Beatpath example 12
This is an example of a beatpath order with 12 candidates. It has been constructed to illustrate a variety of situations that can occur in terms of beatpaths, the beatpath order, and pair-wise wins, losses, and ties. The example purposely has a lot of pair-wise ties and illustrates what is possible, not necessarily what might be typical.
This example could occur with the following ranked ballots involving 12 candidates A - L:
- 6: A>B
- 1: A>C>H>D>B>G>F>E>L>I>K>J
- 6: C>D
- 3: E>A>C>H>D>B
- 3: E>B>D>C>A
- 2: E>F>G>H>A>B
- 2: E>G>F>H>B>C
- 4: F>D>G>H>A
- 1: F>G>E>H>B>D>C>A>L
- 4: G>B>F>H>C
- 1: H>G>F>E>D>C>B>J>I>K>L>A
- 9: H>G>F>E>D>C>B>L>I>J>K>A
- 10: K>J>I>L>A>B>C>D>E>F>G>H
These ballots are evaluated in pair-wise contests using a typical procedure. For example, the first line says there are 6 votes for A in each contest involving A, there are six votes for B in each contest involving B except for the one against A, and there are no votes for either candidate in any other pair-wise contest.
A | B | C | D | E | F | G | H | I | J | K | L | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | -- | 26 | 26 | 22 | 21 | 23 | 23 | 23 | 20 | 20 | 20 | 20 |
B | 20 | -- | 28 | 28 | 21 | 27 | 23 | 23 | 32 | 32 | 32 | 32 |
C | 26 | 20 | -- | 26 | 21 | 23 | 23 | 23 | 30 | 30 | 30 | 30 |
D | 24 | 24 | 18 | -- | 21 | 23 | 27 | 23 | 28 | 28 | 28 | 28 |
E | 21 | 21 | 21 | 21 | -- | 20 | 20 | 21 | 22 | 22 | 22 | 22 |
F | 23 | 19 | 23 | 23 | 20 | -- | 17 | 23 | 24 | 24 | 24 | 24 |
G | 23 | 23 | 23 | 19 | 20 | 17 | -- | 23 | 24 | 24 | 24 | 24 |
H | 23 | 23 | 23 | 23 | 19 | 14 | 14 | -- | 27 | 27 | 27 | 27 |
I | 20 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | -- | 10 | 11 | 11 |
J | 20 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 11 | -- | 10 | 11 |
K | 20 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 11 | -- | 11 |
L | 20 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 11 | 11 | 11 | -- |
Pair-wise Win | Pair-wise Tie | Pair-wise Loss | No Contest |
A | B | C | D | E | F | G | H | I | J | K | L | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | -- | 6 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
B | -6 | -- | 8 | 4 | 0 | 8 | 0 | 0 | 22 | 22 | 22 | 22 |
C | 0 | -8 | -- | 8 | 0 | 0 | 0 | 0 | 20 | 20 | 20 | 20 |
D | 2 | -4 | -8 | -- | 0 | 0 | 8 | 0 | 18 | 18 | 18 | 18 |
E | 0 | 0 | 0 | 0 | -- | 0 | 0 | 2 | 12 | 12 | 12 | 12 |
F | 0 | -8 | 0 | 0 | 0 | -- | 0 | 9 | 14 | 14 | 14 | 14 |
G | 0 | 0 | 0 | -8 | 0 | 0 | -- | 9 | 14 | 14 | 14 | 14 |
H | 0 | 0 | 0 | 0 | -2 | -9 | -9 | -- | 17 | 17 | 17 | 17 |
I | 0 | -22 | -20 | -18 | -12 | -14 | -14 | -17 | -- | -1 | 1 | 0 |
J | 0 | -22 | -20 | -18 | -12 | -14 | -14 | -17 | 1 | -- | -1 | 0 |
K | 0 | -22 | -20 | -18 | -12 | -14 | -14 | -17 | -1 | 1 | -- | 0 |
L | 0 | -22 | -20 | -18 | -12 | -14 | -14 | -17 | 0 | 0 | 0 | -- |
Pair-wise Win | Pair-wise Tie | Pair-wise Loss | No Contest |
A | B | C | D | E | F | G | H | I | J | K | L | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | |
B | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | |
C | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | |
D | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | |
E | Y | Y | Y | Y | Y | |||||||
F | Y | Y | Y | Y | Y | |||||||
G | Y | Y | Y | Y | Y | |||||||
H | Y | Y | Y | Y | ||||||||
I | Y | Y | Y | |||||||||
J | Y | Y | Y | |||||||||
K | Y | Y | Y | |||||||||
L |
Pair-wise Win | Pair-wise Tie | Pair-wise Loss | No Contest |
Cec | A | E | F | G | H | I | L |
---|---|---|---|---|---|---|---|
A | = | > | > | > | > | > | |
E | = | > | > | > | |||
F | < | = | > | > | > | ||
G | < | = | > | > | > | ||
H | < | < | < | < | = | > | > |
I | < | < | < | < | < | = | |
L | < | < | < | < | < | = |
Beatpath from-to | In a Cycle | Beatpath to-from | No Beatpaths |
Note that the cycle equivalence class for A is {A,B,C,D} and the cycle equivalence class for I is {I,J,K}.
- Beat Path Order:
{A,B,C,D} {E}
/ \ /
{F} {G} /
\___ | /
\|/
{H}
/ \
{I,J,K} {L}
- The weak Condorcet winner is E.
- The Schwartz set = {A,B,C,D,E}.
- The Smith set = the set of all candidates.
Winner under different methods of ambiguity resolution
- Copeland: B
- Schulze: B
- Ranked Pairs: B
- Borda/Black: B
- Approval voting/Llull, assuming every ranked candidate is approved: B
- Minimax(winning votes): H
- MMPO: E
- IRV: E
- Raynaud (Gross Looser): E and G tie
- Plurality: E, H and K tie