Selected topics in finite mathematics/Dropout criterion
[Give a very very brief overview of the criteria?]
Objectives
[edit | edit source]Determine whether or not Plurality, Borda Count, Condorcet, Sequential Pairwise, or Sequential Run off can prove or disprove the given statement
Details
[edit | edit source]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
[edit | edit source]
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
[edit | edit source]
Several methods fail to satisfy the dropout criterion. The following election will be used for several examples.
| 1st choice | A | B | C | D | E |
|---|---|---|---|---|---|
| 2nd choice | B | C | A | E | B |
| 3rd choice | C | A | E | B | D |
| 4th choice | D | E | B | A | A |
| 5th choice | E | D | D | C | C |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
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.
First a pairwise election with E and B, we see that E wins.
| 1st choice | B | E | |||
|---|---|---|---|---|---|
| 2nd choice | B | E | B | ||
| 3rd choice | E | B | |||
| 4th choice | E | B | |||
| 5th choice | E | ||||
| Number of votes | 10 | 7 | 6 | 5 | 7 |
Next we have a pairwise election for E versus C. C wins.
| 1st choice | C | E | |||
|---|---|---|---|---|---|
| 2nd choice | C | E | |||
| 3rd choice | C | E | |||
| 4th choice | E | ||||
| 5th choice | E | C | C | ||
| Number of votes | 10 | 7 | 6 | 5 | 7 |
The next challenger is D, whom is defeated.
| 1st choice | C | D | |||
|---|---|---|---|---|---|
| 2nd choice | C | ||||
| 3rd choice | C | D | |||
| 4th choice | D | ||||
| 5th choice | D | D | C | C | |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
The last pairwise race is C versus A, where A wins, and thus wins the whole election.
| 1st choice | A | C | |||
|---|---|---|---|---|---|
| 2nd choice | C | A | |||
| 3rd choice | C | A | |||
| 4th choice | A | A | |||
| 5th choice | C | C | |||
| Number of votes | 10 | 7 | 6 | 5 | 7 |
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.
The first candidate to be eliminated is D, because D has the least first-place votes.
| 1st choice | A | B | C | D | E |
|---|---|---|---|---|---|
| 2nd choice | B | C | A | E | B |
| 3rd choice | C | A | E | B | D |
| 4th choice | D | E | B | A | A |
| 5th choice | E | D | D | C | C |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
The second candidate to be eliminated is C, as C now has the least first-place votes.
| 1st choice | A | B | C | E | E |
|---|---|---|---|---|---|
| 2nd choice | B | C | A | B | B |
| 3rd choice | C | A | E | A | A |
| 4th choice | E | E | B | C | C |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
The third candidate to eliminate is B.
| 1st choice | A | B | A | E | E |
|---|---|---|---|---|---|
| 2nd choice | B | A | E | B | B |
| 3rd choice | E | E | B | A | A |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
Finally, we see that A is the winner.
| 1st choice | A | A | A | E | E |
|---|---|---|---|---|---|
| 2nd choice | E | E | E | A | A |
| Number of votes | 10 | 7 | 6 | 5 | 7 |
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 | A | B |
|---|---|---|
| 2nd choice | D | A |
| 3rd choice | C | D |
| 4th choice | B | C |
| 5th choice | E | E |
| Number of votes | 2 | 5 |
FAQ
[edit | edit source]Homework
[edit | edit source]Consider the preference schedule below. If this election is held using plurality, is the dropout criterion satisfied?
| 1st choice | A | B | C |
|---|---|---|---|
| 2nd choice | B | C | B |
| 3rd choice | C | A | A |
| Number of votes | 10 | 5 | 8 |
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?
| 1st choice | A | B | C |
|---|---|---|---|
| 2nd choice | B | A | A |
| 3rd choice | C | C | B |
| Number of votes | 10 | 5 | 8 |
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?
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.
| 1st choice | A | B | C |
|---|---|---|---|
| 2nd choice | B | C | A |
| 3rd choice | C | A | B |
| Number of votes | 2 | 2 | 3 |