Beatpath examples 3

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The following are examples of beatpath orders for elections with three candidates: A, B, and C. For each example, the pair-wise beat relation is given in a tabular format, and the beatpath order is given in both a tabular and graphical format. In addition, the Schwartz set, the Smith Set, and any Condorcet winner or Weak Condorcet winners are noted.

The tables are color coded as follows:

Color Code for Pair-wise Beat Relation
Pair-wise Win Pair-wise Tie Pair-wise Loss No Contest


Color Code for Beatpath Order
Beatpath from-to In a Cycle Beatpath to-from No Beatpaths


Example 1:  Condorcet Winner

A beats B and C. B beats C.

Beat Relation
     A   B   C 
A -- Y Y
B   -- Y
C     --
Beatpath Order
Cec  A   B   C 
A = > >
B < = >
C < < =

Beat Path Order

    {A}
     |
    {B}
     |
    {C}



  • The Condorcet winner is A.
  • The Schwartz set = {A}.
  • The Smith set = {A}.


Example 2:  Weak Condorcet Winner

A beats B and ties C. B beats C.

Beat Relation
     A   B   C 
A -- Y  
B   -- Y
C     --
Beatpath Order
Cec  A   B   C 
A = > >
B < = >
C < < =

Beat Path Order

    {A}
     |
    {B}
     |
    {C}



  • The weak Condorcet winner is A.
  • The Schwartz set = {A}.
  • The Smith set = {A,B,C}.


Example 3:  Leaders Pair-wise Tie

A ties B. A and B beat C.

Beat Relation
     A   B   C 
A --   Y
B   -- Y
C     --
Beatpath Order
Cec  A   B   C 
A =   >
B   = >
C < < =

Beat Path Order

    {A}   {B}
       \ /
       {C}




  • The weak Condorcet winners are A and B.
  • The Schwartz set = {A,B}, each in their own component.
  • The Smith set = {A,B}.


Example 4:  Three-way Tie

A ties B and C. B ties C.

Beat Relation
     A   B   C 
A --    
B   --  
C     --
Beatpath Order
Cec  A   B   C 
A =    
B   =  
C     =

Beat Path Order

    {A}   {B}   {C}





  • The weak Condorcet winners are A, B, and C.
  • The Schwartz set = {A,B,C}, each in their own component.
  • The Smith set = {A,B,C}.


Example 5:  Cycle

A beats B. B beats C. C beats A.

Beat Relation
     A   B   C 
A -- Y  
B   -- Y
C Y   --
Beatpath Order
Cec  A 
A =

Beat Path Order

    {A, B, C}





  • No Condorcet winner. No weak Condorcet winner.
  • The Schwartz set = {A,B,C}, all in the same component.
  • The Smith set = {A,B,C}.

See also