[EM] FWD - On determining who gets eliminated

[Instant Runoff Voting supporter]

Greetings EM List,

     The following letter belongs on this list, lots of numbers, examples,
and other EM type good stuff.

Don

  ------------ Forwarded Letter ---------------
From: "Edward Schaefer" <ems57@
Date: Mon, 18 Dec 2000
Subject:  On determining who gets eliminated

Although I like the idea of the instant-runoff, I must question the
rule of eliminating the person with the fewest first-choice votes.
This is because there could be a universal second-choice candidate
who would win if he continued in the process, but instead is being
prematurely eliminated.  For example, let there be a contest with 3
candidates and 100 voters whose choices go like this:

       A     B     C
1st   40    40    20
2nd   20    10    70
3rd   40    50    10

In this case, C gets second choice support from 35 of both A's and
B's supporters, meaning that in either an A-C or a B-C runoff, C wins
55-45.  However, as the smallest first-place vote getter, C is
elimianted, and A will win becuase he gets 15 of C's votes while B
gets only 5.

To prevent such an occurance, I suggest that all of the votes a
candidate gets be considered in the elimination decision, with the
votes being weighted based on the choice number as follows:

1st = 1, 2nd = 0.7, 3rd = 0.4,  4th = 0.2, 5th = 0.1.

(There is a mathematical formula behind this.  I will present it
later in this message.)

In this case, for elimination puposes, the weighted totals are:

For A: (40 * 1) + (20 * .7) + (40 * .4) = 40 + 14 + 16 = 70.
For B: (40 * 1) + (10 * .7) + (50 * .4) = 40 +  7 + 20 = 67.
For C: (20 * 1) + (70 * .7) + (10 * .4) = 20 + 49 +  4 = 73.

So in this scheme, B is eliminated and C goes on to win.

The result is more dramatic when a rule is adopted that in an n
candidate round the n-th place votes are weighted at 0.  Then A's
wighted total becomes 54, B's 47, and C's 69.  It also turns out that
in either variety of this scheme, a decrease in second choice support
will not eliminate C in the first round until after A has become the
winner of the A-C runoff.  Since the A-B winner is always B in this
case, that is proper and desirable.

I hope that this can be incorporated into the IRV system in the
future.

                            EMS

Notes:

The mathematical formula for the wieghts is as follows:

w = n sqrt(c!), where

w is a weghted sub-total,
n is the number of c-th choice votes cast for a candidate,
c is the chice number (first = 1, second = 2, etc.)
sqrt is the sqaure root function, and
! id the factorial function (which is the produce of all numbers 1
through n inclusive.  So 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc.)

The single vote wieghts (to 4 decimal places) under this formula are:

1st = 1 (exactly), 2nd = 0.7071, 3rd = 0.4083, 4th = 0.2041,
5th = 0.0913, etc.

The goal was to find a scheme that did not underwieght higher
preference or overwieght the lower ones.  I belive that this rule
achieves that.


When an n-th place choice is elminated, the following choices are
moved up for elimination purposes:  So if you lose your second place
choice, your third place choice gets the second choice weight in the
next round.


I would discourage duplicate choice voting (such as two first
choices).  If it does occur, then I would count *neither* vote for
selection purposes in a round where both candidates are still alive,
since no choice has been made; and for elimination purposes I would
treat both as second choices.  (Note that a second choice vote on
that ballot then initially acts as a third-place choice or worse.)
However, do note that in a round where the other 1st place choices
have been eliminated, the remaining candidate is given a first place
vote from that ballot.

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