The main disadvantage of the plurality method is that it does not allow each voter to express her full preference order: the top-ranked candidate receives 1 "point" and all other candidates receive 0. In the *positional method*, the second-ranked candidate receives a fractional amount of votes. We have a parameter `s` that is between 0 and 1, inclusive, and each voter's three candidates receives 1, `s`, and 0 points, in order of preference.

When `s` = 0, the positional method reduces to the plurality method that we have already discussed.

When `s` = 1, the voter's first and second place candidates both receive an equal vote, and the third place receives zero. In effect, a vote has been placed **against** the third-place candidate. This method is called *antiplurality. *

When `s` = 0.5, the positional method reduces to the *Borda count*, a method developed by Jean-Charles de Borda in 1770. An equivalent way to perform the Borda count is to assign 2 points for each first-place vote, 1 point for each second-place vote, and 0 points for third-place votes.

In the interactive mathlet below, experiment with changing the value of `s`. Can you construct profiles where you can change the winner of the election, not by adding or removing voters, but by simply changing the value of `s`?

Consider another voter profile:

Votes | Preference |
---|---|

6 | A > B > C |

4 | A > C > B |

0 | B > A > C |

8 | B > C > A |

3 | C > A > B |

4 | C > B > A |

Who should win this election? We will analyze this election using the positional method, but we will not choose a value of *s* right away.

Preference | A | B | C |
---|---|---|---|

A > B > C | 6 | 6s |
0 |

A > C > B | 4 | 0 | 4s |

B > A > C | 0 | 0 | 0 |

B > C > A | 0 | 8 | 8s |

C > A > B | 3s |
0 | 3 |

C > B > A | 0 | 4s |
4 |

Totals | 10 + 3s |
8 + 10s |
7 + 12s |

Performing the calculations this way allows us to easily change the value of *s* and determine if the winner of the election changes. In this example, for different values of *s*, any of the three candidates could win this election:

s |
A | B | C |
---|---|---|---|

0.1 | 10.3 | 9 | 8.2 |

0.4 | 11.2 | 12 | 11.8 |

0.8 | 12.4 | 16 | 16.6 |

So which candidate should win? In this case there is no easy answer.

Journal of Online Mathematics and its Applications