Lexicographical Excellence

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.

Usage

lexcelScores(powerRelation, elements = powerRelation$elements)

lexcelRanking(powerRelation)

dualLexcelRanking(powerRelation)

Arguments

powerRelation: A PowerRelation object created by PowerRelation() or as.PowerRelation()
elements: Vector of elements of which to calculate their scores. By default, the scores of all elements in powerRelation$elements are considered.

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

References

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.

Examples

# note that the coalition {1} appears twice
# 123 > 12 ~ 13 ~ 1 ~ {} > 23 ~ 1 ~ 2
# E = {123} > {12, 13, 1, {}} > {23, 1, 2}
pr <- suppressWarnings(as.PowerRelation(
  "123 > (12 ~ 13 ~ 1 ~ {}) > (23 ~ 1 ~ 2)"
))

# lexcel scores for all elements
# `1` = c(1, 3, 1)
# `2` = c(1, 1, 2)
# `3` = c(1, 1, 1)
lexcelScores(pr)
#> $`1`
#> [1] 1 3 1
#> 
#> $`2`
#> [1] 1 1 2
#> 
#> $`3`
#> [1] 1 1 1
#> 
#> attr(,"class")
#> [1] "LexcelScores"

# lexcel scores for a subset of all elements
lexcelScores(pr, c(1, 3))
#> $`1`
#> [1] 1 3 1
#> 
#> $`3`
#> [1] 1 1 1
#> 
#> attr(,"class")
#> [1] "LexcelScores"
lexcelScores(pr, 2)
#> $`2`
#> [1] 1 1 2
#> 
#> attr(,"class")
#> [1] "LexcelScores"

# 1 > 2 > 3
lexcelRanking(pr)
#> 1 > 2 > 3

# 3 > 1 > 2
dualLexcelRanking(pr)
#> 3 > 1 > 2