# Selected topics in finite mathematics/Monotonic

A voting system is monotonic if it satisfies the monotonicity criterion: In an election, X is the winner. If one of the voters had previously not voted for X and were to change their ballot to rank X higher without changing the relative position of other candidates, then X must still win.

## Objectives[edit]

- Phrase the monotonicity condition criterion as a logical argument
- Give examples of different voting schemes that are monotonic, meaning that they satisfy the fairness criterion
- Give examples of different voting schemes that are NOT monotonic, meaning that they do NOT satisfy the fairness criterion

## Details[edit]

Monotonicity is when an election is held and there is a winner and another election is held and the voter changes their ballot to rank the winner of the previous election higher and the candidate X will remain the winner.

If there in an election held between candidates and X is the winner and a new election is held in a voter changes his or her ballot from not voting for X to voting for X, then X will remain the winner in the election.

A system of voting with three candidates or more will satisfy monotonicity if in every election, X is a winner and in a new election the only change is made for a voter to move the winner X higher on his or her ballot, that X will remain the winner of the election (if these are the only changes made to the voters ballot).

## Examples[edit]

**Plurality** satisfies the Monotonic criterion. It satisifes the criterion because if a voter ranks the winner higher, the winner will just be ranked higher, which would allow the winner to win with a greater margin.

**The Condorcet Method** satisfies the Monotonic criterion. If A is the winner in the first election and one person changes their vote from D to A, A is still the winner. The winner will still win in each election against any of the other contestants.

**Sequential Pairwise** satisfies the Monotonic criteria. X wins against all other candidates in a certain election. In the next election, if a voter were to rank X higher, it would not matter because X would still be winning.

**The Borda Count** satisfies the Monotonic criterion. In Borda Count, the votes are given a point value based on ranking and those are then multiplied by the number of people who voted that contestant in that specific position. A winner will still win in this instance because they are only receiving a higher amount of points to help them win.

**The Sequential Runoff** does not satisfy the Monotonic Criterion.

## Nonexamples[edit]

**Plurality** satisfies this criteria.

**The Condorcet Method** satisfies this criteria.

**Sequential Pairwise** satisfies this criteria.

**The Borda Count** satisfies this criteria.

**The Sequential Runoff** fails to satisfy the Monotonic Criterion. In sequential runoff, the voter eliminates the lowest ranked voter and the other voters receive a higher rank. This would not satisfy the monotonic criterion because there is no guarantee that the winner will win once the voters are all ranked higher.

## FAQ[edit]

## Homework[edit]

**EXAMPLE:**

Borda Count is an example that **satisfies** the Monotonic Criterion:
We created an example to show how this works. Our point system is 1 point for 3rd place, 2 points for 2nd place, and 3 points for 1st place. We created a situation where contestant C was always our winner.

In the first election, two of the voters ranked contestant B over contestant C.

First Place | A | C | B |

Second Place | C | A | C |

Third Place | B | B | A |

Total # of Voters |
3 | 5 | 2 |

A = 3*3 + 2*5 + 1*A = 21

B = 1*3 + 1*5 + 3*2 = 14

C = 2*3 + 3*5 + 2*2 = **25**

**Contestant C won the first election**.

However, in the second election, the two voters who had previously voted contestant B over contestant C changed their mind and flipped their preference to contestant C over contestant B.

First Place | A | C | C |

Second Place | C | A | B |

Third Place | B | B | A |

Total # of Voters |
3 | 5 | 2 |

A = 3*3 + 2*5 + 1*A = 21

B = 1*3 + 1*5 + 2*2 = 12

C = 2*3 + 5*3 + 2*3 = **27**

**Contestant C wins the second election.**

This shows that Borda Count satisfies the Monotonic Criterion. A voter changed their mind, changing their preference from contestant B to contestant C. However, with their change in vote, the winner is still the same in both elections. The criterion is satisfied because both part a and b are satisfied in the criterion. Because C still wins in both elections even though in the second election two voters voted for C that had previously not done so in the first election.

**NONEXAMPLE:**

Sequential Runoff is an example that **fails to satisfy** the Monotonic Criterion. Here is our example:

Rank | Number of Votes | |||
---|---|---|---|---|

10 | 8 | 6 | 2 | |

First | A | C | B | B |

Second | B | B | C | A |

Third | C | A | A | C |

If this is the case, both **B** and **C** are eliminated in the first Round and **A** wins the election. However, if he voter in the last column changes his first place vote to **A** everything changes.

Rank | Number of Votes | |||
---|---|---|---|---|

10 | 8 | 6 | 2 | |

First | A | C | B | A |

Second | B | B | C | B |

Third | C | A | A | C |

This time, only **B** is eliminated in Round 1, changing the landscape of Round 2 to the following:

Rank | Number of Votes | |||
---|---|---|---|---|

10 | 8 | 6 | 2 | |

First | A | C | C | A |

Second | C | A | A | C |

Now, **A** is on top of 12 lists and **C** is on top of 14 lists, eliminating **A** and making **C** the winner. The only change made in the ballots favored **A** and **A** lost in the next election. This shows that **The Sequential Runoff** system fails to satisfy monotonicity.