L1Scores {socialranking} | R Documentation |
L1 Ranking
Description
Calculate the L^{(1)}
scores.
Usage
L1Scores(powerRelation, elements = powerRelation$elements)
L1Ranking(powerRelation)
lexcel1Scores(powerRelation, elements = powerRelation$elements)
lexcel1Ranking(powerRelation)
Arguments
powerRelation |
A |
elements |
Vector of elements of which to calculate their scores.
By default, the scores of all elements in |
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 allp < p^0
-
(M^\succsim_i)_{p,q} = (M^\succsim_j)_{p,q}
for allq < q^0
andp \in \{1, \dots, |N|\}
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.
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, RĂ©mila E, Solal P (2021). “Lexicographic solutions for coalitional rankings.” Social Choice and Welfare, 57(4), 1–33.
See Also
Other ranking solution functions:
L2Scores()
,
LPSScores()
,
LPScores()
,
copelandScores()
,
cumulativeScores()
,
kramerSimpsonScores()
,
lexcelScores()
,
ordinalBanzhafScores()
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
L1Ranking(pr)
# 2 > 1 > 3