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[Bruce Anderson]

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BACKGROUND PAPER
SOME  RELATIONSHIPS  AMONG  VOTING  
METHODS  AND  THE  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
Each voter casts one ballot by ranking (1st, 2nd, 3rd, etc.) as many 
of the candidates as the voter wishes, with ties allowed, and with any 
unranked candidates on a voter's ballot being considered as tied for last 
by that voter.  Each voter casts only one such ballot, and (with the 
exception of Approval voting as discussed in Section D), the winners 
according to each of the voting methods considered below can be 
determined solely from these ballots.  Each criterion considered below is 
defined in Section B.  Table 1 then gives relationships involving these 
criteria and 34 different voting methods.  The voting methods themselves 
are not defined here, and no proofs or examples are given.  See the 
publications cited in Reference [1], and their references, for additional 
information on voting methods.
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
The criteria in this section involve an alternative "winning or 
losing the pairing" between it and individual other alternatives.  These 
terms are defined as follows.  Given the voters ballots (as described 
above), all possible distinct pairings of the candidates are considered.  
For any two candidates, say A and B:
A wins (and B loses) their pairing if more voters rank A ahead of B than 
rank B ahead of A,
B wins (and A loses) their pairing if more voters rank B ahead of A than 
rank A ahead of B,
and the pairing is a tie if the same number of voters rank A ahead of B as 
rank B ahead of A.
Accordingly, a "pairing" between two alternatives can be viewed as a 
"head-to-head matchup" between them, which necessarily results in one 
(and only one) of the three outcomes just described.
a.	The Condorcet (Winner) Criterion (CW)
If a particular alternative would win all of the 
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 
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 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.
To see that these two ways of stating the monotonicity criterion 
are equivalent, note that if a voting method fails either one then it 
necessarily must fail the other.  This criterion is abbreviated by Mono. 
below.
C.	RELATIONSHIPS AMONG THESE CRITERIA AND 
SELECTED VOTING METHODS
Whether or not Dodgson's voting method satisfies ML or MG 
remains an open question here.  Except for these two cases, Table 1 on 
page 4 states which of the listed voting methods satisfy which of these 
seven criteria.  Those that satisfy all seven are listed in bold italics.
D.	APPROVAL  VOTING
Approval voting requires an additional input beyond just a 
ranking from each voter.  That is, a valid Approval ballot is a valid 
(ranked) ballot, as defined here, because ties are allowed in voter's 
rankings.  Thus, any voter who believes that the alternatives can be put 
into one of two categories (either approved or non-approved) can always 
cast a valid ranked ballot as well as a valid Approval ballot.  Conversely, 
though, a valid ranked ballot does not generally correspond to a valid 
Approval ballot because it contains insufficient information_the 
additional input of where in a voter's ranking that voter separates his or 
her approved alternatives from the non-approved ones is also needed 
from each voter.  Table 1 assumes that both the voter's rankings and this 
additional input are on the voter's ballot under Approval voting.

REFERENCE

1.	Anderson, L.B., An Introductory Bibliography on Single-Winner 
Voting Methods, Background Paper, March 1996.


Table 1.   Voting Methods vs. the Majority, Condorcet, and 
Monotonicity Criteria

Criteria

Voting Method
MW
ML
MG
CW
CL
CG
Mono.
All

Anderson
yes
yes
yes
yes
yes
yes
yes
yes

Approval
no
no
no
no
no
no
yes
no

Arrow-Raynaud
no
yes
no
no
yes
no
no
no

Black 
yes
no
no
yes
yes
no
yes
no

Borda
no
no
no
no
yes
no
yes
no

Bucklin
yes
yes
yes
no
no
no
yes
no

Complete-Champion
yes
yes
yes
yes
yes
yes
yes
yes

Condorcet
yes
no
no
yes
no
no
yes
no

/Condorcet/1a/
yes
yes
yes
yes
yes
yes
no
no

Consensus-
Champion
yes
yes
yes
yes
yes
yes
yes
yes

Coombs
yes
yes
no
no
yes
no
no
no

Copeland
yes
yes
yes
yes
yes
yes
yes
yes

/Copeland/2a/
yes
yes
yes
yes
yes
yes
no
no

Copeland-mod
yes
yes
yes
yes
yes
yes
yes
yes

Dodgson
yes
?
?
yes
no
no
no
no

Fishburn
yes
yes
yes
yes
yes
yes
yes
yes

Hare
yes
yes
yes
no
yes
no
no
no

Kemeny
yes
yes
yes
yes
yes
yes
yes
yes

Max-Tourneys
yes
yes
yes
yes
yes
yes
yes
yes

Nanson
yes
yes
yes
yes
yes
yes
no
no

Niemi-Riker
yes
yes
yes
yes
yes
yes
yes
yes

Plurality
yes
no
no
no
no
no
yes
no

Plurality-w/runoff
yes
yes
no
no
yes
no
no
no

Regular-Champion
yes
yes
yes
yes
yes
yes
yes
yes

Schwartz
yes
yes
yes
yes
yes
yes
yes
yes

Sister
yes
yes
yes
yes
yes
yes
yes
yes

Smith
yes
yes
yes
yes
yes
yes
yes
yes

Smith//Borda
yes
yes
yes
yes
yes
yes
yes
yes

Smith//Bucklin
yes
yes
yes
yes
yes
yes
yes
yes

Smith//Condorcet
yes
yes
yes
yes
yes
yes
yes
yes

Smith//Plurality
yes
yes
yes
yes
yes
yes
yes
yes

Qualified-Champion
yes
yes
yes
yes
yes
yes
yes
yes

Qualified-Kemeny
yes
yes
yes
yes
yes
yes
yes
yes

Young
yes
no
no
yes
no
no
yes
no


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