decided simply by those N*(N-1)/
2 numbers and the pair ordering that comes with them.
i think the stupid rectangular Defeat Matrix should be ditched and
replaced with a Defeat Triangle with N*(N-1)/2 cells and three numbers
in each cell: Winner total voters, Loser total votes, and the W-L
margin.
whether it's Schulze, Ranked Pairs, or MinMax (using margins), the
determination of the election result is based on the same N*(N-1)/2
numbers.
--
r b-j [email protected]
"Imagination is more important than knowledge."
----
Election-Methods mailing list - see http://electorama.com/em for list info
