LPSScores {socialranking} | R Documentation |
LP* Ranking
Description
Calculate the scores.
Usage
LPSScores(powerRelation, elements = powerRelation$elements)
LPSRanking(powerRelation)
lexcelPSScores(powerRelation, elements = powerRelation$elements)
lexcelPSRanking(powerRelation)
Arguments
powerRelation |
A |
elements |
Vector of elements of which to calculate their scores.
By default, the scores of all elements in |
Details
Let be a set of elements,
a power relation,
and
its corresponding quotient order.
For an element , construct a matrix
with
columns and
rows.
Whereas each column
represents an equivalence class, each row
corresponds to the coalition size.
For , the social ranking solution
then ranks
strictly above
if one of the following conditions hold:
-
;
-
and there exists a row
and column
such that:
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 .
From this, we get the following three matrices:
in this context refers to the value in the second row and third column of element 2, in this case
.
In the example, will be immediately put above
and
because
and
.
Since
, we next consider the coalitions of size 2. Here, it turns out that
,
setting
to be the least preferred option (this is opposed to the
relation, which has no strict preference of
over
).
As alluded to, is similar to
,
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 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