Cumulative scores

Calculate cumulative score vectors for each element.

Usage

cumulativeScores(powerRelation, elements = powerRelation$elements)

cumulativelyDominates(powerRelation, e1, e2, strictly = FALSE)

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.

e1, e2

Elements in powerRelation$elements

strictly

If TRUE, check if p1 strictly dominates p2

Value

Score function returns a list of type CumulativeScores and length of powerRelation$elements

(unless parameter elements is specified). Each index contains a vector of length powerRelation$eqs, cumulatively counting up the number of times the given element appears in each equivalence class.

cumulativelyDominates() returns TRUE if e1 cumulatively dominates e2, else FALSE.

Details

An element's cumulative score vector is calculated by cumulatively adding up the amount of times it appears in each equivalence class in the powerRelation. I.e., in a linear power relation with eight coalitions, if element 1 appears in coalitions placed at 1, 3, and 6, its score vector is [1, 1, 2, 2, 2, 3, 3, 3].

Dominance

\(i\) dominates \(j\) if, for each index \(x, \textrm{Score}(i)_x \geq \textrm{Score}(j)_x\).

\(i\) strictly dominates \(j\) if there exists an \(x\) such that \(\textrm{Score}(i)_x > \textrm{Score}(j)_x\).

References

Moretti S (2015). “An axiomatic approach to social ranking under coalitional power relations.” Homo Oeconomicus, 32(2), 183--208.

Moretti S, Öztürk M (2017). “Some axiomatic and algorithmic perspectives on the social ranking problem.” In International Conference on Algorithmic Decision Theory, 166--181. Springer.

See also

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

Examples

pr <- as.PowerRelation("12 > 1 > 2")

# `1`: c(1, 2, 2)
# `2`: c(1, 1, 2)
cumulativeScores(pr)
#> $`1`
#> [1] 1 2 2
#> 
#> $`2`
#> [1] 1 1 2
#> 
#> attr(,"class")
#> [1] "CumulativeScores"

# calculate for selected number of elements
cumulativeScores(pr, c(2))
#> $`2`
#> [1] 1 1 2
#> 
#> attr(,"class")
#> [1] "CumulativeScores"

# TRUE
d1 <- cumulativelyDominates(pr, 1, 2)

# TRUE
d2 <- cumulativelyDominates(pr, 1, 1)

# FALSE
d3 <- cumulativelyDominates(pr, 1, 1, strictly = TRUE)

stopifnot(all(d1, d2, !d3))