Election Methods Poll (final draft?)
Steve Eppley
seppley at alumni.caltech.edu
Sat Sep 28 12:41:53 PDT 1996
Which single-winner voting method(s) should electoral reformers advocate?
This report was produced by the subscribers of election-methods-list.
--Steve Eppley
--------------------------------------------------------
Message Contents:
1. Intro: election-methods-list and its poll about voting methods
2. The seven responses to the poll
3. Compact summary of responses
4. Tallying of the responses by Condorcet, Smith-Condorcet,
and Regular-Champion
5. Tallying of the responses by MPV (a.k.a. Instant Runoff)
6. Glossary: Definitions of some of the single-winner voting methods
--------------------------------------------------------
1. Introduction
The election-methods-list at eskimo.com (EM) maillist spun off from
the elections-reform (ER) maillist early in 1996 because some ER
subscribers didn't want to receive discussions about the pros and
cons of particular voting methods. Most of the discussion in EM has
been about voting methods for "single-winner" elections: elections
where one candidate wins all, such as single-member legislative
districts, presidents, senators, governors, mayors, rival ballot
initiatives, rival proposals in committees and legislatures, etc.
Some ER and EM subscribers believe that reforming the voting method
used in single-winner elections rivals proportional representation as
an important and urgent goal, that it may be easier to reform
single-winner elections than win proportional representation at the
national level (and maybe easier also at the state level since
switching to a better voting method is probably less controversial
than restructuring the government), and that reforming single-winner
elections could help pave the way for proportional representation.
Like proportional representation, good voting methods make multiparty
competition more feasible, since third party candidates are freed to
run without fear of throwing the election to a greater evil and
voters are freed to vote their true preferences without throwing the
election to a greater evil. (It's this "lesser of evils" dilemma
which holds the two "big tent" parties together.)
After extensive discussion of single-winner voting methods, the
subscribers of EM 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; one subscriber 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 subscribers of EM, 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; expect to find the pro and con arguments in that
message.
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 MPV (a.k.a. Instant Runoff). Few reformers
know about Condorcet or Smith-Condorcet, however, since "pairwise"
methods were impractical for large elections with many candidates
before 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,
these methods support majority rule and eliminate the "lesser of evils"
dilemma better than Instant Runoff, and therefore are best at promoting
multiparty competition and electing the popular choice.)
During the tallying phase, 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 last with any other methods they left unranked).
------------------------------------------------------------------
2. 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 and 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 REST------
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 REST--------
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 REST-------
6 Instant-Runoff-1
7 Instant Runoff
8 Approval
9 all the methods omitted from this response
10 Runoff
11 Double Complement
12 Plurality
-----------------------------------------------------------
3. Compact Summary of Responses
Here's a more compact notation, 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
NOTR None of the Rest (This is not a method; it marks one's
dividing line between approval and 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".
Here are the responses using the compact notation:
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 > NOTR
Rob: SC > C > RC > SCw > Cw > IR > DC > A > NOTR > P > SR
Steve: SC > SCw > C > Cw > NOTR > IR1 > IR > A > the_unranked > R > DC > P
-----------------------------------------------------------
4. 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.) See the next section for the
tally performed using Instant Runoff.
There are 16 choices (15 methods plus the NOTR approval/disapproval
dividing line) included in our ballots, so that means there are
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 NOTR
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.
-------------------------------------------------------------
5. 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, NOTR, 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 > NOTR
Rob: C > RC > SCw > Cw > IR > DC > A > NOTR > P > SR
Steve: SCw > C > Cw > NOTR > IR1 > IR > A > the_unranked > R > DC > P
Round 1:
Eliminate D2, IR1, SR, P, R, DC, RC, NOTR, 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.
------------------------------------------------------------------
6. Glossary: Definitions of single-winner voting methods and terms
[This section is incomplete. It includes definitions of the most
important methods and terms, however.]
Approval:
Each voter can approve (vote Yes) as few or as many candidates
as s/he likes. The winner is the candidate with the most approvals.
Condorcet:
This is one of the pairwise methods. See the definition of
"Pairwise Methods" 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 3 candidates X, Y, and Z are running and
100 voters cast ballots. Suppose the ballots are:
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
(a message in the EM list)
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:
Similar to the Runoff and Plurality methods, each voter may vote
for only one candidate. If a candidate wins a majority, s/he is
elected. A plurality is also enough to win if the number of votes
for the second-place finisher is sufficiently small; but if the
second-place finisher receives substantial votes then a two-candidate
runoff election will be held in which the top two finishers compete.
Instant Runoff and Instant-Runoff-1:
The voters use ranked ballots. The tally iterates, successively
eliminating the candidate(s) ranked by fewest ballots as the most
preferred of the remaining candidates, until one candidate has a
"majority".
The difference between Instant Runoff and Instant-Runoff-1 is what
to do with ballots in which some candidates are ranked as equals:
Instant Runoff splits the weight of the ballot between the equals,
and Instant-Runoff-1 gives a full vote to each of the equals.
(Another variation would not allow voters to rank candidates as
equals, and invalidate any ballots that violate this restriction.)
None of the Rest (NOTR):
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 NOTR and disapproves of all the choices which s/he
ranked less preferred than NOTR. Its use by some of the respondents
in this poll does not imply a recommendation that NOTR 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 "virtual 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.
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. For example, suppose the ballots
are:
46 voters: 1=X 2=Y 3=Z
20 voters: 1=Y 2=X=Z
34 voters: 1=Z 2=Y 3=X
There are 54 (20+34) voters who prefer Y more than X.
There are 46 voters who prefer X more than Y.
Since 54 is greater than 46, Y pairwise-defeats X.
If there is one candidate who pairwise-defeats all the other
candidates, this candidate is the winner.
For example, suppose the ballots are as above:
46 voters: 1=X 2=Y 3=Z
20 voters: 1=Y 2=X=Z
34 voters: 1=Z 2=Y 3=X
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 a candidate who pairwise-defeated all the others. See
the definitions of the particular methods (Condorcet, Regular-
Champion, Smith//Condorcet, Smith//Random, etc.) for those details.
Plurality (a.k.a. vote-for-only-one, a.k.a. First Past The Post)
Each voter may vote for only one candidate. The winner is the
candidate who receives the most votes.
Rated Ballot methods: See "Weighted Ballot methods" below.
Regular-Champion:
This is one of the pairwise methods. See the definition
of "Pairwise Methods".
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, Regular-Champion elects
the candidate with the largest number of pairing-wins less
pairing-defeats. If more than one candidate ties for first
according to this measure, the tie-breaker is to pick the one
who is the first choice of the most voters (plurality).
For example, suppose 3 candidates X, Y, and Z are running and
100 voters cast ballots. Suppose the ballots are:
46 voters: 1.X 2.Y=Z
20 voters: 1.Y 2.X=Z
34 voters: 1.Z 2.Y 3.X
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.
All three candidates have one pairing-win and one pairing-defeat,
so the plurality tie-breaker is used:
X is the first choice of 46 voters.
Y is the first choice of 20 voters.
Z is the first choice of 34 voters.
Since the largest of these is 46, X wins.
Note that these are the same ballots as in the example used above in
the description of Condorcet, but a different candidate is elected.
Runoff:
Each voter may vote for only one candidate. If a candidate
receives the votes of a majority of voters, that candidate wins.
Otherwise, a new election is held in which only the top two
finishers compete.
Smith:
This is a part of several pairwise methods:
Smith-Condorcet
Smith-Condorcet with weighted (rated) ballots
Smith-Condorcet-Tobin
Smith-Random
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 this group. The "best" group will 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
contention. If the "best" group includes only one candidate, it
means that this candidate pairwise-defeated all the other candidates,
so this candidate is the winner. If the "best" group contains more
than one candidate, the winner is chosen according to the particular
method:
Smith-Condorcet: The winner is the candidate in the "best" group
whose largest pair-defeat is the smallest. (See Condorcet.)
Smith-Condorcet with weighted (rated) ballots
Smith-Condorcet-Tobin
Smith-Random:
The winner is randomly selected from the "best" group.
Weighted Ballot methods:
The voters are asked to rate each candidate using a numeric scale,
such as 0 to 100, -10 to +10, etc. The simplest way to tally such
ballots is to sum for each candidate the voters' ratings of the
candidate, and elect the one with the largest sum. None of the
seven respondents approves of that method, however: any algorithm
which tries to elect based on the "intensity" of the voters'
preferences will inescapably create an incentive for voters to
exaggerate the intensity of their preferences.
The only weighted ballot methods under consideration in this poll
are ones which ignore the "intensities" by converting the ratings
to ranked ballots.
--------------- End of Report -----------------
More information about the Election-Methods
mailing list