LPSScores {socialranking}R Documentation

LP* Ranking

Description

Calculate the LpL^{p^*} scores.

Usage

LPSScores(powerRelation, elements = powerRelation$elements)

LPSRanking(powerRelation)

lexcelPSScores(powerRelation, elements = powerRelation$elements)

lexcelPSRanking(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.

Details

Let NN be a set of elements, T(P)\succsim \in \mathcal{T}(\mathcal{P}) a power relation, and Σ1Σ2Σm\Sigma_1 \succ \Sigma_2 \succ \dots \succ \Sigma_m its corresponding quotient order.

For an element iNi \in N, construct a matrix MiM^\succsim_i with mm columns and N|N| rows. Whereas each column qq represents an equivalence class, each row pp corresponds to the coalition size.

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

For i,jNi, j \in N, the social ranking solution LpL^{p^*} then ranks ii strictly above jj if one of the following conditions hold:

  1. {i}{j}\lbrace i \rbrace \succ \lbrace j \rbrace;

  2. {i},{j}Σk\lbrace i \rbrace, \lbrace j \rbrace \in \Sigma_k and there exists a row p0{2,,N}p_0 \in \lbrace 2, \dots, |N|\rbrace and column q0{1,,k1}q_0 \in \lbrace 1, \dots, k-1\rbrace such that:

    (Mi)p,q=(Mj)p,qp<p0,q<k,(M^\succsim_i)_{p,q} = (M^\succsim_j)_{p,q}\quad \forall p < p_0, q < k,

    (Mi)p0,q=(Mj)p0,qq<q0, and(M^\succsim_i)_{p_0,q} = (M^\succsim_j)_{p_0,q}\quad \forall q < q_0,\text{ and}

    (Mi)p0,q0>(Mj)p0,q0.(M^\succsim_i)_{p_0,q_0} > (M^\succsim_j)_{p_0,q_0}.

Value

Score function returns a list of type LP*Scores and length of powerRelation$elements (unless parameter elements is specified). Each index contains a matrix with length(powerRelation$elements) rows and a variable number of columns, depending on the equivalence class index containing the singleton coalition of that element (matrix can have 0 columns).

Ranking function returns corresponding SocialRanking object.

Example

Let :(123122)(1323)(13{})\succsim: (123 \sim 12 \sim 2) \succ (13 \sim 23) \succ (1 \sim 3 \sim \{\}). From this, we get the following three matrices:

M1=[001110100]M2=[100101100]M3=[001020100] M^\succsim_1 = \begin{bmatrix} 0 & 0 & 1\\ 1 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix} M^\succsim_2 = \begin{bmatrix} 1 & 0 & 0\\ 1 & 0 & 1\\ 1 & 0 & 0 \end{bmatrix} M^\succsim_3 = \begin{bmatrix} 0 & 0 & 1\\ 0 & 2 & 0\\ 1 & 0 & 0 \end{bmatrix}

(M2)2,3(M^\succsim_2)_{2,3} in this context refers to the value in the second row and third column of element 2, in this case 11.

In the example, 22 will be immediately put above 11 and 33 because {2}{1}\lbrace 2 \rbrace \succ \lbrace 1 \rbrace and {2}{3}\lbrace 2 \rbrace \succ \lbrace 3 \rbrace. Since {1}{3}\lbrace 1 \rbrace \sim \lbrace 3 \rbrace, we next consider the coalitions of size 2. Here, it turns out that (M1)2,1=1>0=(M3)2,1(M^\succsim_1)_{2,1} = 1 > 0 = (M^\succsim_3)_{2,1}, setting 33 to be the least preferred option (this is opposed to the LpL^p relation, which has no strict preference of 11 over 33).

As alluded to, LpL^{p^*} is similar to LpL^p, LPRanking(), in that it first considers the singleton coalitions, then sequentially every coalition of size 2 and above that ranks better than the corresponding singleton. It can be assumed, however, that LpL^{p^*} is more granular, as it doesn't throw away any information about which equivalence class these bigger coalitions belong to.

Alterations

The matrices as described above and in Béal S, Rémila E, Solal P (2022). “Lexicographic solutions for coalitional rankings based on individual and collective performances.” Journal of Mathematical Economics, 102, 102738. can be investigated with the L1Scores() function.

LPSScores() discards some redundant information, most notably all columns from each element's singleton class and the ones thereafter. The first row is also removed, as all values there are guaranteed to be 0.

For the example above, this would actually result in the matrices

matrix(c(1,1, 1,0), nrow=2)
matrix(numeric(), nrow=2)
matrix(c(0,1, 2,0), nrow=2)

Aliases

For better discoverability, lexcelPSScores() and lexcelPSRanking() serve as aliases for LPSScores() and LPSRanking(), respectively.

References

Béal S, Rémila E, Solal P (2022). “Lexicographic solutions for coalitional rankings based on individual and collective performances.” Journal of Mathematical Economics, 102, 102738.

See Also

Other ranking solution functions: L1Scores(), L2Scores(), LPScores(), copelandScores(), cumulativeScores(), kramerSimpsonScores(), lexcelScores(), ordinalBanzhafScores()

Examples

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

scores$`2`
#
# [1,]
# [2,]

LPSRanking(pr)
# 2 > 1 > 3


[Package socialranking version 1.2.0 Index]