Calculate cumulative score vectors for each element.

e1, e2

(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))