Selected topics in finite mathematics/Dropout criterion

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

Objectives
Determine whether or not Plurality, Borda Count, Condorcet, Sequential Pairwise, or Sequential Run off can prove or disprove the given statement

Details
Dropout Criterion: Occurs when the the individual who wins does not change when a loser drops out of the election. A 'fair' voting system should meet this criterion.

If then form: If there is a winner, and the loser drops out, then the winner does not change.

[Give a prose-explanation of the criteria?]

Examples


The Condorcet method does satisfy the dropout criterion. Consider that under the Condorcet method a winning candidate must defeat each other candidate in a one-on-one election. If one of the losing candidates drops out, then the winning candidate would still beat the remaining candidates in one-on-one elections.

Nonexamples
Several methods fail to satisfy the dropout criterion. The following election will be used for several examples.

Plurality fails to satisfy the dropout criteria. A will win as it is. But if B drops out, then C becomes the winner.

Sequential pairwise elections fails to satisfy this criteria. Consider the above election with agenda E B C D A. Currently A is set to win, but if E drops out, then B becomes the winner. Explanation: sequential pairwise elections First a pairwise election with E and B, we see that E wins.

Next we have a pairwise election for E versus C. C wins.

The next challenger is D, whom is defeated.

The last pairwise race is C versus A, where A wins, and thus wins the whole election.

Sequential runoffs fail to satisfy this criteria. Consider the above election. A will win it as is. But if E drops out, then B wins. Explanation: sequential runoffs The first candidate to be eliminated is D, because D has the least first-place votes.

The second candidate to be eliminated is C, as C now has the least first-place votes.

The third candidate to eliminate is B.

Finally, we see that A is the winner.

For Borda count we turn to another election. Below we have an election where A is winning (23-22), but if C drops out then B becomes the winner (17-16).

Homework
Consider the preference schedule below. If this election is held using plurality, is the dropout criterion satisfied?

Solution No, the dropout criterion is not satisfied in this election. For instance if B drops, then the winner changes (to C).

Consider the preference schedule below. If this election is held using plurality, is the dropout criterion satisfied? Solution Yes, the dropout criterion is satisfied in this election. This is because of the following two facts. If B drops, A still wins. If C drops, A still wins.

Consider the following voting system. It is similar to sequential runoffs, but instead of eliminating the candidate least favored in first, we'll sequentially eliminate the candidate with the most last-place votes. The last candidate left is declared the winner. Does this voting system satisfy the dropout criterion? Solution No, consider the election below. As it is, C will be the winner. But if A drops out, then B becomes the winner. Because there exists an election in which the criterion is not satisfied, the voting system as a whole does not satisfy the criterion.