User:DCary/WIP/Beatpath Example 12

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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.

Vote Totals
     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 --
Color Code for Vote Totals
Pair-wise Win Pair-wise Tie Pair-wise Loss No Contest



Vote Margins
     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 --
Color Code for Vote Margins
Pair-wise Win Pair-wise Tie Pair-wise Loss No Contest



Beatpath Exists
     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                        
Color Code for BeatPath Exists
Pair-wise Win Pair-wise Tie Pair-wise Loss No Contest



Beatpath Order on Cycle Equivalence Classes
Cec  A   E   F   G   H   I   L 
A =   > > > > >
E   =     > > >
F <   =   > > >
G <     = > > >
H < < < < = > >
I < < < < < =  
L < < < < <   =
Color Code for Beatpath Order on Cycle Equivalence Classes
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.