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The Ceteris Paribus-majority relation compares the relative success between two players joining a coalition.

cpMajorityComparisonScore() only returns two numbers, a positive number of coalitions where e1 beats e2, and a negative number of coalitions where e1 is beaten by e2.

Usage

cpMajorityComparison(
  powerRelation,
  e1,
  e2,
  strictly = FALSE,
  includeEmptySet = TRUE
)

cpMajorityComparisonScore(
  powerRelation,
  e1,
  e2,
  strictly = FALSE,
  includeEmptySet = TRUE
)

Arguments

powerRelation

A PowerRelation object created by newPowerRelation()

e1, e2

Elements in powerRelation$elements

strictly

Only include \(D_{ij}(\succ)\) and \(D_{ji}(\succ)\), i.e., coalitions \(S \in 2^{N \setminus \lbrace i,j\rbrace}\) where \(S \cup \lbrace i\rbrace \succ S \cup \lbrace j\rbrace\) and vice versa.

includeEmptySet

If TRUE, check \(\lbrace i \rbrace \succeq \lbrace j \rbrace\) even if empty set is not part of the power relation.

Value

cpMajorityComparison() returns a list with elements described in the details.

cpMajorityComparisonScore() returns a vector of two numbers, a positive number of coalitions where e1 beats e2

(\(d_{ij}(\succeq)\)), and a negative number of coalitions where e1 is beaten by e2 (\(-d_{ji}(\succeq)\)).

Details

Given two elements \(i\) and \(j\), go through each coalition \(S \in 2^{N \setminus \lbrace i, j \rbrace}\). \(D_{ij}(\succeq)\) then contains all coalitions \(S\) where \(S \cup \lbrace i \rbrace \succeq S \cup \lbrace j \rbrace\) and \(D_{ji}(\succeq)\) contains all coalitions where \(S \cup \lbrace j \rbrace \succeq S \cup \lbrace i \rbrace\).

The cardinalities \(d_{ij}(\succeq) = |D_{ij}|\) and \(d_{ji}(\succeq) = |D_{ji}|\) represent the score of the two elements, where \(i \succ j\) if \(d_{ij}(\succeq) > d_{ji}(\succeq)\) and \(i \sim j\) if \(d_{ij}(\succeq) == d_{ji}(\succeq)\).

cpMajorityComparison() tries to retain all that information. The list returned contains the following information. Note that in this context the two elements \(i\) and \(j\) refer to element 1 and element 2 respectively.

  • $e1: list of information about element 1

    • $e1$name: name of element 1

    • $e1$score: score \(d_{ij}(\succeq)\). \(d_{ij}(\succ)\) if strictly == TRUE

    • $e1$winningCoalitions: list of coalition sets::sets \(S \in D_{ij}(\succeq)\). \(S \in D_{ij}(\succ)\) if strictly == TRUE

  • $e2: list of information about element 2

    • $e2$name: name of element 2

    • $e1$score: score \(d_{ji}(\succeq)\). \(d_{ji}(\succ)\) if strictly == TRUE

    • $e1$winningCoalitions: list of coalition sets::sets \(S \in D_{ji}(\succeq)\). \(S \in D_{ji}(\succ)\) if strictly == TRUE

  • $winner: name of higher scoring element. NULL if they are indifferent.

  • $loser: name of lower scoring element. NULL if they are indifferent.

  • $tuples: a list of coalitions \(S \in 2^{N \setminus \lbrace i, j \rbrace }\) with:

    • $tuples[[x]]$coalition: sets::set, the coalition \(S\)

    • $tuples[[x]]$included: logical, TRUE if \(S \cup \lbrace i \rbrace\) and \(S \cup \lbrace j \rbrace\) are in the power relation

    • $tuples[[x]]$winner: name of the winning element \(i\) where \(S \cup \lbrace i \rbrace \succ S \cup \lbrace j \rbrace\). It is NULL if \(S \cup \lbrace i \rbrace \sim S \cup \lbrace j \rbrace\)

    • $tuples[[x]]$e1: index \(x_1\) at which \(S \cup \lbrace i \rbrace \in \sum_{x_1}\)

    • $tuples[[x]]$e2: index \(x_2\) at which \(S \cup \lbrace j \rbrace \in \sum_{x_2}\)

The much more efficient cpMajorityComparisonScore() only calculates $e1$score.

Unlike Lexcel, Ordinal Banzhaf, etc., this power relation can introduce cycles. For this reason the function cpMajorityComparison() and cpMajorityComparisonScore() only offers direct comparisons between two elements and not a ranking of all players. See the other CP-majority based functions that offer a way to rank all players.

References

Haret A, Khani H, Moretti S, Öztürk M (2018). “Ceteris paribus majority for social ranking.” In 27th International Joint Conference on Artificial Intelligence (IJCAI-ECAI-18), 303--309.

Fayard N, Escoffier MÖ (2018). “Ordinal Social ranking: simulation for CP-majority rule.” In DA2PL'2018 (From Multiple Criteria Decision Aid to Preference Learning).

See also

Other CP-majority based functions: copelandScores(), kramerSimpsonScores()

Examples

pr <- newPowerRelationFromString("ac > (a ~ b) > (c ~ bc)")

# a > b
# D_ab = {c, {}}
# D_ba = {{}}
# Score of a = 2
# Score of b = 1
scores <- cpMajorityComparison(pr, "a", "b")
stopifnot(scores$e1$name == "a")
stopifnot(scores$e2$name == "b")
stopifnot(scores$e1$score == 2)
stopifnot(scores$e2$score == 1)
stopifnot(scores$e1$score == length(scores$e1$winningCoalitions))
stopifnot(scores$e2$score == length(scores$e2$winningCoalitions))

# get tuples with coalitions S in 2^(N - {i,j})
emptySetTuple <- Filter(function(x) x$coalition == sets::set(), scores$tuples)[[1]]
playerCTuple  <- Filter(function(x) x$coalition == sets::set("c"), scores$tuples)[[1]]

# because {} u a ~ {} u b, there is no winner
stopifnot(is.null(emptySetTuple$winner))
stopifnot(emptySetTuple$e1 == emptySetTuple$e2)

# because c u a > c u b, player "a" gets the score
stopifnot(playerCTuple$winner == "a")
stopifnot(playerCTuple$e1 < playerCTuple$e2)
stopifnot(playerCTuple$e1 == 1L)
stopifnot(playerCTuple$e2 == 3L)

cpMajorityComparisonScore(pr, "a", "b") # c(1,0)
#> [1]  2 -1
cpMajorityComparisonScore(pr, "b", "a") # c(0,-1)
#> [1]  1 -2