Actually not all Condocet methods are nicely summable, although all
typical (commonly used) ones are. For example a method that uses IRV
if there is no Condorcet winner is not "nicely summable". Maybe
Condorcet methods that can find the winner based on the pairwise
comparison matrix only should be classified as one specific subset of
Condorcet methods. Some methods may add e.g. an explicit (approval)
cutoff to this, but in most cases this kind of additions are still
"nicely summable".
Juho
On Feb 3, 2010, at 11:31 PM, robert bristow-johnson wrote:
it might be the case that my "ASCII math" didn't translate okay
through the EM list server or in whatever mail client program. i
meant for it to be viewed with a mono-spaced font.
it looks okay on my client (which seems to know how to undo word-
wrapping that it put in), but i took a look the email returned by
the EM list server, and it appears to have wrapped some of the
longer lines of text. in the future, i have to limit my line length
to 70 characters.
the case below allows equality only for the unranked candidates who
are tied for last place. for IRV or STV it's pretty hard to
consider equally ranked candidates. if one marked two candidates
equally and their ranking was eventually promoted to the top, where
it gets counted, how much does it count for each candidate? 1 vote
or 1/2 vote? i think that is why it is not allowed in the IRV
method i am familiar with.
if the ranked ballots were used for Borda (which i don't like) or
Condorcet (which i do like), then there are more natural (and fewer)
"precinct summable" tallies to keep track of.
this all just started with my anal-retentive need to establish how
many IRV piles one would have to maintain to have "precinct
summability". i am still convinced that for 3 candidates, the
number is 9 (not 15) and for 4 candidates, you would need 40 piles
(it *does* grow pretty rapidly).
--
r b-j [email protected]
"Imagination is more important than knowledge."
-----Original Message-----
From: "Kristofer Munsterhjelm" [[email protected]]
Date: 02/03/2010 14:24
To: "robert bristow-johnson" <[email protected]>
CC: "election-methods List" <[email protected]>
Subject: Re: [EM] IRV ballot pile count (proof of closed form)
robert bristow-johnson wrote:
On Feb 2, 2010, at 2:28 PM, robert bristow-johnson wrote:
Warren tells me that
C-1
SUM{ C!/n! }
n=1
has a closed form, but didn't tell me what it is. does someone have
the closed form for it? i fiddled with it a little, and i can
certainly see an asymptotic limit of
(e-1)(C!)
as C gets large, but i don't see an exact closed form for it. if
someone has such a closed form, would you mind sharing it?
Okay, I spent a little time working on this and figgered it out. The
fact that the number of distinct piles needed to represent all
possible
manners of *relatively* ranking C candidates (no ties except unranked
candidates are tied for lowest rank) is
C-1
SUM{ C!/n! } = floor( (e-1) C! ) - 1
n=1
Now I wonder if there's a closed form for the number of orders with
both
equality and truncation permitted. Since I don't quite get the
proof, I
can't answer, though!
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