Selected topics in finite mathematics/Pareto condition

Overview
The Pareto condition measures a voting system's fairness. According to the Pareto Fairness Criterion, in a fair election, a candidate should not win if every voter prefers another candidate. Common sense suggests that a fair election system should satisfy this criterion.

Objectives
To apply the Pareto Fairness Criterion for all the different voting methods and determine which voting method is fair based on the Pareto Fairness Criterion.

Details
Pareto Condition:A fairness criterion that states if a Candidate A is favored over Candidate B by every voter, then Candidate B cannot win.

If candidate A is favored over candidate B by every voter, then candidate B will not win. However we can't say that candidate A is going to be the winner, we can only say that candidate B will not win.

The Fairness Criterion that is presented in the Pareto Condition seems logical, that if a candidate is favored over another candidate by every voter, then that candidate should never win the election. The problem is that the candidate that isn't favored by every voter can be set to win in a voting method.

Examples
Plurality elections do satisfy the criteria. Why? Plurality requires that the winner is the candidate with the most first place votes. If candidate B is ranked below candidate A by every voter then B cannot have any first place votes so B cannot be the winner. Thus, satisfying the Pareto fairness criterion and making Plurality fair according to Pareto.

Condorcet elections satisfies this criteria. Why? In order to win, a Candidate must win all of his elections one on one. Say if Candidate A is favored over Candidate C by every voter, then Candidate C can never win the election. Candidate C will never win because Candidate C has to win ever single one of his elections, but Candidate C can't beat Candidate A if all of the votes go to Candidate A. This means that the Condorcet method is fair according to Pareto's condition.

Sequential Pairwise elections do not satisfy the criteria.Why? Sequential Pairwise elections uses an agenda, which is a sequence of the candidates that will go against each other. The candidate that is left standing wins the entire election. Because Sequential Pairwise voting uses an agenda, it can be set up so that a candidate will win even if it violates the Pareto Fairness Criterion which will be shown in the nonexamples below.

Borda Count does satisfy the criterion.Why? In this method, whoever has the highest number of points wins the election. If Candidate B is favored over Candidate C by every voter, then Candidate C will never be able to get more points than Candidate B and making Candidate C never the winner. This makes Borda Count fair in the eyes of the Pareto Condidtion.

Sequential Runoffs do satisfy the Pareto condition. Why? In this method, the candidate who has the lowest number of first place votes is eliminated in each round until the last candidate standing is declared the winner. When a candidate is eliminated, his or her first place votes are given to the next candidate in the preference schedule. If Candidate B is ranked above Candidate C by every voter then Candidate C must be eliminated before Candidate B. Thus Candidate C cannot win.

Nonexamples
Plurality does satisfy the criterion.

Condorcet does satisfy the criterion

Sequential Pairwise does not satisfy the criteria. Here is an example why,

Using the chart below, we can see that according to Pareto's condition, Candidate A should never win the election because every voter favors Candidate C over Candidate A. However if we set the agenda just right, then we can make Candidate A the winner of the election, thus breaking Pareto's fairness criteria.

If we set the agenda to D, C, B, A, we will get Candidate A as the winner. As we can see, Candidate D goes up against Candidate C first and we can conclude Candidate D will win 2 to 1. So Candidate D moves on to face Candidate B. Candidate B wins this fight with 2 to 1 also and allows him to move on to Candidate A, the candidate that should not win. However if we look closely, Candidate A comes out to be winner with a 2 to 1 victory. Since Candidate A wasn't supposed to win but won anyway, it makes this voting method not fair.

Borda Count: satisfies the criterion

Sequential Runoffs does satisfy the criterion

Homework
Which chart will satisfy the criterion, using Sequential Pairwise;

In chart 1 of the Homework Section use Agenda BACD

In chart 2 use Agenda BADC

Solution ['' In chart one, it does satisfy the fairness criterion because A is favored over B so be will never win. B doesn't end up as the winnder so it does satisfy the criterion. '']

Solution [In chart 2, it does not satisfy the criterion because A is favored over C by every voter but C ends up winning the election, thus making it unfair.]