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 |
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 |
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 |