# Majority Judgment

Majority Judgment is an electoral process with which a single winner is determined on the basis of predicates that the voters assign to the candidates.

The voters rate each candidate with a predicate from a given selection (e.g. “Excellent”, “Good”, “Passable”, “Poor”, “Bad”). Majority Judgment then ranks the candidates based on the median of the ratings received.

## Description of the voting method

The electoral winner is determined by means of majority judgment in three steps: First, the candidates are evaluated by the voters; this is followed by the determination of the predicate for each candidate. Finally, the candidates are placed in an order according to their predicates and, in the event of a tie, a tie resolution rule is applied.

### Voting

Voting under Majority Judgment is performed similarly as in the valuation option . Each voter evaluates all candidates independently of one another and has various ratings available for evaluation. In contrast to the assessment choice, however, these are not values ​​from a value range such as 0-10, but natural language predicates similar to the grades of a school system such as “Excellent”, “Average” or “Poor”. The predicates have a strict, transitive total order among themselves, i. H. there is a clearly “best” rating, a clearly second best, a clearly worst etc.

Example of the choices in a Majority Judgment election:

A rating is now determined for each candidate from the ratings submitted. For this purpose, all ratings that a candidate has received are sorted and then the median of the ratings is determined. This is the predicate assigned to the candidate. For example, if a candidate has received the evaluations {"Excellent", "Good", "Bad"}, the median is "Good" and the candidate receives the predicate "Good".

If the median lies between two ratings, the candidate is assigned the worse one, for example the ratings {"Good", "Passable"} would lead to the predicate "Passable" for the candidate.

Comment : This clearly means that every candidate is assigned the best rating for which there is an absolute majority of voters who have at least set them to this rating. In the above example with the median “good”, two thirds of the voters rate the candidate as “good” or better (but only 33% as “excellent” or better). In the example with the median “Passable”, 100% rate it as “Passable” or better (but only 50% as “Good” or better).

### winner

The winner under Majority Judgment is the candidate who received the best grade. If several candidates have been assigned the best predicate, one occurrence of the predicate assigned to them is removed from their ratings until their median changes and the candidates with a tie are compared according to the new median. This process continues until a clear winner exists.

For example, if one candidate has the ratings {"Excellent", "Good", Good "} and another {" Excellent "," Good "," Passable "}, both are assigned the predicate" Good ". In order to break the tie "Good" removed from both. The first candidate now has the ratings {"Excellent", Good "} and thus continues to be rated" Good ". The second candidate now has the ratings {"Excellent", "Passable"} and thus the rating "Passable". The first candidate would thus win against the second candidate.

Note : Since the predicate assigned to a candidate is carried by an absolute majority and the candidate with the best predicate is elected, Majority Judgment selects a winner, so that the best predicate, which is carried by an absolute majority, determines the winner.

### Alternative description

Majority judgment can also be seen as a modification of the Bucklin election , in which equal and omitted ranks are allowed. Accordingly, the algorithm for the Bucklin election can also be used, with slight modification, to determine the winner under Majority Judgment:

To do this, the following two steps must be applied to each predicate (starting with the best and then descending) until the victory condition is met:

1. Counting : The number of voters assigned to the current predicate is added to each candidate's previous votes.
2. Victory condition : If a candidate has a vote from the absolute majority of voters, he is the winner and the process is terminated.

If several candidates have an absolute majority at this point, the tie will be resolved as above.

## example

Consider an election with four candidates A, B, C and D and the following ratings by the 10 voters:

number 1 2 3 4th Median
reviews A. Excellent Passable Well Well Well
B. Poor Well Excellent Well Well
C. Poor Well Excellent Poor Poor
D. Well Passable Poor Well Passable

The sorted reviews would be as follows:

candidate
 ↓ Median point
A.
B.
C.
D.

 Excellent Well Passable Poor

The medians of candidates A and B are both “good”, C is given the rating “poor” and D “passable”. In order to break the tie between A and B, both “good” ratings are removed until the median changes for one of the two. If you remove 5 "good" ratings, the following ratings remain:

candidate
 ↓ Median point
A.
B.

 Excellent Well Passable Poor

The median of B is now “Excellent”, while the median of A remains “Good”. Hence, B is the winner.

## properties

In social election theory, there are a few criteria for determining the quality of an electoral system, among which majority judgment scores as follows:

Majority judgment meets the monotony criterion, the independence from clone alternatives, the favorite betrayal criterion and the independence from 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, the reversal symmetry criterion and the later no harm criterion.

Since these criteria were designed for preference voting systems, their interpretation with regard to evaluation procedures such as majority judgment is partly ambiguous. In fact, Majority Judgment fulfills some of these criteria adapted to evaluation procedures: the modified (mutual) majority criterion and the predicate consistency criterion.

### Consistency criterion

The consistency criterion says: If one divides the electorate into two groups and if the electoral process for both groups selects the same candidate as the winner, then this candidate must also be chosen as the winner for the entire electorate.

Majority judgment violates the consistency criterion. This is illustrated by the following example with two candidates A and B and 6 voters with the following ratings:

Candidates /
# of voters
A. B.
1 Well Excellent
1 Well Passable
1 Poor Passable
1 Passable Well
1 Passable Poor

The line marks the separation of the two groups of voters. The top three voters belong to voter group I, the bottom three to voter group II.

#### Result for voter group I.

The voter group I gives the following ratings to the candidates:

Candidates /
# of voters
A. B.
1 Well Excellent
1 Well Passable
1 Poor Passable

The sorted reviews would be as follows:

candidate
 ↓ Median point
A.
B.

Result : A is assigned the predicate "Good", B gets the predicate "Passable". Therefore, A is the Majority Judgment winner of voter group I.

#### Result for voter group II

The electoral group II gives the following ratings to the candidates:

Candidates /
# of voters
A. B.
1 Passable Well
1 Passable Poor

The sorted reviews would be as follows:

candidate
 ↓ Median point
A.
B.

Result : A is assigned the predicate “Passable”, B gets the predicate “Poor”. Therefore, A is the Majority Judgment winner of voter group II.

#### Result for a united electorate

The united electorate evaluates the candidates as follows:

Candidates /
# of voters
A. B.
1 Well Excellent
1 Well Passable
1 Poor Passable
1 Passable Well
1 Passable Poor

The sorted reviews would be as follows:

candidate
 ↓ Median point
A.
B.

Both A and B are given the rating “Passable”. In fact you can see that the sorted ratings are only different at the edges, with B being preferred at both ends. So if you remove the same ratings until a difference occurs (twice "Passable", once "Poor" and once "Good"), the following remains:

candidate
 ↓ Median point
A.
B.

Result : After removing identical entries, A is assigned the predicate "Bad", B gets the predicate "Poor". Hence, B is the Majority Judgment winner of the united electorate.

#### Conclusion

Majority Judgment selects A as the winner for both the first and second group of voters; However, if you combine the two groups of voters, Majority Judgment makes B the winner. Therefore, Majority Judgment violates the consistency criterion.

#### Predicate consistency criterion

Majority judgment fulfills a modified consistency criterion, which states: If the electorate is divided into two groups and the electoral process assigns the same rating to a candidate for both groups, this rating must also be assigned to this candidate for the entire electorate.

This can easily be seen by means of an informal proof: Let us consider a permanent candidate. Let us be a set of voters, a predicate, the number of voters from V who gave the candidate a rating better than and, analogously, the number of voters from V who gave the candidate a rating worse than . ${\ displaystyle V}$${\ displaystyle P}$${\ displaystyle P _ {>} (V)}$${\ displaystyle P}$${\ displaystyle P _ {<} (V)}$${\ displaystyle P}$

Then Majority Judgment assigns the predicate to a candidate if and only if ${\ displaystyle P}$

• Less than half of the voters rated the candidate better than : and${\ displaystyle P}$${\ displaystyle P _ {>} (V) <50 \, \% \ cdot | V |}$
• at most half of the voters rated the candidate worse than :${\ displaystyle P}$${\ displaystyle P _ {<} (V) \ leq 50 \, \% \ cdot | V |}$

If the electorate is divided into two parts and the candidate has been assigned the predicate P in both parts , then the following applies: ${\ displaystyle V = V_ {1} {\ dot {\ cup}} V_ {2}}$

(I) ${\ displaystyle P _ {>} (V_ {1}) <50 \, \% \ cdot | V_ {1} |}$
(II) ${\ displaystyle P _ {<} (V_ {1}) \ leq 50 \, \% \ cdot | V_ {1} |}$
(III) ${\ displaystyle P _ {>} (V_ {2}) <50 \, \% \ cdot | V_ {2} |}$
(IV) ${\ displaystyle P _ {<} (V_ {2}) \ leq 50 \, \% \ cdot | V_ {2} |}$

The number of voters from the entire electorate who gave the candidate a better rating is then less than half, because ${\ displaystyle P}$

${\ displaystyle P _ {>} (V) = P _ {>} (V_ {1} {\ dot {\ cup}} V_ {2}) = P _ {>} (V_ {1}) + P _ {>} ( V_ {2}) {\ stackrel {(I), (III)} {<}} 50 \, \% \ cdot | V_ {1} | +50 \, \% \ cdot | V_ {2} | = 50 \, \% (| V_ {1} | + | V_ {2} |) = 50 \, \% \ cdot | V |}$

Similarly, it follows that at most half of the voters of the entire electorate rated the candidate worse than . ${\ displaystyle P}$

Thus, Majority Judgment assigns the predicate to the candidate also with regard to the entire electorate .${\ displaystyle P}$${\ displaystyle qed}$

### Majority criterion

The majority criterion expresses that a candidate who is preferred by the majority of voters over all other candidates must win. Majority judgment violates the majority criterion. This is illustrated by the following example with two candidates A and B and three voters with the following ratings:

Voters x y z Median
reviews A. Excellent Passable Poor Passable

The sorted reviews would be as follows:

candidate
 ↓ Median point
A.
B.

 Excellent Well Respectable Passable Poor Bad

The median of A is "Passable", the median of B is "Eightful". Thus, B is Majority Judgment winner, although an absolute majority of 2 out of 3 voters favors A. Majority judgment thus violates the majority criterion.

In fact, the example can be extended to any number of voters if one adds a type x voter (who rates both candidates above the median) and a type z voter (who rates both candidates below the median). The good predicates given by type x voters are balanced out by the bad predicates given by type z voters, so that the one type y voter is always decisive for the result.

Majority judgment fulfills a weakened majority criterion, which states that a candidate who receives the best rating from the majority of voters as the only candidate must be the winner.