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.
| 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 Pair-wise 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: Three-way 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}.