Report from the Single-Winner Committee (draft)
Steve Eppley
seppley at alumni.caltech.edu
Fri Aug 16 13:41:21 PDT 1996
The subscribers of election-methods-list at eskimo.com (a.k.a. the EM
list), after extensive discussion of voting methods which could be
used in single-winner elections, polled ourselves on the question:
"Which single-winner voting method(s) should electoral reformers
advocate?"
Seven subscribers posted responses. Six ranked their recommendations
from most preferred to least preferred; the seventh listed only one
recommendation. Some subscribers didn't post a response.
Below is a summary of the responses, followed by tallies of the
responses using several of the single-winner algorithms which were
themselves advocated by members of the EM list, followed by
definitions of the voting methods which were the subject of the
poll. An additional message will follow (soon, hopefully) containing
commentary from any subscriber of EM who desires his/her comments to
be included.
No matter how the responses were tallied, the top two recommendations
are the same:
1. Smith-Condorcet
2. Condorcet
(See the definitions section at the bottom if you don't yet know how
these methods work.) The recommendation order of the rest of the
methods depends on how the responses are tallied.
Though only seven people provided responses, the results may be
significant: the poll followed months of detailed discussion about
the pros and cons of many methods. Most electoral reformers who have
learned a little about single-winner reforms know about and advocate
a method called Instant Runoff. Few reformers know about Condorcet
or Smith-Condorcet, however, since "pairwise" methods haven't been
practical for large elections with many candidates until the computer
age. This report may serve as a wakeup call to reformers, since there
are reasons why most of the respondents prefer Condorcet and
Smith-Condorcet more than Instant Runoff. (Briefly, they support
majority rule and eliminate the "lesser of evils" dilemma better
than Instant Runoff.)
During the tallying a new method was identified and dubbed
Instant-Runoff-1. (See the definitions section for more info.)
Two subscribers modified their responses to rank Instant-Runoff-1
more preferred than Instant Runoff (and more preferred than some
other methods, as well). The subscribers who didn't modify their
responses made no comments about Instant-Runoff-1; their original
responses were tallied as if they ranked Instant-Runoff-1 last
(equally with any other methods they left unranked).
------------------------------------------------------------------
Message Contents:
1. The seven responses
2. Compact summary of responses
3. Tallying of the responses by Condorcet, Smith-Condorcet,
and Regular-Champion
4. Tallying of the responses by Instant Runoff
5. Glossary: Definitions of the single-winner voting methods
The seven responses:
Here are the responses, listed in alphabetical order by first name:
Demorep1 at aol.com, 7-8-96 22:11:04 -0700 (PDT)
1 Demorep-1
2 Demorep-2
3 Instant Runoff
Donald Eric Davison, 7-4-96 11:01:33 -0700 (PDT)
1 Instant Runoff
Hugh R. Tobin, 7-7-96 00:21:19 -0700 (PDT)
1 Smith-Condorcet-Tobin
2 Smith-Condorcet
3 Smith-Random
4 Instant Runoff
5 Condorcet
6 Regular-Champion
7 Double Complement
8 Approval
9 all other methods
Kevin Hornbuckle, 7-8-96 22:12:01
1 Condorcet, Smith-Condorcet
2 Approval
3 Smith-Random
Mike Ossipoff, 5-26-96 23:41:56 -0700 (PDT)
1 Condorcet
2 Smith-Condorcet
3 Instant-Runoff-1
3 Approval
4 Smith-Random
5 -----NONE OF THE BELOW------
Rob Lanphier, 7-7-96 00:54:23 -0700 (PDT)
1 Smith-Condorcet
2 Condorcet
3 Regular-Champion
4 Smith-Condorcet using rated, not ranked, ballots
5 Condorcet using rated, not ranked, ballots
6 Instant Runoff
7 Double Complement
8 Approval
9 ------NONE OF THE BELOW--------
10 Plurality
11 Smith-Random
Steve Eppley, 6-2-96 04:40:44 -0700 (PDT)
1 Smith-Condorcet
2 Smith-Condorcet using rated, not ranked, ballots
3 Condorcet
4 Condorcet using rated, not ranked, ballots
5 ------NONE OF THE BELOW-------
6 Instant-Runoff-1
7 Instant Runoff
8 Approval
9 all the methods omitted from this response
10 Runoff
11 Double Complement
12 Plurality
* *
Here's a more compact summary, using the following table of abbreviations:
Abbr. Method
----- --------------------------
A Approval
C Condorcet
Cw Condorcet with weighted (a.k.a. rated), not ranked, ballots
D1 Demorep-1 (Approved-Condorcet-MostApproved ??)
D2 Demorep-2 (Approval of top n-1 ranked choices)
DC Double Complement
IR Instant Runoff (a.k.a. MPV, a.k.a. Hare's method)
IR1 Instant-Runoff-1
NOTB [not a method: it marks one's threshold of approval/disapproval.]
P Plurality (a.k.a. Vote-For-Only-One, a.k.a. First Past The Post)
R Runoff
RC Regular-Champion (aka Copeland-Plurality)
SC Smith-Condorcet
SCt Smith-Condorcet-Tobin
SCw Smith-Condorcet with weighted (a.k.a. rated), not ranked, ballots
SR Smith-Random
Notes:
The symbol '>' means "is preferred more than".
The symbol '=' means "is preferred the same as".
Demorep: D1 > D2 > IR
Don: IR
Hugh: SCt > SC > SR > IR > C > RC > DC > A
Kevin: C = SC > A > SR
Mike: C > SC > IR1 > A > SR > NOTB
Rob: SC > C > RC > SCw > Cw > IR > DC > A > NOTB > P > SR
Steve: SC > SCw > C > Cw > NOTB > IR1 > IR > A > the_unranked > R > DC > P
* *
Tallying the responses by Condorcet, Smith-Condorcet, and
Regular-Champion (some "pairwise" methods)
These three pairwise methods were heavily discussed and were most
favored by at least one subscriber, so the responses were tallied
using all three. (The subscriber who favors Regular-Champion did
not provide a response for this poll.)
There are 16 choices (15 methods plus the NOTB approval/disapproval
dividing line) included in our ballots, so that means 120 = 16*(16-1)/2
pairings to compute. All three pairwise tallies can make use of the
same 16x16 array. This section shows the array and the results
using Smith-Condorcet, Condorcet, and Regular-Champion.
The "Pairwise" Array
Notes:
1. The number in each cell of the array shows how many voters prefer
the choice at the row's left more than the choice at the column's
top. A blank cell means zero.
2. The suffix 'L' after a number denotes that the choice at the
column's top lost the pairing.
3. The suffix '=' after a number denotes a pairwise tie between the
choice at the row's left and the choice at the column's top.
IR1 IR RC C Cw SC SCw SCt SR A P DC R D1 D2 NOTB
IR1 - 2 2= 1 1 2L 2 2 2L 2= 2L 2L 2L 1
IR 4L - 4L 3 3L 2 3L 4L 4L 5L 5L 5L 5L 4L 4L 4L
RC 2= 1 - 2L 2L 1= 1 2 3L 3L 3L 2L 2L 2L
C 5L 4L 5L - 5L 1 4L 4L 4L 5L 5L 5L 5L 5L 5L 5L
Cw 2L 2 1 - 2L 2 2 2L 2L 2L 2L 2L 2L
SC 5L 5L 5L 3L 5L - 5L 4L 5L 5L 5L 5L 5L 5L 5L 5L
SCw 2L 2 1 1 2L - 2L 2 2 2L 2L 2L 2L 2L 2L
SCt 1 1 1= 1 1 1 1 - 1 1 2L 2L 2L 1= 1= 1
SR 3L 3 3L 1 3L 3L 3L - 1 4L 4L 5L 4L 4L 3L
A 4L 2 3L 3L 3L 4L 4L - 5L 3L 5L 5L 5L 4L
P 1 1 - 1=
DC 2= 1 1 1= 1 2 3L - 2L 2= 2= 2L
R 1= 1 -
D1 1 1 1 1 1 1 1 1= 1 1 2L 2= 2L - 1L 1
D2 1 1 1 1 1 1 1 1= 1 1 2L 2= 2L - 1
NOTB 2L 1 1 2L 2 1 2L 1 2L 2L 2L -
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
Wins: 5 13 8 14 8 15 9 2 11 12 1 3 3 2 6
Losses: 8 2 5 1 7 6 9 4 3 13 9 14 10 11 9
Ties: 2 2 4 1 3 1 2 2
LargestLoss: 5 5 5 3 5 5 4 5 5 5 5 5 5 5 5
Smith-Condorcet defeated all other choices pairwise, so it wins when our
responses are tallied by Smith-Condorcet, Condorcet, and Regular-Champion.
Condorcet finishes second according to Smith-Condorcet, Condorcet,
and Regular-Champion. Its "Condorcet score" (its largest pairwise
loss) of 3 is smaller than any but Smith-Condorcet. Its "Copeland
score" (pairwise wins minus pairwise losses) of 12 is larger than
any but Smith-Condorcet. And if the Smith-Condorcet choice were
removed from all the responses, Condorcet would defeat all the other
choices pairwise.
* *
Tallying the responses by Instant Runoff
(Note: During this tally, the question arose about how to treat a
ballot where more than one choice is listed in first place. Two
ways were identified: give an equal fraction to each choice, or give
each choice a whole vote. Instant Runoff gives a fraction, and the
variation which gives a whole vote has been dubbed Instant-Runoff-1.
The two EM subscribers who posted comments about this question
like Instant-Runoff-1 more than Instant Runoff; however, the
tally shown here uses Instant Runoff.)
Round 1:
Eliminate D2, IR1, SR, P, R, DC, RC, NOTB, A, Cw, SCw since they
are the first choice of nobody:
Demorep: D1 > IR
Don: IR
Hugh: SCt > SC > IR > C
Kevin: C = SC
Mike: C > SC
Rob: SC > C > IR
Steve: SC > C > IR > the_unranked
Round 2:
Eliminate SCt, IR, and D1 since they tie for least first-ranked votes:
Demorep: --wasted--
Don: --wasted
Hugh: SC > C
Kevin: C = SC
Mike: C > SC
Rob: SC > C
Steve: SC > C
Round 3:
Eliminate C since it has fewer first-rank votes than SC:
Demorep: --wasted--
Don: --wasted
Hugh: SC
Kevin: SC
Mike: SC
Rob: SC
Steve: SC
Result: Smith-Condorcet finishes 1st, with 5 votes, when the
responses are tallied by Instant Runoff.
Finding the second place finisher using Instant Runoff:
Method 1: Find the method eliminated last (see above):
C was eliminated last, so it finishes second.
Method 2: Delete the winner (SC) from the initial rankings:
Demorep: D1 > D2 > IR
Don: IR
Hugh: SCt > SR > IR > C > RC > DC > A
Kevin: C > A > SR
Mike: C > IR1 > A > SR > NOTB
Rob: C > RC > SCw > Cw > IR > DC > A > NOTB > P > SR
Steve: SCw > C > Cw > NOTB > IR1 > IR > A > the_unranked > R > DC > P
Round 1:
Eliminate D2, IR1, SR, P, R, DC, RC, NOTB, A, Cw, since they
are the first choice of nobody:
Demorep: D1 > IR
Don: IR
Hugh: SCt > IR > C
Kevin: C
Mike: C
Rob: C > IR
Steve: SCw > C > IR > the_unranked
Round 2:
Eliminate SCt, IR, D1, SCw since they tie for least first-ranked votes:
Demorep: --wasted--
Don: --wasted
Hugh: C
Kevin: C
Mike: C
Rob: C
Steve: C
Result: Condorcet finishes 2nd, with 5 votes, when the responses
are tallied by Instant Runoff.
* *
Glossary: Definitions of single-winner voting methods and terms
[This section needs to be completed before the report is posted to ER.]
Approval
Condorcet:
This is one of the pairwise methods (see below).
As in all pairwise methods, if there is one candidate who pairwise-
defeated all the other candidates, then this candidate is the winner.
If there isn't such a candidate, Condorcet elects the candidate
whose largest pairing-loss is the smallest (where the size of each
pairing-loss is the number of voters who prefer the pair-winner more
than the pair-loser).
For example, suppose the ballots were:
46 voters: 1=X 2=Y=Z
20 voters: 1=Y 2=X=Z
34 voters: 1=Z 2=Y 3=Z
There are 3*(3-1)/2 = 3 pairings:
X is pairwise-defeated by Y (46 to 54).
Z is pairwise-defeated by X (34 to 46).
Y is pairwise-defeated by Z (20 to 34).
There is no candidate who pairwise-defeated all the others.
The size of X's one pairing-loss is 54.
The size of Y's one pairing-loss is 34.
The size of Z's one pairing-loss is 46.
Since Y's largest pairing-loss (34) is smaller than X's (54) and
smaller than Z's (46), Y is the winner.
Condorcet with weighted (rated) ballots:
Demorep-1:
A mixture of rank voting, majority yes (approval), Condorcet
for the majority yes candidates, limited approval tie breaker
(limited combined choices).
See: Subject: Single Person Elections 6/27/96
Date: Fri, Jun 28, 1996 12:20 AM EDT
Demorep-2:
Approval with rankings (dropping the last ranked choice--such as
having 5 candidates, only top 4 choices of a voter would count--
highest majority candidate wins)
Double Complement
Instant Runoff
Instant-Runoff-1
NOTB:
This choice is not a method. Its insertion into a voter's rankings
indicates that the voter approves of all the choices s/he ranked more
preferred than NOTB and disapproves of all the choices which s/he
ranked less preferred than NOTB. Its use by some of the respondents
in this poll does not imply a recommendation that NOTB ought to be
allowable as a choice in elections for public offices; in principle,
good voting methods will encourage more good candidates to run (since
they won't face a spoiler dilemma), and if all the candidates are
disapproved by the electorate it might create a dangerous power
vacuum.
Pairwise Methods:
In all pairwise methods, the info contained in the ranked ballots
is used to calculate what the results would be in all the possible
2-candidate "runoffs", called pairings. If there are N candidates
running, there are N*(N-1)/2 possible pairings. For example,
suppose there are three candidates named X, Y, and Z. The number
of possible pairings is 3*(3-1)/2 = 3:
X vs. Y
X vs. Z
Y vs. Z
A ranked ballot such as 1=Y,2=X,3=Z is counted as a vote preferring
Y more than X in the X vs Y pairing, and a vote preferring X more
than Z in the X vs Z pairing, and a vote preferring Y more than Z
in the Y vs Z pairing (since a voter's rankings are assumed to be
transitive).
A candidate X "pairwise-defeats" another candidate Y if the number
of voters who prefer X more than Y is greater than the number of
voters who prefer Y more than X.
If there is one candidate who pairwise-defeated all the other
candidates, this candidate is the winner in all pairwise methods.
For example, suppose the ballots were:
46 voters: 1=X 2=Y 3=Z
20 voters: 1=Y 2=X=Z
34 voters: 1=Z 2=Y 3=Z
There are 3*(3-1)/2 = 3 pairings:
X is pairwise-defeated by Y (46 to 54).
Z is pairwise-defeated by X (34 to 46).
Z is pairwise-defeated by Y (34 to 66).
Y pairwise-defeated both X and Z, so Y is the winner.
Each pairwise method differs in how it determines the winner when
there isn't one candidate who pairwise-defeated all the others.
Plurality (a.k.a. vote-for-only-one, a.k.a. First Past The Post)
Rated Ballot methods: See "Weighted Ballot methods" below.
Regular-Champion
Runoff
Smith:
This is a part of several pairwise methods (below). The Smith step
separates the candidates into two groups ("best" and "worst"): all
the candidates in the "best" group pairwise-defeated all the candidates
in the "worst" group. (To be more precise, the "best" group is the
smallest group which can be found such that all the candidates in
the group pairwise-defeated all the candidates not in the group.
It must have at least one candidate in it, and it may have all the
candidates in it.) All the candidates not in the "best" group are
eliminated from further consideration.
Smith-Condorcet
Smith-Condorcet with weighted (rated) ballots
Smith-Condorcet-Tobin
Smith-Random
Weighted Ballot methods
--------------- End of Report (Draft) -----------------
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