# Beatpath examples 3

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:

 Pair-wise Win Pair-wise Tie Pair-wise Loss No Contest

 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}.