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

• 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

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

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

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.

Homework

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*2 = 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*2 = 21

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

C = 2*3 + 3*5 + 3*2 = 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 the 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.