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:

1. $$(M^\succsim_i)_{p^0,q^0} > (M^\succsim_j)_{p^0,q^0}$$

2. $$(M^\succsim_i)_{p,q^0} = (M^\succsim_j)_{p,q^0}$$ for all $$p < p^0$$

3. $$(M^\succsim_i)_{p,q} = (M^\succsim_j)_{p,q}$$ 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.

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