L1 Ranking

Calculate the $L^{(1)}$ scores.

Usage

L1Scores(powerRelation, elements = powerRelation$elements)

L1Ranking(powerRelation)

lexcel1Scores(powerRelation, elements = powerRelation$elements)

lexcel1Ranking(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 L1Scores 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

Similar to lexcelRanking(), the number of times an element appears in each equivalence class is counted. In addition, we now also consider the size of the coalitions.

Let $N$ be a set of elements, $\succsim \in \mathcal{T}(\mathcal{P})$ a power relation, and $\Sigma_1 \succ \Sigma_2 \succ \dots \succ \Sigma_m$ its corresponding quotient order.

For an element $i \in N$, construct a matrix $M^\succsim_i$ with $m$ columns and $|N|$ rows. Whereas each column $q$ represents an equivalence class, each row $p$ corresponds to the coalition size.

$$(M^\succsim_i)_{p,q} = |\lbrace S \in \Sigma_q: |S| = p \text{ and } i \in S\rbrace|$$

The $L^{(1)}$ rewards elements that appear in higher ranking coalitions as well as in smaller coalitions. When comparing two matrices for a power relation, if $M^\succsim_i >_{L^{(1)}} M^\succsim_j$, this suggests that there exists a $p^0 \in \{1, \dots, |N|\}$ and $q^0 \in \{1, \dots, m\}$ such that the following holds:

$(M^\succsim_i)_{p^0,q^0} > (M^\succsim_j)_{p^0,q^0}$
$(M^\succsim_i)_{p,q^0} = (M^\succsim_j)_{p,q^0}$ for all $p < p^0$
$(M^\succsim_i)_{p,q} = (M^\succsim_j)_{p,q}$ for all $q < q^0$ and $p \in \{1, \dots, |N|\}$

Example

Let $\succsim: (123 \sim 13 \sim 2) \succ (12 \sim 1 \sim 3) \succ (23 \sim \{\})$. From this, we get the following three matrices:

$$ M^\succsim_1 = \begin{bmatrix} 0 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix} M^\succsim_2 = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 1\\ 1 & 0 & 0 \end{bmatrix} M^\succsim_3 = \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 0 \end{bmatrix} $$

From $(M^\succsim_2)_{1,1} > (M^\succsim_1)_{1,1}$ and $(M^\succsim_2)_{1,1} > (M^\succsim_3)_{1,1}$ it immediately follows that $2$ is ranked above $1$ and $3$ according to $L^{(1)}$.

Comparing $1$ against $3$ we can set $p^0 = 2$ and $q^0 = 2$. Following the constraints from the definition above, we can verify that the entire column 1 is identical. In column 2, we determine that $(M^\succsim_1)_{1,q^0} = (M^\succsim_3)_{1,q^0}$, whereas $(M^\succsim_1)_{p^0,q^0} > (M^\succsim_3)_{p^0,q^0}$, indicating that $1$ is ranked higher than $3$, hence $2 \succ 1 \succ 3$ according to $L^{(1)}$.

Aliases

For better discoverability, lexcel1Scores() and lexcel1Ranking() serve as aliases for L1Scores() and L1Ranking(), respectively.

References

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

Examples

pr <- as.PowerRelation("(123 ~ 13 ~ 2) > (12 ~ 1 ~ 3) > (23 ~ {})")
scores <- L1Scores(pr)
scores$`1`
#>      [,1] [,2] [,3]
#> [1,]    0    1    0
#> [2,]    1    1    0
#> [3,]    1    0    0
#      [,1] [,2] [,3]
# [1,]    0    1    0
# [2,]    1    1    0
# [3,]    1    0    0

L1Ranking(pr)
#> 2 > 1 > 3
# 2 > 1 > 3