cpMajorityComparison: CP-Majority relation

In socialranking: Social Ranking Solutions for Power Relations on Coalitions

CP-Majority relation

Description

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 PowerRelation() or as.PowerRelation()
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 \succsim \lbrace j \rbrace even if empty set is not part of the power relation.

Details

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

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

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}(\succsim). d_{ij}(\succ) if strictly == TRUE

  • ⁠$e1$winningCoalitions⁠: list of coalition vectors S \in D_{ij}(\succsim). 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}(\succsim). d_{ji}(\succ) if strictly == TRUE

  • ⁠$e1$winningCoalitions⁠: list of coalition vectors S \in D_{ji}(\succsim). 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: vector, 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.

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

References

\insertRef

2018CPMajoritysocialranking

\insertRef

2018CPMajoritySimssocialranking

See Also

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

Examples

pr <- as.PowerRelation("ac > (a ~ b) > (c ~ bc)")

scores <- cpMajorityComparison(pr, "a", "b")
scores
# a > b
# D_ab = {c, {}}
# D_ba = {{}}
# Score of a = 2
# Score of b = 1

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) identical(x$coalition, c()), scores$tuples)[[1]]
playerCTuple  <- Filter(function(x) identical(x$coalition, "c"), scores$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)
cpMajorityComparisonScore(pr, "b", "a") # c(0,-1)

socialranking documentation built on May 29, 2024, 2:10 a.m.