7.Example 4 (page 447) showed that the Borda count is manipulable if there are five candidates and six voters. Mimic what was done there to construct an example with seven candidates and eight voters.
7.
One way is to alter the elections in Example 4 of the text by adding and to the bottom of each of the six ballots in both elections, and then adding the two rightmost columns as shown.
| Election 1 | ||||||||
| Rank | Number of voters (8) | |||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| First | ||||||||
| Second | ||||||||
| Third | ||||||||
| Fourth | ||||||||
| Fifth | ||||||||
| Sixth | ||||||||
| Seventh | ||||||||
has the highest Borda score and is the winner.
The voter on the far left prefers to . By casting a disingenuous ballot (still preferring to , though), the outcome of the election is altered.
| Election 2 | ||||||||
| Rank | Number of voters (8) | |||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| First | ||||||||
| Second | ||||||||
| Third | ||||||||
| Fourth | ||||||||
| Fifth | ||||||||
| Sixth | ||||||||
| Seventh | ||||||||
Now has the highest Borda score and is the winner.