[EM] IRV ballot pile count (proof of closed form)

[Abd ul-Rahman Lomax]

At 01:08 PM 2/10/2010, Dave Ketchum wrote:
>Condorcet does an N*N matrix showing for EACH pair of candidates which
>is better liked - used in counting and usable by others to help plan
>their future.  Often there is a CW which wins for winning in all of
>its pairs; else a cycle in which each would be CW if other cycle
>members were not candidates.
>      How best to resolve a cycle is debatable, but a simple method
>could be used unless others are demonstrated to be much better:
>Delete weakest pair used to define the cycle; repeat until remainder
>defines a CW.
>      Note that N*Ns show progress, or lack of such, among non-winners.

Often overlooked is that Condorcet methods, if truncation is allowed 
(and voting without truncation being allowed tends to input a lot of 
noise), is that it they are plurality methods, unless used with 
special rules, which I've never seen anyone buy myself propose.

Consider the following votes:

34 A
33 B>C
33 C>B.

The Condorcet winner is A, because in the two pairwise elections 
involving A, A wins

A>B, 34:33
A>C, 34:33.

However, A certainly does not have a majority. This is a problem 
entirely apart from the issue of cycles. Note that a majority winner 
is always a Condorcet winner.

In the election above, almost two-thirds of the voters are actually 
voting against A. A could be a *lousy* result. Or not. Can't tell.

I just noticed that while Wikipedia has many articles on voting 
systems, it doesn't list as a voting system what is commonly used by 
democratic organizations, probably most commonly! Repeated ballot 
until a majority is found for the winner. No eliminations, the 
election process is repeated, with new nominations allowed -- and, of 
course, withdrawals are also allowed.

Basically, seeking a majority and not insisting on finding a winner 
in a single ballot, can make Condorcet almost irrelevant.

(But I find it quite relevant in determining featured candidates in 
runoff elections; in my view, a Condorcet winner should *always* be, 
if not the winner, at least featured in a runoff election, for 
optimal overall results. But some algorithms may make a runoff 
unnecessary, i.e., the possible improvement in social utility from 
holding a runoff *might* be so small as to make it unnecessary. And 
I'd vastly prefer much more collection of data on real elections that 
do collect much more information than is on a "plurality" ballot, 
than coming to some fixed conclusion about that, snatched out of thin air.)