Selected topics in finite mathematics/Independent of irrelevant alternatives

[Give a very very brief overview of the criteria?]

Overview
According to the Independent of Irrelevant Alternatives Criterion, in a fair election, it is impossible for a candidate to move from a non-winner position to being a winner unless one voter reverses the order in which the non-winning candidate and the winning candidate were ranked.

Objectives
To apply the Independent of Irrelevant Alternatives Criterion for all of the different voting methods.

The main objective is to determine which voting systems are satisfied or not satisfied by the Independence of Irrelevant Alternatives Criterion.

Details
The definition of the criterion for Independence of Irrelevant Alternatives is if it is impossible for a candidate, let's call him John, to move from nonwinner status to winner status unless another voter rearranges/reverses the order in which John and the winning candidate ranked are moved.

If it is impossible for candidate X to move from nonwinner status to winner status unless at least one voter reverses the order in which she had candidate X and the winning candidate ranked, then this satisfies independence of irrelevant alternatives (IIA).

Examples
Condorcet does satisfy independence of irrelevant alternatives. - If there was a Condorcet winner, then that winner will inevitably beat every other candidate, in a one-on-one contest since the winner will have the highest amount. As long as no voter reverses their order in which they ranked the winner and a non-winner, the winner will defeat every other nonwinner

Example: If A has 10 votes, B has 7, C with 4 and D with 8, this yields A as the Condorcet winner and B,C, and D as the non-winners. Therefore A defeats every other candidate. Even if a voter reverses the order in which they have, such as D versus A one on one, then D still is a non-winner.

Nonexamples
Plurality does not satisfy the criterion on Independent of Irrelevant Alternatives

Let's say there are 3 candidates: A, B, C.

In this election, A wins because it has the most votes, 4. However, if a recount occurs where the votes for candidates A and B were in fact counted incorrectly, then the new election would appear like...

Now, B would win the election as it receives 7 votes to A's 0. Therefore, IIA is not satisfied as B moves from the losers bracket to become a winner.

Sequential pairwise elections does not satisfy the independent of irrelevant alternatives criterion.

A vs. B: A

A vs. C: A

A vs. D: A

A vs. E: E

If the last ranking of votes was changed so that candidate D is now ranked first instead of last the election would now look like this-

The agenda stays the same A, B, C, D, E

A vs. B: A

A vs. C: A

A vs. D: D

D vs. E: D

Because the ranking was changed, D was moved from a non-winner position to a winning position. The original winner E did not stay the winner when the ranking was changed therefore Sequential Pairwise voting system does not satisfy the Independent of Irrelevant Alternatives Criterion.

Borda Count fails to satisfy independence of irrelevant alternatives. Suppose there are 5 voters in this system. If A has 6 votes, B has 5, and C has 4, then the winner is A.

However, if C moves down between A and B, then A now has 6 votes, B has 7 votes, and C has 2 votes. The Borda count now yields B as the winner. Since B went from being a nonwinner to a winner, then Borda count fails to satisfy IIA.

Sequential Runoffs does not satisfy the independence of irrelevant alternatives criteria.

In this election, C would be eliminated first because it has the least number of first place votes. In the second round, B has only 2 first place votes, while A has 3. B is eliminated and A is declared the winner.

Suppose B and C (two non-winners) are switched in the first column during the same election as above. In this case, B would be eliminated first. C would be declared the winner after A was eliminated second round. Two irrelevant alternatives were swapped, leading to a change in the winner. Sequential Runoffs there fails to satisfy IIA.

Homework
Election: College SGA Election (Using Borda Count) Carson, Dylan and Emily are all running for SGA President this year. It's a small school, and only 5 people vote:

According to this chart, Dylan wins with a Borda Score of 6 while Carson and Emily are non-winners with Borda Scores of 4 and 5. However, Emily decides to increase her marketing towards the student body, which increases her popularity. tThere is a new election with 5 voters still:

Now, Emily wins with a Borda Score of 7, even though Carson, not Dylan was really affected by the changes in Emily's popularity in her marketing plan. Therefore, Borda Count does not satisfy IIA.

100 People vote in a (Condorcet) election to determine which of the four TV shows is preferred. The preference schedule is shown below:

Solution - Now Emily wins, even though she had the fewest votes from the previous election, therefore this does not satisfy independence of Irrelevant Alternatives criteria.

Solution W vs. X: 67 vs. 33= W wins

W vs. Y: 65 vs. 35= W wins

W vs. Z: 84 vs. 16= W wins

Even if we were to rearrange the candidates below the winner, W would still win all the elections. Therefore, Condorcet satisfies the criterion.