Voting paradox
Encyclopedia
The voting paradox is a situation noted by the Marquis de Condorcet
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
), even if the preferences of individual voters are not. This is paradox
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
.
When a Condorcet method
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
), even if the preferences of individual voters are not. This is paradox
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
.
When a Condorcet method
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
), even if the preferences of individual voters are not. This is paradox
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
.
When a Condorcet method
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
), even if the preferences of individual voters are not. This is paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
| Voter | First preference | Second preference | Third preference |
|---|---|---|---|
| Voter 1 | A | B | C |
| Voter 2 | B | C | A |
| Voter 3 | C | A | B |
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
Majority rule
Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
Kingmaker scenario
A kingmaker scenario, in a game of three or more players, is an endgame situation where a player unable to win has the capacity to determine which player among others is the winner. Said player is referred to as the kingmaker or spoiler...
.
When a Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
See also
- Kenneth ArrowKenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, Section 1 with an example of a distributional difficulty of intransitivity + majority rule - Arrow's impossibility theoremArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
- Gibbard-Satterthwaite theoremGibbard-Satterthwaite theoremThe Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...
- Independence of irrelevant alternativesIndependence of irrelevant alternativesIndependence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
- Nakamura numberNakamura numberIn cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
- Smith setSmith setIn voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome...
- Instant-runoff votingInstant-runoff votingInstant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
- Discursive dilemmaDiscursive dilemmaDiscursive dilemma or doctrinal paradox is a paradox in social choice theory which states that aggregating judgements with majority voting might result in a self-contradictory judgement....
The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
), even if the preferences of individual voters are not. This is paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
| Voter | First preference | Second preference | Third preference |
|---|---|---|---|
| Voter 1 | A | B | C |
| Voter 2 | B | C | A |
| Voter 3 | C | A | B |
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
Majority rule
Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
Kingmaker scenario
A kingmaker scenario, in a game of three or more players, is an endgame situation where a player unable to win has the capacity to determine which player among others is the winner. Said player is referred to as the kingmaker or spoiler...
.
When a Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
See also
- Kenneth ArrowKenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, Section 1 with an example of a distributional difficulty of intransitivity + majority rule - Arrow's impossibility theoremArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
- Gibbard-Satterthwaite theoremGibbard-Satterthwaite theoremThe Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...
- Independence of irrelevant alternativesIndependence of irrelevant alternativesIndependence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
- Nakamura numberNakamura numberIn cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
- Smith setSmith setIn voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome...
- Instant-runoff votingInstant-runoff votingInstant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
- Discursive dilemmaDiscursive dilemmaDiscursive dilemma or doctrinal paradox is a paradox in social choice theory which states that aggregating judgements with majority voting might result in a self-contradictory judgement....
The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...
in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
), even if the preferences of individual voters are not. This is paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
ical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):
| Voter | First preference | Second preference | Third preference |
|---|---|---|---|
| Voter 1 | A | B | C |
| Voter 2 | B | C | A |
| Voter 3 | C | A | B |
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. The requirement of majority rule
Majority rule
Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...
then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter
Kingmaker scenario
A kingmaker scenario, in a game of three or more players, is an endgame situation where a player unable to win has the capacity to determine which player among others is the winner. Said player is referred to as the kingmaker or spoiler...
.
When a Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this trivial example because each candidate is in an exactly symmetrical situation.
See also
- Kenneth ArrowKenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, Section 1 with an example of a distributional difficulty of intransitivity + majority rule - Arrow's impossibility theoremArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
- Gibbard-Satterthwaite theoremGibbard-Satterthwaite theoremThe Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference...
- Independence of irrelevant alternativesIndependence of irrelevant alternativesIndependence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
- Nakamura numberNakamura numberIn cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....
- Smith setSmith setIn voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome...
- Instant-runoff votingInstant-runoff votingInstant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...
- Discursive dilemmaDiscursive dilemmaDiscursive dilemma or doctrinal paradox is a paradox in social choice theory which states that aggregating judgements with majority voting might result in a self-contradictory judgement....