# 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.

1st choice 2nd choice 3rd choice 4th choice 5th choice A B C D E B C A E B C A E B D D E B A A E D D C C

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.

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.

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).

1st choice 2nd choice A B D A C D B C E E

## Homework

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

1st choice 2nd choice 3rd choice A B C B C B C A A

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

1st choice 2nd choice 3rd choice A B C B A A C C B

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?