Attachment COND-FLA
Bruce Anderson
landerso at ida.org
Thu Jun 6 02:49:45 PDT 1996
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POSITION PAPER
FIVE EXAMPLES THAT SHOW FLAWS OF
CONDORCET'S VOTING METHOD
Lowell Bruce Anderson
Voice: 703-845-2148
FAX: 703-845-2255
e-mail: landerso at ida.org
May 2, 1996
This paper gives five examples concerning flaws of Condorcet's
voting method. Some of these examples also apply to certain other
voting methods, like Young's method. However, the method that merits
scrutiny here is Condorcet's, for the following reasons. First, it is
extremely easy to compute. Indeed, its winner(s) can be read directly
from the pairwise preference array (i.e., the array of the numbers of
voters who prefer any given alternative to any other given alternative,
plus, in some variants, « of any corresponding indifferent voters) by just
looking at the row minimums. Second, it is easily extended in case of
ties by just looking at the second smallest row entries for the alternatives
that are tied with the smallest row minimums. Third, it is not too
difficult to explain. Fourth, it has been proposed for use in the popular
press and on the internet. Fifth, it satisfies the well-established Pareto,
monotonicity, majority, and Condorcet criteria. Admittedly, it fails both
the somewhat more complicated generalized Condorcet criterion and the
relatively less well known majority loser criterion_see References [1]
and [2]. But, with all of these advantages, one might ask: "How bad can
it be to fail these two criteria?"
Example 1 gives a relatively simple voting situation in which a
strict majority of the voters (64%) rank Alternative D dead last, yet
Condorcet's method selects D as its unique winner. Of all the voting
methods listed, only Plurality and Young agree with Condorcet in that
example. If the point is to do better than Plurality, then Example 1
suggests doing better than Condorcet also. See Reference [3] for an
introductory bibliography on single-winner voting methods.
Of course, when things can go wrong, they frequently can go very
wrong. Example 2 gives a more complex voting situation in which
Condorcet's method selects a overwhelming loser, Alternative Z, as its
unique winner. In that example, Z receives the least number of first
place votes of any of the alternatives being considered, it is ranked alone
in last place by 84% of the voters, and it loses its pairwise contest with
each other alternative by 91 to 9. Alternative Z is so weak in that
example that Young's method is the only other listed method that selects
Z as a winner, not even Plurality (with all its flaws), or Schwartz or
Smith (which are relatively indecisive), select Z. In Examples 2 and 5,
ties in Plurality and Plurality-with-runoff voting are resolved by
considering most seconds, then most thirds, and so on until the tie is
broken; ties in Hare voting are resolved by considering least seconds,
then least thirds, and so on; and ties in Coombs voting are resolved by
considering most seconds-to-last and so on. Alternative A is listed as the
Niemi/Riker winner in Example 2 because it is both a Copeland winner
and the Borda winner.
If an alternative that is ranked first by a strict majority of the
voters ought to win the vote being taken; then surely if an alternative is
ranked last by a strict majority of the voters, then that alternative ought
not to win that voting. Examples 1 and 2 show that Condorcet's method
fails this simple majority-rule test.
As discussed in [1] and [2], the Condorcet loser criterion says
that if one of the alternatives would be beaten by every other alternative
under consideration in the head-to-head race between them, then that
alternative must never be a winner. Examples 1 and 2 show that
Condorcet's method also fails the Condorcet loser criterion. Example 3
shows that the winner by Condorcet's method can both lose to every
other alternative in the head-to-head race between them and also receive
absolutely no first-place votes.
Examples 1, 2, and 3 indicate that Condorcet's voting method is
completely unacceptable for use in most, if not all, voting situations. Its
proponents might claim that Condorcet's method could be modified
somehow to address the flaws demonstrated here. Perhaps, but
Examples 4 and 5 indicate that this may not be simple to do. These
examples show that Condorcet's method can elect pretty strange winners
even when every alternative would beat at least one of the other
alternatives in the head-to-head pairing between them. In particular, note
that, in Example 5, the winner according to Condorcet's voting method
receives the least number of first-place votes of all of the candidates, has
a lower average ranking than any other candidate, wins fewer of its head-
to-head pairings than any other candidate, and loses more of its head-to-
head pairings than any other candidate.
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., A Partial Ranking of Selected Voting Methods Based
on Majority, Condorcet, and Monotonicity Criteria, Position Paper,
May 1996.
3. Anderson, L.B., An Introductory Bibliography on Single-Winner
Voting Methods, Background Paper, March 1996.
Example 1. Condorcet's Method Elects a Majority Loser
4 Alternatives: A, B, C, and D
100 Voters
Voter's Preferences:
Preference
Position _______Orders________
1st: A B D D D
2nd: B C A B C
3rd: C A B C A
4th: D D C A B
#: 33 31 2 2 32
# = Number Of The 100 Voters With This Preference Order
Corresponding Number Of The
100 Voters Who Prefer The
Row Alternative To The Paired
Column Alternative Below: Wins-
Losses; Row Avg. #
# # #
A B C D Ties Min Rank 1st
2nd 3rd 4th
A * -- 67 35 64 2-1;0 35 2.34 33
2 63 2
B * 33 -- 68 64 2-1;0 33 2.35 31
35 2 32
C * 65 32 -- 64 2-1;0 32 2.39 0
63 35 2
D * 36 36 36 -- 0-3;0 36 2.92 36
0 0 64
Voting Resulting
Method Winner(s)
Arrow/Raynaud A
Black A
Borda A
Bucklin B
Condorcet D
Coombs A
Copeland-all A, B, C
Dodgson A
Fishburn A, B, C
Hare A
Kemeny A
Max-Tourneys A, B, C
Nanson A
Niemi/Riker A
Plurality D
Pl/runoff A
Regular-Champion A
Schwartz A, B, C
Smith A, B, C
Young D
Copeland-all = Anderson, Copeland, Copeland-mod, Sister
Example 2. Condorcet's Method Elects a Landslide Loser
15 Alternatives: A, B, C, D, E, F, G, H, I, J, K, M, N,
and Z
100 Voters
Voter's Preferences:
Position ________Preference Orders________
1st: A A A A A A A Z
Z
2nd: Z Z Z Z Z Z Z A
B
3rd: D E F G H I J C
A
4th: C D E F G H I B
N
5th: B C D E F G H N
M
6th: N B C D E F G M
L
7th: M N B C D E F L
K
8th: L M N B C D E K
J
9th: K L M N B C D J
I
10th: J K L M N B C I
H
11th: I J K L M N B H
G
12th: H I J K L M N G
F
13th: G H I J K L M F
E
14th: F G H I J K L E
D
15th: E F G H I J K D
C
#: 1 1 1 1 1 1 1 1
1
Position _______________Preference Orders_______________
__
1st: B C D E F G H I J K
L M N
2nd: A B C D E F G H I J
K L M
3rd: N A B C D E F G H I
J K L
4th: M N A B C D E F G H
I J K
5th: L M N A B C D E F G
H I J
6th: K L M N A B C D E F
G H I
7th: J K L M N A B C D E
F G H
8th: I J K L M N A B C D
E F G
9th: H I J K L M N A B C
D E F
10th: G H I J K L M N A B
C D E
11th: F G H I J K L M N A
B C D
12th: E F G H I J K L M N
A B C
13th: D E F G H I J K L M
N A B
14th: C D E F G H I J K L
M N Z
15th: Z Z Z Z Z Z Z Z Z Z
Z Z A
#: 7 7 7 7 7 7 7 7 7 7
7 7 7
# = Number Of The 100 Voters With This Preference Order
Continued
Example 2. Condorcet's Method Elects a Landslide Loser
(continued)
Corresponding Number Of The 100 Voters Who Prefer
The Row Alternative To The Column Alternative Below:
Z A B C D E F G H I J K L M
N
Z * -- 9 9 9 9 9 9 9 9 9 9 9 9 9
9
A * 91 -- 8 16 23 30 37 44 51 58 65 72 79 86
93
B * 91 92 -- 8 16 24 32 40 48 56 64 72 79 86
93
C * 91 84 92 -- 8 16 24 32 40 48 56 64 71 78
85
D * 91 77 84 92 -- 8 16 24 32 40 48 56 63 70
77
E * 91 70 76 84 92 -- 8 16 24 32 40 48 55 62
69
F * 91 63 68 76 84 92 -- 8 16 24 32 40 47 54
61
G * 91 56 60 68 76 84 92 -- 8 16 24 32 39 46
53
H * 91 49 52 60 68 76 84 92 -- 8 16 24 31 38
45
I * 91 42 44 52 60 68 76 84 92 -- 8 16 23 30
37
J * 91 35 36 44 52 60 68 76 84 92 -- 8 15 22
29
K * 91 28 28 36 44 52 60 68 76 84 92 -- 7 14
21
L * 91 21 21 29 37 45 53 61 69 77 85 93 -- 7
14
M * 91 14 14 22 30 38 46 54 62 70 78 86 93 --
7
N * 91 7 7 15 23 31 39 47 55 63 71 79 86 93
--
Paired
Wins- |
Losses; Row Avg. # # | Voting
Resulting
Alt. Ties Min Rank 1st 15th | Method
Winner(s)
A 8- 6;0 8 7.47 7 7 | Arrow/Raynaud
B, C, D, E, F,
B 8- 6;0 8 6.99 7 0 |
G, H, I, J
C 8- 6;0 8 7.11 7 1 | Black
B
D 8- 6;0 8 7.22 7 1 | Borda
G
E 8- 6;0 8 7.33 7 1 | Condorcet
Z
F 8- 6;0 8 7.44 7 1 | Coombs
J
G 8- 6;0 8 7.55 7 1 | Copeland-all
A, B, C, D,
H 7- 7;0 8 7.66 7 1 |
E, F, G
I 7- 7;0 8 7.77 7 1 | [Copeland-all]
G
J 7- 7;0 8 7.88 7 1 | [Fishburn]
all but Z & N
K 7- 7;0 7 7.99 7 1 | Hare
J
L 7- 7;0 7 7.97 7 0 | Nanson
J
M 7- 7;0 7 7.95 7 0 | Niemi-Riker
B
N 7- 7;0 7 7.93 7 0 | Ossipoff
all but Z
Z 0-14;0 9 13.74 2 84 | Plurality-ext
A
| Pl-runoff
all but Z
| Regular-Champion
G
| Schwartz
all but Z
| Smith
all but Z
| Young
Z
Copeland-all = Anderson, Copeland, Copeland-mod, Sister
Coombs, Hare, Plurality, and Pl/runoff break ties as
stated in the text
Example 3. Condorcet's Method Elects a Condorcet Loser
With No First-Place Support
6 Alternatives: A, B, C, D, E, and F
100 Voters
Voter's Preferences:
Position __________Preference Orders__________
1st: A A F F F E E
E
2nd: B B B D E D D
F
3rd: C C C B D F F
D
4th: D E A A C B C
C
5th: E D D C A C A
A
6th: F F E E B A B
B
#: 3 30 31 2 2 2 1
29
# = Number Of The 100 Voters With This Preference Order
Corresponding Number Of The
100 Voters Who Prefer The
Row Alternative To The Paired
Column Alternative Below: Wins-
Losses; Row
Avg. #
F B A C E D Ties Min
Rank 1st
F * -- 67 67 67 35 64 4-1;0 35
3.00 35
B * 33 -- 35 68 66 64 3-2;0 35
3.34 0
A * 33 65 -- 35 66 64 3-2;0 33
3.37 33
C * 33 32 65 -- 66 64 3-2;0 32
3.40 0
E * 65 34 34 34 -- 64 2-3;0 34
3.69 32
D * 36 36 36 36 36 -- 0-5;0 36
4.20 0
Voting Resulting
Method Winner(s)
Arrow/Raynaud F
Black F
Borda F
Bucklin F
Condorcet D
Coombs F
Copeland-all F
\Copeland-all\2a\ F
Fishburn all but D win
Hare F
Kemeny F
Nanson F
Niemi/Riker F
Plurality F
Pl/runoff F
Regular-Champion F
Schwartz all but D win
Smith all but D win
Copeland-all = Anderson, Copeland, Copeland-mod, Sister
Example 4. Condorcet's Method Elects a "Strange" Winner
9 Alternatives: A, B, C, D, E, F, G, H, and I
100 Voters
Voter's Preferences:
Position Preference Orders
1st: A B E F
2nd: B D I A
3rd: F H C B
4th: C I G C
5th: G E H G
6th: H F D H
7th: D G A D
8th: I C B I
9th: E A F E
#: 1 33 33 33
# = Number Of The 100 Voters With This Preference Order
Corresponding Number Of The 100 Voters
Who Prefer The Row Alternative To The Paired
Column Alternative Below: Wins-
Losses;
Row Avg. #
B H I C D E F G A Ties
Min Rank 1st
B * -- 67 67 67 67 67 67 67 33 7-1;0
33 3.98 33
H * 33 -- 67 33 67 67 66 33 66 5-3;0
33 4.68 0
I * 33 33 -- 66 33 67 66 66 66 5-3;0
33 4.70 0
C * 33 67 34 -- 67 34 33 67 66 4-4;0
33 4.99 0
D * 33 33 67 33 -- 67 66 33 66 4-4;0
33 5.02 0
E * 33 33 33 66 33 -- 66 66 66 4-4;0
33 5.04 33
F * 33 34 34 67 34 34 -- 67 66 3-5;0
33 5.31 33
G * 33 67 34 33 67 34 33 -- 66 3-5;0
33 5.33 0
A * 67 34 34 34 34 34 34 34 -- 1-7;0
34 5.95 1
Voting Resulting
Method Winner(s)
Arrow-Raynaud A
Black & Borda & Bucklin B
Condorcet & Coombs A
Copeland-all & \Copeland-all\2a\ B
Fishburn & [Fishburn] A, B, C, H, I
Hare & Kemeny & Nanson & Niemi-Riker B
Plurality B, E, F
Pl-runoff B, E
Plurality-ext & [Plurality] & [Pl-runoff] B
Regular-Champion B
Ossipoff & Schwartz & Smith all
Copeland-all = Anderson, Copeland, Copeland-mod, Sister
Example 5. Condorcet's Method Elects a "Strange" Winner
With No First-Place Support
10 Alternatives: A, B, C, D, E, F, G, H, I, and J
100 Voters
Voter's Preferences:
Position _____Preference Orders_____
1st: C F B E E J
2nd: J C D I I A
3rd: A B H C G F
4th: F D I G H C
5th: B H E H D B
6th: G I F D B G
7th: H E G B A H
8th: D G C A F D
9th: I A J J C I
10th: E J A F J E
#: 1 1 32 32 1 33
# = Number Of The 100 Voters With This Preference Order
Corresponding Number Of The 100 Voters
Who Prefer The Row Alternative To The
Paired
Column Alternative Below:
Wins-
Losses; Row Avg.
B C H I D E G F J A
Ties Min Rank
B * -- 33 67 67 67 67 67 65 66 66 8-
1;0 33 4.35
C * 67 -- 67 35 67 35 67 33 67 66 6-
3;0 33 4.96
H * 33 33 -- 67 67 67 33 65 66 66 6-
3;0 33 5.03
I * 33 65 33 -- 33 67 66 65 66 66 6-
3;0 33 5.06
D * 33 33 33 67 -- 67 33 65 66 66 5-
4;0 33 5.37
E * 33 65 33 33 33 -- 66 65 66 66 5-
4;0 33 5.40
G * 33 33 67 34 67 34 -- 33 66 66 4-
5;0 33 5.67
F * 35 67 35 35 35 35 67 -- 34 33 2-
7;0 33 6.24
J * 34 33 34 34 34 34 34 66 -- 66 2-
7;0 33 6.31
A * 34 34 34 34 34 34 34 67 34 -- 1-
8;0 34 6.61
Voting Resulting
Method Winner(s)
Black & Borda B
Bucklin C
Condorcet A
Coombs C
Copeland-all B
\Copeland-all\2a\ C
Fishburn & [Fishburn] B, C, I
Hare B
Nanson C
Niemi-Riker B
Plurality E, J
Pl-runoff E
Regular-Champion B
Ossipoff & Schwartz & Smith all
Copeland-all = Anderson, Copeland, Copeland-mod, Sister
15
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