9. Show that the Borda count is manipulable if there are four candidates and five voters. (Hint: Start with the ballots in the previous exercise, and then add two ballots that cancel each other out.)
9.
The desired ballots (obtained as suggested in the statement of the exercise) are as follows:
| Election 1 | |||||
| Rank | Number of voters (5) | ||||
| 1 | 1 | 1 | 1 | 1 | |
| First | |||||
| Second | |||||
| Third | |||||
| Fourth | |||||
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 (5) | ||||
| 1 | 1 | 1 | 1 | 1 | |
| First | |||||
| Second | |||||
| Third | |||||
| Fourth | |||||
Now, has the highest Borda score and is the winner.