Beatpath examples 3
The following are examples of beatpath orders for elections with three candidates: A, B, and C. For each example, the pairwise 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:
Pairwise Win  Pairwise Tie  Pairwise Loss  No Contest 
Beatpath fromto  In a Cycle  Beatpath tofrom  No Beatpaths 
Example 1: Condorcet Winner
A beats B and C. B beats C.
A  B  C  

A    Y  Y 
B    Y  
C   
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.
A  B  C  

A    Y  
B    Y  
C   
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 Pairwise Tie
A ties B. A and B beat C.
A  B  C  

A    Y  
B    Y  
C   
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: Threeway Tie
A ties B and C. B ties C.
A  B  C  

A    
B    
C   
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.
A  B  C  

A    Y  
B    Y  
C  Y   
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}.