Create relation matrix

For a given PowerRelation object create a relations::relation() object.

Usage

powerRelationMatrix(
  powerRelation,
  domainNames = c("pretty", "numericPrec", "numeric")
)

# S3 method for PowerRelation
as.relation(x, ...)

Arguments

powerRelation

A PowerRelation object created by PowerRelation() or as.PowerRelation()

domainNames

How should the row and column names be formatted?

• pretty: Coalitions such as c(1,2) are formatted as 12. To ensure that it's correctly sorted alphabetically, every name is preceded by a certain amount of the invisible Unicode character \u200b

• numericPrec: Coalitions such as c(1,2) are formatted as 1{12}, the number in front of the curly brace marking its sorted spot. While less pretty, it won't use Unicode characters.

• numeric: Drop coalition names, only count from 1 upwards. Each number corresponds to the index in TODO powerRelation$rankingCoalitions

• function(x): A custom function that is passed a number from 1 through length(powerRelation$rankingCoalitions). Must return a character object.

x

A PowerRelation object

...

Further parameters (ignored)

Value

relations::relation() object to the corresponding power relation.

Details

Turn a PowerRelation object into a relations::relation() object. The incidence matrix can be viewed with relations::relation_incidence().

The columns and rows of a PowerRelation object are ordered by TODO powerRelation$rankingCoalitions. The relations package automatically sorts the columns and rows by their domain names, which is the reason the parameter domainNames is included. This way we ensure that the columns and rows are sorted by the order of the power relation.

Cycles

A PowerRelation object is defined as being transitive. If a power relation includes a cycle, meaning that the same coalition appears twice in the ranking, all coalitions within that cycle will be considered to be indifferent from one another.

For example, given the power relation \(1 \succ 2 \succ 3 \succ 1 \succ 12\), the relation is somewhat equivalent to \(1 \sim 2 \sim 3 \succ 12\). There is no way to check for cycles in the incidence matrix only.

Call transitiveClosure() to remove cycles in a PowerRelation object.

See also

relations::as.relation()

Examples

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

# do relation stuff
# Incidence matrix
# 111
# 011
# 001
relations::relation_incidence(relation)
#> Incidences:
#>    12 ​1 ​​2
#> 12  1 1 1
#> ​1   0 1 1
#> ​​2   0 0 1

# all TRUE
stopifnot(all(
  relations::relation_is_acyclic(relation),
  relations::relation_is_antisymmetric(relation),
  relations::relation_is_linear_order(relation),
  relations::relation_is_complete(relation),
  relations::relation_is_reflexive(relation),
  relations::relation_is_transitive(relation)
))


# a power relation where coalitions {1} and {2} are indifferent
pr <- as.PowerRelation("12 > (1 ~ 2)")
relation <- powerRelationMatrix(pr)

# Incidence matrix
# 111
# 011
# 011
relations::relation_incidence(relation)
#> Incidences:
#>    12 ​1 ​​2
#> 12  1 1 1
#> ​1   0 1 1
#> ​​2   0 1 1

# FALSE
stopifnot(!any(
  relations::relation_is_acyclic(relation),
  relations::relation_is_antisymmetric(relation),
  relations::relation_is_linear_order(relation)
))
# TRUE
stopifnot(all(
  relations::relation_is_complete(relation),
  relations::relation_is_reflexive(relation),
  relations::relation_is_transitive(relation)
))


# a pr with cycles
pr <- suppressWarnings(as.PowerRelation("12 > 1 > 2 > 1"))
relation <- powerRelationMatrix(pr)

# Incidence matrix
# 1111
# 0111
# 0111
# 0111
relations::relation_incidence(relation)
#> Incidences:
#>    12 ​1 ​​2 ​​​1
#> 12  1 1 1 1
#> ​1   0 1 1 1
#> ​​2   0 1 1 1
#> ​​​1   0 1 1 1

# custom naming convention
relation <- powerRelationMatrix(
  pr,
  function(x) paste0(letters[x], ":", paste(pr$rankingCoalitions[[x]], collapse = "|"))
)

relations::relation_incidence(relation)
#> Incidences:
#>    a: b: c: d:
#> a:  1  1  1  1
#> b:  0  1  1  1
#> c:  0  1  1  1
#> d:  0  1  1  1
# Incidences:
#       a:1|2 b:1 c:2 d:1
# a:1|2     1   1   1   1
# b:1       0   1   1   1
# c:2       0   1   1   1
# d:1       0   1   1   1