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[Bruce Anderson]

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POSITION PAPER
A  PARTIAL  RANKING  OF  SELECTED  VOTING  
METHODS  BASED  ON  MAJORITY,  CONDORCET,  
AND  MONOTONICITY  CRITERIA

Lowell Bruce Anderson

Voice:  703-845-2148
  FAX:  703-845-2255
   e-mail:  landerso at ida.org
May 2, 1996

A.	INTRODUCTION
Reference [1] defines 7 criteria that can be applied to ranked-
ballot voting methods; it lists (but does not define) 34 voting methods, 
and, with two exceptions, it tabulates which voting methods satisfy, and 
which fail, each of these 7 criteria.  An essentially equivalent tabulation 
is given on page 6, below.  Nineteen of these 34 voting methods satisfy 
all 7 criteria; the other 15 fail one or more of them.  Arguably, the most 
interesting and important question [1] raises is how to choose among the 
many voting methods (whether listed in [1] or not) that satisfy all of 
these criteria.  However, several of the 15 voting methods that fail one or 
more of these criteria are quite well known, and [1] allows a comparison 
and ranking of these methods based on which criteria they pass and 
which they fail.  Of course, such a comparison and ranking is still 
judgmental, but the judgment is applied to criteria, not (directly) to 
voting methods, and then the comparison and ranking follow from 
applying the criteria to the various voting methods.  See the publications 
cited in Reference [2], and their references, for additional information on 
voting methods.
Section B, next, repeats the definitions of the criteria given in [1].  
Section C grades the voting methods based on which criteria they pass 
and which, if any, they fail.  Those that pass all 7 criteria are said to 
receive a grade somewhere between C and A+, inclusive.  This range 
allows future comparisons to more finely grade those voting methods.  
This lumping of those 19 voting methods into one "C-A+" category is 
why the ranking given here is called a partial ranking.
With one exception, the voting methods that fail one or more of 
these criteria will receive a specific grade between F- and C-, inclusive.  
The one exception is Approval voting, which is the only one of these 34 
voting methods whose winners cannot be determined directly from 
ranked ballots.  However, Approval voting would receive an F- if:  1) the 
voters cast ballots that contained sufficient information to determine 
Approval-voting winners,  2) the voters marked their ballots in a manner 
consistent with their true rank-order preferences, and  3) the criteria 
below are applied to these voters true rank-order preferences.  To 
represent this situation, Approval voting is given a grade of WF- below.
B.	DEFINITIONS OF CRITERIA
To avoid trivialities, assume that two or more alternatives are 
being considered.
1.	Majority Criteria
a.	The Majority (Winner) Criterion (MW)
If a strict majority of the voters rank a particular 
alternative as their unique first choice, then the 
voting method must select that alternative as the 
unique winner.
This criterion is abbreviated by MW below.
b.	The Majority Loser Criterion (ML)
If a strict majority of the voters rank a particular 
alternative as their unique last choice, then the 
voting method must not select that alternative as a 
winner.
This criterion is abbreviated by ML below.
c.	The Generalized Majority Criterion (MG)
If there is a nonempty subset of the alternatives, 
say S, such that, for some strict majority subset of 
the voters, say M, each voter in M ranks every 
alternative in S ahead of every alternative not in S, 
then all of winners selected by the voting method 
must be in S.
This criterion is abbreviated by MG below.  Note that if a voting 
method satisfies MG then it must satisfy both MW and ML, but not vice 
versa.
2.	Condorcet Criteria
a.	The Condorcet (Winner) Criterion (CW)
If a particular alternative would win all of the 
head-to-head pairings between it and each other 
alternative, then the voting method must select 
that alternative as the unique winner.
This criterion is abbreviated by CW below.
b.	The Condorcet Loser Criterion (CL)
If a particular alternative would lose all of the 
head-to-head pairings between it and each other 
alternative, then the voting method must not select 
that alternative as a winner.
This criterion is abbreviated by CL below.
c.	The Generalized Condorcet Criterion (CG)
If there is a nonempty subset of the alternatives, 
say S, such that every alternative in S would win 
all of the head-to-head pairings between it and 
each of the alternatives not in S, then all of 
winners selected by the voting method must be in 
S.
This criterion is also known as Smith's generalized Condorcet 
criterion.  It is abbreviated by CG below.  Note that if a voting method 
satisfies CG then it must satisfy both CW and CL, but not vice versa.  
Also note that if a voting method satisfies CG then it must satisfy MG 
(and, hence, MW and ML), but not vice versa.
3.	The Monotonicity Criterion (Mono.)
If an alternative would be a winner according to 
the voting method with some particular set of 
voter's rankings, and then one or more voters 
change their rankings in a way favorable to that 
alternative (without changing the relative order in 
which they rank any other alternatives), then that 
alternative must still be a winner.
A completely equivalent way of stating this criterion is as 
follows:
If an alternative would not be a winner according 
to the voting method with some particular set of 
voter's rankings, and then one or more voters 
change their rankings in a way unfavorable to that 
alternative (without changing the relative order in 
which they rank any other alternatives), then that 
alternative must still not be a winner.
This criterion is abbreviated by Mono. below.
C.	A  CRITERIA-BASED  RANKING  OF  SELECTED  
VOTING  METHODS
1.	Some Grades Based Solely on the Majority, 
Monotonicity, and Condorcet Criteria
As discussed in Section A, the 19 voting methods that pass all 7 
criteria receive unspecified grades between C and A+, inclusive, which is 
denoted by "C-A+" below.  Also as discussed in Section A, Approval 
voting is the only voting method considered here whose winners cannot 
be determined directly from ranked ballots.  However, if the voters were 
to cast ranked ballots along with consistent information needed to 
determine Approval-voting winners, then Approval voting would receive 
an F- for failing to pass MC.  To represent this situation, Approval 
voting is given a grade of WF- below.  The grades between F- and C- 
below are based solely on the majority, monotonicity, and Condorcet 
criteria.  The rationale for these grades is as follows.
Voting situations can be viewed as lying on a line between highly 
partisan political voting at one end, and altruistic judging at the other.  
Judging Olympic competitors in diving, gymnastic, and figure skating 
events are examples of voting situations near the judging end.  Political 
elections and votes taken by legislatures are examples near or at the other 
end.  If, in a voting situation, there is agreement on the specifics of the 
goals to be achieved, on the proper techniques to achieve those goals, 
and on scales that measure how accomplished the various alternatives are 
with respect to those techniques, then that situation is likely to belong at 
or near the judging end of this voting spectrum.  If there isn't agreement 
on scales that measure the alternatives with respect to their achievement 
of (or their capability to achieve) common goals, or there isn't even 
agreement on the particular goals to be achieved, then the voting 
situation clearly belongs near the partisan end.  Several qualitatively 
different types of voting methods might be successfully used in voting 
situations located at or near the judging end of this spectrum.  Arguably, 
however, democratic societies have strongly and consistently required the 
majority criterion to hold in voting situations located at or near the 
partisan end of this spectrum.  This fundamental requirement is imposed 
here, and so any voting method that fails MC receives an F- in the 
grading tabulated below.
If an alternative that is ranked first by a strict majority of the 
voters ought to win a voting situation, then surely an alternative that is 
ranked last by a strict majority of the voters ought not to win that voting 
situation.  Accordingly, any voting method that passes MC but fails ML 
receives an F in the grading tabulated below.  MG is the natural 
generalization of MW and ML to those situations in which five or more 
alternatives are being voted upon.  It's hard, if not impossible, to picture 
any plausible argument for MW and ML when four or fewer alternatives 
are being considered, that would not also correspond to a quite plausible 
argument for MG when there are five or more alternatives on the ballot.  
Accordingly, any voting method that passes MC and ML, but fails MG, 
receives an F+ in the grading tabulated below.
The first way that the monotonicity criterion is stated above can 
be pictured as follows.  Suppose that a group of voters, who hadn't 
known much about Alternative A and so had not ranked A first, were to 
learn enough about A to persuade them to move A to first leaving all else 
unchanged.  Then this action by those voters certainly shouldn't cause A 
to lose an election that A would otherwise have won.  The second way of 
stating the monotonicity criterion can be pictured as follows.  Suppose 
that voter v ranks A first and makes a special effort to go out to vote, 
while voter w, who would have ranked A lower (but agrees with v 
otherwise), stays home.  Further, suppose that A loses.  Then A shouldn't 
have won had v (who likes A) stayed home and w (who doesn't like A) 
gone out to vote instead.  If informed voting is to mean something, then 
the monotonicity criterion surely ought to hold.  Fortunately, there are 
many voting methods that satisfy the monotonicity criterion; but there are 
also many that don't.  Any voting method that passes MG but fails 
Mono. receives a D- in the grading tabulated below.
The concept of considering who would win the pairing (i.e., the 
head-to-head matchup) between each alternative and each other 
alternative (in a two-way race between them) leads to the three 
Condorcet criteria stated above.  These criteria are logical and useful 
extensions of the corresponding three majority criteria to such head-to-
head pairings.  Accordingly, a voting method that passes MG and Mono., 
but fails CW, receives a D in the grading tabulated below.  A voting 
method that would pass MG, Mono., and CW, but fails CL, would 
receive a D+ in the grading tabulated below.  And a voting method that 
would pass MG, Mono., CW, and CL, but fails CG, would receive a C- 
in that grading.
2.	A Ranking of Voting Methods According to These 
Grades
Applying these grades to the voting methods listed in [1] gives 
Table 1 on page 6.  This table is structured as follows.
The voting methods are listed and ranked in order by the grade 
each receives.  The dark horizontal lines on that table separate the sets of 
methods that receive different grades.  For example, Black, Condorcet, 
Young, and Plurality each receive an F, and these methods are separated 
from those above and those below by dark horizontal lines.
The light horizontal lines separate methods that receive the same 
grades but do not pass identical sets of criteria.  For example, Black, 
Condorcet, Young, and Plurality receive the F because each passes MW 
but each fails ML.  However, Black passes CL, while Condorcet and 
Young fail CL; so Black is ranked above Condorcet and Young and is 
separated from them by a light horizontal line.  Similarly, Condorcet and 
Young are ranked above (and separated by a light horizontal line from) 
Plurality because those methods satisfy CW while Plurality doesn't.
Voting methods that pass identical sets of criteria (and so also 
receive the same grade) are listed in alphabetical order and are not 
separated from each other by horizontal lines.
Dodgson's voting method passes MW, so it receives at least an F; 
and it fails Mono., so it receives no more than a D-.  Whether it receives 
an F, F+, or D- depends on whether or not it passes ML or both ML and 
MG.
None of the voting methods listed in [1] receive either a C- or a 
D+.  Of course, this does not mean that no such voting methods exist.  To 
allow ranking such voting methods, a C- row and a D+ row are included 
on Table 1 with a "?" listed under the Voting Method column.
The following three points are well worth remembering.  First, 
there are many other judgmentally evaluated characteristics (attributes) 
and rigorously defined characteristics (criteria) that could be applied to 
voting methods, in addition to the seven criteria discussed here.  Such 
characteristics include computability, decisiveness, and simplicity.  
Second, there are very many other ranked-ballot voting methods, in 
addition to the 33 ranked-ballot methods considered here.  Third, every 
ranked-ballot voting method has multiple versions that depend on 
whether ties, truncations, both, or neither are allowed on voter's ballots 
and, if so, how such ties and/or truncations are treated.  The way that 
such ties and truncations are counted can significantly enhance or reduce 
the quality and usefulness of various ranked-ballot voting methods.  Less 
significantly, but still potentially important, is whether write-ins are 
allowed and, if so, how they are treated.



REFERENCES

1.	Anderson, L.B., Some Relationships Among Voting Methods and the 
Majority, Condorcet, and Monotonicity Criteria, Background Paper, 
May 1996.
2.	Anderson, L.B., An Introductory Bibliography on Single-Winner 
Voting Methods, Background Paper, March 1996.



Table 1.  A Partial Ranking of Voting Methods Based on Majority, 
Condorcet, and Monotonicity Criteria

Criteria


Voting Method
MW
ML
MG
Mono.
CW
CL
CG
Grade

Anderson
yes
yes
yes
yes
yes
yes
yes
C-A+

Complete-Champion
yes
yes
yes
yes
yes
yes
yes
C-A+

Consensus-
Champion
yes
yes
yes
yes
yes
yes
yes
C-A+

Copeland
yes
yes
yes
yes
yes
yes
yes
C-A+

Copeland-mod
yes
yes
yes
yes
yes
yes
yes
C-A+

Fishburn
yes
yes
yes
yes
yes
yes
yes
C-A+

Kemeny
yes
yes
yes
yes
yes
yes
yes
C-A+

Max-Tourneys
yes
yes
yes
yes
yes
yes
yes
C-A+

Niemi-Riker
yes
yes
yes
yes
yes
yes
yes
C-A+

Regular-Champion
yes
yes
yes
yes
yes
yes
yes
C-A+

Schwartz
yes
yes
yes
yes
yes
yes
yes
C-A+

Sister
yes
yes
yes
yes
yes
yes
yes
C-A+

Smith
yes
yes
yes
yes
yes
yes
yes
C-A+

Smith//Borda
yes
yes
yes
yes
yes
yes
yes
C-A+

Smith//Bucklin
yes
yes
yes
yes
yes
yes
yes
C-A+

Smith//Condorcet
yes
yes
yes
yes
yes
yes
yes
C-A+

Smith//Plurality
yes
yes
yes
yes
yes
yes
yes
C-A+

Qualified-Champion
yes
yes
yes
yes
yes
yes
yes
C-A+

Qualified-Kemeny
yes
yes
yes
yes
yes
yes
yes
C-A+

?
yes
yes
yes
yes
yes
yes
no
C-

?
yes
yes
yes
yes
yes
no
no
D+

Bucklin
yes
yes
yes
yes
no
no
no
D 

/Condorcet/1a/
yes
yes
yes
no
yes
yes
yes
D-

/Copeland/2a/
yes
yes
yes
no
yes
yes
yes
D-

Nanson
yes
yes
yes
no
yes
yes
yes
D-

Hare
yes
yes
yes
no
no
yes
no
D-

Coombs
yes
yes
no
no
no
yes
no
F+

Plurality-w/runoff
yes
yes
no
no
no
yes
no
F+

Dodgson
yes
?
?
no
yes
no
no
F-D-

Black 
yes
no
no
yes
yes
yes
no
F 

Condorcet
yes
no
no
yes
yes
no
no
F 

Young
yes
no
no
yes
yes
no
no
F 

Plurality
yes
no
no
yes
no
no
no
F 

Arrow-Raynaud
no
yes
no
no
no
yes
no
F-

Borda
no
no
no
yes
no
yes
no
F-

Approval
no
no
no
yes
no
no
no
WF-


9



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