Majority Judgment
Majority Judgment is an elective procedure by which a single winner on the basis of predicates which assign voters to the candidates is determined.
Voters rate each candidate with a predicate from a given selection (eg " Excellent ", " Good ", " Passable ", " Poorly ", " bad" ). Majority Judgment ranks the candidates then using the median of the reviews received.
- 3.1 Consistency criterion 3.1.1 Result for voters Group I
- 3.1.2 Result for electoral group II
- 3.1.3 Results for united electorate
- 3.1.4 Conclusion
- 3.1.5 predicates consistency criterion
Description of the selection method
The determination of the election winner by Majority Judgment takes place in three steps: First, the evaluation of the candidates is conducted by the voters; to this, the determination of the predicate follows each candidate. Finally, the candidates are placed according to their predicates in a sequence and used in a tie of candidates resolution rule for DC stands.
Voting
Voting under Majority Judgment is very similar as in the evaluation choice. Each voter evaluates all candidates independently and has to evaluate different predicates available. In contrast to vote choice, these are, however, not to values from a range of values such as 0-10, but natural language predicates similar to the notes of a school system as "excellent", " Fair " or " Poorly ". The predicates have among themselves a strict, transitive total order, ie there is a clear "best" predicate, a clearly second best, a clearly worst, etc.
Candidate predicate
From the ratings given a predicate is then determined for each candidate. For this purpose all ratings received by a candidate, sorted and then determined the median of the reviews. This is the predicate that is assigned to the candidate. If a candidate has, for example, the ratings {" excellent", "good", "bad "} get, so the median is "good" and therefore the candidate is awarded the title "Good".
If the median between two predicates, so the candidate is assigned to the worse, for example, the reviews {" Good ", " Passable "} would lead to the predicate " passable " for the candidate.
Note: This means concretely that each candidate gets assigned to the best predicate for which an absolute majority of voters there that has put him at least on this predicate. In the example above, with median "good" rate two-thirds of the voters the candidate as "good" or better (but only 33 % as "excellent" or better). In the example with median " Passable " rate it 100 % with " passable " or better ( but only 50 % as "good " or better).
Winner
The winner among Majority Judgment is the candidate who got assigned the best predicate. If several candidates are associated with the best predicate, so each one occurrence of their associated predicate is removed until you change the median and compared the candidates with a tie, according to the new median of their reviews. This process is continued until a clear winner exists.
For example, if a candidate's reviews {" Excellent ", " Good," Good "} and another {" excellent ", " good ", " Passable "}, then both the predicate " good " assigned. Shall be able to resolve a tie away with both a "good". the first candidate now has the ratings { "excellent " Good " }, and therefore kept the title of" Good ". The second candidate is now the ratings {" excellent", " Passable "}, thus the title " passable ". The first candidate would thus win against the second candidate.
Remark: This a candidate associated predicate is supported by an absolute majority, the candidate is elected with the best predicate Majority Judgment selects a winner, so the best predicate, which is carried by an absolute majority, determines the winner.
Alternative description
Majority Judgment can also be regarded as a modification of Bucklin - election, are allowed in the same ranks and exuberant. According to the algorithm for the choice Bucklin can be used with slight modification also for determining the winner under Majority Judgment:
For this purpose, the following two steps on each predicate are (starting with the best and then descending ) to be applied to the victory condition is met:
If several candidates have at this point an absolute majority, the tie is resolved as above.
Example
Consider an election with four candidates A, B, C and D and the following assessments by the 10 voters:
The assorted reviews were as follows:
The medians of candidates A and B are both "good", C is awarded the title " Poorly " and D " passable ". To resolve the tie between A and B, so many "good" reviews are now removed until the median changes from two in a at both. Removes one each 5 "good" reviews, so remain following ratings:
The median of B is now in "excellent", while the median of A at "good" remains. Therefore, B is the winner.
Properties
In social choice theory, there are some criteria to determine the quality of the electoral system, under which Majority Judgment cuts as follows:
Majority Judgment satisfies the monotonicity criterion, the independence of clone alternatives, the Favorite - betrayal - criterion and the independence of irrelevant alternatives.
Majority Judgment violates the Condorcet criterion, the Condorcet loser criterion, the majority criterion, the mutual majority criterion, the participation criterion, the consistency criterion that reversal symmetry criterion and the later- no-harm criterion.
Since these criteria were designed for preferential voting systems, their interpretation regarding assessment procedures as Majority Judgment is partially ambiguous. In fact Majority Judgment met on assessment procedures adapted versions of some of these criteria: the modified (mutual ) majority criterion and the predicate consistency criterion.
Consistency criterion
The consistency criterion states: If you divide the electorate into two groups and selects the electoral process with respect to both groups, the same candidate as the winner, so it must also choose with respect to the total electorate as the winner this candidate.
Majority Judgment violates the consistency criterion. This is illustrated by the following example of two candidates A and B, and 6 voters with the following ratings:
The line marks the separation of the two groups of voters. The top three voters belong to the group of voters I, the lower three voter group II
Result for voters Group I
Voters Group I therefore award the following ratings to the candidates:
The assorted reviews were as follows:
Result: A receives the rating " Good " is assigned, B gets the title of " passable ". Therefore, the A Majority Judgment is the winner of the electoral group I.
Result for voters Group II
The electoral group II so will award the following ratings to the candidates:
The assorted reviews were as follows:
Result: A receives the title of " passable " is assigned, B gets the title of " Poorly ". Therefore, the A Majority Judgment is the winner of the electoral group II
Result for united electorate
The combined electorate assesses the candidate as follows:
The assorted reviews were as follows:
Both A and B will receive the title of " passable ". In fact, one sees that the assorted reviews are different only at the edges, on both end B is preferred. So if you like reviews removed until a difference occurs ( twice " Passable ", even " Poorly " and even "good" ), then remains:
Result: After removing identical entries, gets A the predicate "bad " associated, B gets the title of " Poorly ". Therefore, B is the Majority Judgment winner is the combined electorate.
Conclusion
Majority Judgment selects both for the first and for the second voter group A as the winner; you combine the two groups of voters as Majority Judgment, however, chooses B the winner. Therefore Majority Judgment violates the consistency criterion.
Predicates consistency criterion
Majority Judgment satisfies a modified consistency criterion, which states: If you divide the electorate into two groups and assigns the election process a candidate with respect to both groups the same predicate, so this candidate this predicate must also be assigned with respect to the total electorate.
This is easy to recognize by means of an informal proof: Consider a solid candidate. Be a voter amount, a predicate, the number of voters in V which have the candidate a score better than analog and given the number of voters in V which have the candidate a score as given worse.
Then Majority Judgment assigns a candidate iff the predicate if
- Less than half of the electorate has the candidate better than rated: and
- More than half of the electorate has the candidate worse than rated:
Thus, if the electorate is divided into two parts and both the candidate has to get assigned to the predicate P, then:
The number of voters out of the total electorate, which have given the candidate a predicate is better then less than half, because
Analog follows that more than half of the voters of the total electorate candidate worse than rated.
Thus, Majority Judgment shall notify the candidates also regarding the total electorate to the predicate.
Majority criterion
The majority criterion expresses that a candidate who is preferred by the majority of voters from all other candidates must win. Majority Judgment violates the majority criterion. This is illustrated by the following example of two candidates A and B, and three voters with the following ratings:
The assorted reviews were as follows:
The median of A is " passable ", the median of B at " Honourable ". Thus B Majority Judgment is the winner, although an absolute majority of voters 2 of 3 A preferable. Majority Judgment thus violates the majority criterion.
Indeed, one can extend the example to any voter numbers when you each a selector of the type x ( which both candidates above the median rated) and a selector of type z ( which both candidates below the median rated) adds. The good predicates, which awarded the voters of type x are compensated from the bad predicates of voters of type z, so that more of a voter is critical of the type y for the result.
Majority Judgment satisfies a weakened majority criterion, which states that a candidate, who is the only candidate gets the best grade of the majority of the voters must be the winner.