To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Condorcet loser criterion

From Wikipedia, the free encyclopedia

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.

A voting system complying with the Condorcet loser criterion will never allow a Condorcet loser to win. A Condorcet loser is a candidate who can be defeated in a head-to-head competition against each other candidate.[1] (Not all elections will have a Condorcet loser since it is possible for three or more candidates to be mutually defeatable in different head-to-head competitions.)

YouTube Encyclopedic

  • 1/3
    Views:
    30 996
    1 798
    955
  • Voting Theory: Plurality Method and Condorcet Criterion
  • The Condorcet Win Criterion (Voting Theory)
  • Avoiding Arrow's Impossibility (Alternative Voting Criteria)

Transcription

- WELCOME TO A LESSON ON THE PLURALITY VOTING METHOD. IN THIS LESSON WE'LL DEFINE THE PLURALITY VOTING METHOD, DETERMINE WINNERS OF ELECTIONS USING THE PLURALITY METHOD, DEFINE THE CONDORCET FAIRNESS CRITERION AND ALSO FIND A CONDORCET WINNER. THE PLURALITY VOTING METHOD IS PROBABLY THE METHOD YOU'RE MOST FAMILIAR WITH, WHERE THE CHOICE WITH THE MOST FIRST PREFERENCE VOTES IS DECLARED THE WINNER. TIES ARE POSSIBLE AND WOULD HAVE TO BE SETTLED THROUGH SOME SORT OF RUN OFF. THIS METHOD IS SOMETIMES MISTAKENLY CALLED THE MAJORITY METHOD, OR MAJORITY RULES, BUT IT IS NOT NECESSARY FOR A CHOICE TO HAVE GAINED A MAJORITY OF VOTES TO WIN, WHERE A MAJORITY IS OVER 50% OF THE VOTES. SO IT IS POSSIBLE FOR A WINNER TO HAVE A PLURALITY WITHOUT HAVING A MAJORITY. LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES. THE SURVEY ASKED TO RANK WHICH WEST COAST STATES PEOPLE PREFER TO LIVE. THE RESULTS ARE BELOW. USE THE PLURALITY METHOD TO SELECT THE WINNER. WE'RE LOOKING AT THE PREFERENCE TABLE HERE, C = CALIFORNIA, O = OREGON, AND W = WASHINGTON. NOTICE THAT WE FIND THE SUM OF THESE VALUES HERE, WE CAN DETERMINE THE TOTAL VOTES IS 300. TO DETERMINE THE PLURALITY WINNER WE'LL DETERMINE HOW MANY FIRST CHOICE VOTES CALIFORNIA RECEIVED, THEN HOW MANY FIRST CHOICE VOTES OREGON RECEIVED, AND THEN HOW MANY FIRST CHOICE VOTES WASHINGTON RECEIVED. WELL, CALIFORNIA RECEIVED 75 + 94 FIRST CHOICE VOTES, WHERE 75 + 94 = 169. OREGON RECEIVED 51 + 12 FIRST CHOICE VOTES, WHICH IS 63 FIRST CHOICE VOTES. AND FINALLY, WASHINGTON RECEIVED 43 + 25 OR 68 FIRST CHOICE VOTES. SO IN THIS CASE, NOTICE THAT CALIFORNIA RECEIVED THE MOST FIRST CHOICE VOTES. AND THEREFORE, CALIFORNIA IS THE PLURALITY WINNER. NOTICE HOW IN THIS CASE CALIFORNIA RECEIVED 169 FIRST CHOICE VOTES OUT OF 300, WHICH IS APPROXIMATELY 56.3%, WHICH IS MORE THAN 50%. AND THEREFORE, CALIFORNIA WOULD ALSO BE THE MAJORITY WINNER. REMEMBER, A WINNER DOES NOT HAVE TO BE A MAJORITY WINNER TO BE THE PLURALITY WINNER. LET'S TAKE A LOOK AT A SECOND EXAMPLE, WHERE HERE A SMALL GROUP OF COLLEGE STUDENTS RANK THE BEST DESTINATION FOR SPRING BREAK WHERE S = SAN DIEGO, L = LAKE HAVASU, AND R = ROCKY POINT. AGAIN, BY FINDING THE SUM OF THESE VALUES HERE WE CAN DETERMINE THERE ARE A TOTAL OF 17 VOTES. NOTICE, SAN DIEGO RECEIVED A TOTAL OF 4 + 4, OR 8, FIRST PLACE VOTES. LAKE HAVASU RECEIVED A TOTAL OF TWO FIRST PLACE VOTES. AND ROCKY POINT RECEIVED A TOTAL OF 5 + 2, OR 7, FIRST PLACE VOTES. AND SINCE SAN DIEGO RECEIVED THE MOST FIRST CHOICE VOTES, OR FIRST PLACE VOTES, SAN DIEGO IS THE WINNER. NOTICE IN THIS CASE, SAN DIEGO RECEIVED A TOTAL OF 8 FIRST PLACE VOTES OUT OF 17, WHICH IS APPROXIMATELY 47.1%. SO NOTICE HOW HERE EVEN THOUGH SAN DIEGO IS NOT THE MAJORITY WINNER, IT STILL IS THE WINNER USING THE PLURALITY METHOD. THIS LEADS US TO A DISCUSSION ABOUT WHAT CAN BE WRONG ABOUT THE PLURALITY VOTING METHOD. IF THERE ARE THREE OR MORE CHOICES IT IS POSSIBLE THAT A CHOICE COULD LOSE, BUT WHEN COMPARED IN A ONE TO ONE COMPARISON IT COULD BE PREFERRED OVER THE PLURALITY WINNER. AND THIS VIOLATES WHAT'S CALLED A FAIRNESS CRITERION WHERE THE FAIRNESS CRITERIA ARE STATEMENTS THAT SEEM LIKE THEY SHOULD BE TRUE IN A FAIR ELECTION. THE FIRST FAIRNESS CRITERION WE'LL CONSIDER IS CALLED THE CONDORCET CRITERION WHERE IF THERE IS A CHOICE, IT IS PREFERRED IN EVERY ONE TO ONE COMPARISON WITH THE OTHER CHOICES. THAT CHOICE SHOULD BE THE WINNER AND WE CALL THIS WINNER THE CONDORCET WINNER OR CONDORCET CANDIDATE. LET'S LOOK AT TWO MORE EXAMPLES. THIS IS THE EXAMPLE THAT WE SAW BEFORE WHERE WE KNOW THE PLURALITY WINNER WAS SAN DIEGO WITH A TOTAL OF 8 VOTES, BUT NOW WE WANT TO FIND THE CONDORCET WINNER. SO TO FIND THE CONDORCET WINNER WE'LL DO A ONE TO ONE COMPARISON WITH OUR THREE OPTIONS. SO WE'LL COMPARE SAN DIEGO VERSUS LAKE HAVASU. WE'LL COMPARE SAN DIEGO VERSUS ROCKY POINT. AND WE'LL COMPARE LAKE HAVASU VERSUS ROCKY POINT. TO DO THE ONE TO ONE COMPARISON WITH SAN DIEGO AND LAKE HAVASU WE WOULD IGNORE ROCKY POINT. SO WE'LL IGNORE ROCKY POINT HERE, HERE, HERE, HERE, AND HERE. REMEMBER, WE HAVE A TOTAL OF 17 VOTES. SO OF THE 17, SAN DIEGO IS PREFERRED OVER LAKE HAVASU 4 + 4 + 5 TIMES, SO THAT WOULD BE 8 + 5 = 13. SO SAN DIEGO WINS OVER LAKE HAVASU 13 TO 4. NOW WE'LL COMPARE SAN DIEGO TO ROCKY POINT SO WE'LL IGNORE LAKE HAVASU. SO NOTICE SAN DIEGO BEATS ROCKY POINT HERE AND HERE, BUT NOTICE HOW ROCKY POINT WINS HERE, HERE, AND HERE. AND THEREFORE, FOR SAN DIEGO VERSUS ROCKY POINT THE VOTE IS 8 TO 9. NOTICE IN THIS ONE TO ONE COMPARISON ROCKY POINT WINS. AND THEN FINALLY, WE WANT TO CONSIDER LAKE HAVASU VERSUS ROCKY POINT. SO NOW WE'LL IGNORE SAN DIEGO. SO LAKE HAVASU'S PREFERRED OVER ROCKY POINT HERE AND HERE AND THEREFORE, LAKE HAVASU VERSUS ROCKY POINT WOULD BE 6 TO 11. NOW, LOOKING AT THESE ONE TO ONE COMPARISONS NOTICE HOW ROCKY POINT BEATS LAKE HAVASU HERE AND ROCKY POINT ALSO BEATS SAN DIEGO HERE. THEREFORE ROCKY POINT ALWAYS WINS IN A ONE TO ONE COMPARISON. AND THEREFORE, ROCKY POINT IS THE CONDORCET WINNER. SO EVEN THOUGH SAN DIEGO WAS THE PLURALITY WINNER, UNDER THE CONDORCET FAIRNESS CRITERION ROCKY POINT SHOULD BE THE WINNER. LET'S TAKE A LOOK AT ONE MORE EXAMPLE. WE WANT TO FIND THE CONDORCET WINNER, OR CONDORCET CANDIDATE, IF THERE IS ONE. SO THE CANDIDATES ARE "A," B, AND C SO WE'LL DO A ONE TO ONE COMPARISON. WE'LL HAVE "A" VERSUS B, "A" VERSUS C, AND B VERSUS C. NOTICE THE PLURALITY WINNER WOULD BE C WITH A TOTAL OF 16 FIRST CHOICE VOTES. SO FOR "A" VERSUS B WE'LL IGNORE C. SO "A" WOULD WIN OVER B HERE AND HERE. SO "A" VERSUS B WOULD BE 31 TO 10. NEXT, FOR "A" VERSUS C WE'LL IGNORE B. NOTICE, "A" WINS ONLY HERE SO "A" VERSUS C WOULD BE 15 TO 26. AND THEN FOR B VERSUS C WE'LL IGNORE "A". NOTICE HOW B WINS HERE AND C WINS HERE AND HERE. SO B VERSUS C WOULD BE 10 TO 31. SO AGAIN, LOOKING AT THESE TWO HERE NOTICE C WINS OVER B AND HERE C ALSO WINS OVER "A" AND THEREFORE CANDIDATE C IS THE CONDORCET WINNER, BUT NOTICE HOW C IS ALSO THE PLURALITY WINNER HERE. I HOPE YOU FOUND THIS HELPFUL.  

Compliance

Compliant methods include: two-round system, instant-runoff voting (AV), contingent vote, borda count, Schulze method, ranked pairs, and Kemeny-Young method. Any voting method that ends in a runoff passes the criterion, so long as all voters are able to express their preferences in that runoff i.e. STAR voting passes only when voters can always indicate their ranked preference in their scores; if there are more than 6 candidates, then this is impossible.

Noncompliant methods include: plurality voting, supplementary voting, Sri Lankan contingent voting, approval voting, range voting, Bucklin voting and minimax Condorcet.

The Smith criterion implies the Condorcet loser criterion, because no candidate in the Smith set can lose a head-to-head matchup against a candidate not in the Smith set.

Examples

Approval voting

The ballots for Approval voting do not contain the information to identify the Condorcet loser. Thus, Approval Voting cannot prevent the Condorcet loser from winning in some cases. The following example shows that Approval voting violates the Condorcet loser criterion.

Assume four candidates A, B, C and L with 3 voters with the following preferences:

# of voters Preferences
1 A > B > L > C
1 B > C > L > A
1 C > A > L > B

The Condorcet loser is L, since every other candidate is preferred to him by 2 out of 3 voters.

There are several possibilities how the voters could translate their preference order into an approval ballot, i.e. where they set the threshold between approvals and disapprovals. For example, the first voter could approve (i) only A or (ii) A and B or (iii) A, B and L or (iv) all candidates or (v) none of them. Let's assume, that all voters approve three candidates and disapprove only the last one. The approval ballots would be:

# of voters Approvals Disapprovals
1 A, B, L C
1 B, C, L A
1 A, C, L B

Result: L is approved by all three voters, whereas the three other candidates are approved by only two voters. Thus, the Condorcet loser L is elected Approval winner.

Note, that if any voter would set the threshold between approvals and disapprovals at any other place, the Condorcet loser L would not be the (single) Approval winner. However, since Approval voting elects the Condorcet loser in the example, Approval voting fails the Condorcet loser criterion.

Majority Judgment

This example shows that Majority Judgment violates the Condorcet loser criterion. Assume three candidates A, B and L and 3 voters with the following opinions:

Candidates/
# of voters
A B L
1 Excellent Bad Good
1 Bad Excellent Good
1 Fair Poor Bad

The sorted ratings would be as follows:

Candidate   
  Median point
L
 
A
   
B
   
   
 
          Excellent      Good      Fair      Poor      Bad  

L has the median rating "Good", A has the median rating "Fair" and B has the median rating "Poor". Thus, L is the Majority Judgment winner.

Now, the Condorcet loser is determined. If all informations are removed that are not considered to determine the Condorcet loser, we have:

# of voters Preferences
1 A > L > B
1 B > L > A
1 A > B > L

A is preferred over L by two voters and B is preferred over L by two voters. Thus, L is the Condorcet loser.

Result: L is the Condorcet loser. However, while the voter least preferring L also rates A and B relatively low, the other two voters rate L close to their favorites. Thus, L is elected Majority Judgment winner. Hence, Majority Judgment fails the Condorcet loser criterion.

Minimax

This example shows that the Minimax method violates the Condorcet loser criterion. Assume four candidates A, B, C and L with 9 voters with the following preferences:

# of voters Preferences
1 A > B > C > L
1 A > B > L > C
3 B > C > A > L
1 C > L > A > B
1 L > A > B > C
2 L > C > A > B

Since all preferences are strict rankings (no equals are present), all three Minimax methods (winning votes, margins and pairwise opposite) elect the same winners:

Pairwise election results
X
A B C L
Y A [X] 3
[Y] 6
[X] 6
[Y] 3
[X] 4
[Y] 5
B [X] 6
[Y] 3
[X] 3
[Y] 6
[X] 4
[Y] 5
C [X] 3
[Y] 6
[X] 6
[Y] 3
[X] 4
[Y] 5
L [X] 5
[Y] 4
[X] 5
[Y] 4
[X] 5
[Y] 4
Pairwise election results (won-tied-lost): 2-0-1 2-0-1 2-0-1 0-0-3
worst pairwise defeat (winning votes): 6 6 6 5
worst pairwise defeat (margins): 3 3 3 1
worst pairwise opposition: 6 6 6 5
  • [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
  • [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

Result: L loses against all other candidates and, thus, is Condorcet loser. However, the candidates A, B and C form a cycle with clear defeats. L benefits from that since it loses relatively closely against all three and therefore L's biggest defeat is the closest of all candidates. Thus, the Condorcet loser L is elected Minimax winner. Hence, the Minimax method fails the Condorcet loser criterion.

Plurality voting

Tennessee and its four major cities: Memphis in the far west; Nashville in the center; Chattanooga in the east; and Knoxville in the far northeast

Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:

  • Memphis, the largest city, but far from the others (42% of voters)
  • Nashville, near the center of the state (26% of voters)
  • Chattanooga, somewhat east (15% of voters)
  • Knoxville, far to the northeast (17% of voters)

The preferences of each region's voters are:

42% of voters
Far-West
26% of voters
Center
15% of voters
Center-East
17% of voters
Far-East
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis


Here, Memphis has a plurality (42%) of the first preferences, so would be the winner under simple plurality voting. However, the majority (58%) of voters have Memphis as their fourth preference, and if two of the remaining three cities were not in the running to become the capital, Memphis would lose all of the contests 58–42. Hence, Memphis is the Condorcet loser.

Range voting

This example shows that Range voting violates the Condorcet loser criterion. Assume two candidates A and L and 3 voters with the following opinions:

Scores
# of voters A L
2 6 5
1 0 10

The total scores would be:

Scores
candidate Sum Average
A 12 4
L 20 6.7

Hence, L is the Range voting winner.

Now, the Condorcet loser is determined. If all informations are removed that are not considered to determine the Condorcet loser, we have:

# of voters Preferences
2 A > L
1 L > A

Thus, L would be the Condorcet loser.

Result: L is preferred only by one of the three voters, so L is the Condorcet loser. However, while the two voters preferring A over L rate both candidates nearly equal and L's supporter rates him clearly over A, L is elected Range voting winner. Hence, Range voting fails the Condorcet loser criterion.

See also

References

  1. ^ https://arxiv.org/pdf/1801.05911 "We say that an alternative is a Condorcet loser if it would be defeated by every other alternative in a kind of one-on-one contest that takes place in a sequential pairwise voting with a fixed agenda4.– Condorcet loser criterion (CLC), [...] we say that a social choice procedure satisfies the Condorcet loser criterion (CLC) provided that a Condorcet loser is never among the social choices."
This page was last edited on 16 February 2024, at 02:23
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.