Calculate the Lexicographical Excellence (or Lexcel) score.

lexcelRanking() returns the corresponding ranking.

dualLexcelRanking() uses the same score vectors but instead of rewarding participation, it punishes mediocrity.

## Value

Score function returns a list of type LexcelScores and length of powerRelation$elements (unless parameter elements is specified). Each index contains a vector of length powerRelation$eqs, the number of times the given element appears in each equivalence class.

Ranking function returns corresponding SocialRanking object.

## Details

An equivalence class $$\sum_i$$ contains coalitions that are indifferent to one another. In a given power relation created with PowerRelation() or as.PowerRelation(), the equivalence classes are saved in \$eqs.

As an example, consider the power relation $$\succsim: 123 \succ (12 \sim 13 \sim 1 \sim \emptyset) \succ (23 \sim 1 \sim 2)$$. The corresponding equivalence classes are:

$$\sum_1 = \lbrace 123 \rbrace, \sum_2 = \lbrace 12, 13, 1, \emptyset \rbrace, \sum_3 = \lbrace 23, 1, 2 \rbrace.$$

The lexcel score of an element is a vector wherein each index indicates the number of times that element appears in the equivalence class. From our example, we would get

$$\textrm{lexcel}(1) = [ 1, 3, 1 ], \textrm{lexcel}(2) = [ 1, 1, 2 ], \textrm{lexcel}(3) = [ 1, 1, 1 ].$$

## Lexcel Ranking

The most "excellent contribution" of an element determines its ranking against the other elements. Given two Lexcel score vectors $$\textrm{Score}(i)$$ and $$\textrm{Score}(j)$$, the first index $$x$$ where $$\textrm{Score}(i)_x \neq \textrm{Score}(j)_x$$ determines which element should be ranked higher.

From the previous example this would be $$1 > 2 > 3$$, because:

$$\textrm{Score}(1)_2 = 3 > \textrm{Score}(2)_2 = \textrm{Score}(3)_2 = 1$$, $$\textrm{Score}(2)_3 = 2 > \textrm{Score}(3)_3 = 1$$.

## Dual Lexcel Ranking

The dual lexcel works in reverse order and, instead of rewarding high scores, punishes mediocrity. In that case we get $$3 > 1 > 2$$ because:

$$\textrm{Score}(3)_3 < \textrm{Score}(2)_3$$ and $$\textrm{Score}(3)_2 < \textrm{Score}(1)_2$$, $$\textrm{Score}(1)_3 < \textrm{Score}(2)_3$$.

Bernardi G, Lucchetti R, Moretti S (2019). “Ranking objects from a preference relation over their subsets.” Social Choice and Welfare, 52(4), 589--606.

Algaba E, Moretti S, Rémila E, Solal P (2021). “Lexicographic solutions for coalitional rankings.” Social Choice and Welfare, 57(4), 1--33.

Serramia M, López-Sánchez M, Moretti S, Rodríguez-Aguilar JA (2021). “On the dominant set selection problem and its application to value alignment.” Autonomous Agents and Multi-Agent Systems, 35(2), 1--38.