Make Power Relation monotonic

Given a powerRelation object, make its order monotonic.

Usage

makePowerRelationMonotonic(powerRelation, addMissingCoalitions = TRUE)

Arguments

powerRelation

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

addMissingCoalitions

If TRUE, also include all coalitions in the power set of powerRelation$elements that are not present in the current power relation.

Value

PowerRelation object containing the following values:

• $elements: vector of elements

• $eqs: equivalence classes. Nested list of lists, each containing vectors representing groups of elements in the same equivalence class

• $coalitionLookup: function(v) taking a coalition vector v and returning the equivalence class it belongs to. See coalitionLookup() for more.

• $elementLookup: function(e) taking an element e and returning a list of 2-sized tuples. See elementLookup() for more.

Details

A power relation is monotonic if

$$T \subset S \Leftrightarrow S \succsim T.$$

for every coalition \(S \subseteq N\).

Calling makePowerRelationMonotonic() on some PowerRelation object moves or adds coalitions to certain equivalence classes so that the power relation becomes monotonic.

See also

Other helper functions for transforming power relations: appendMissingCoalitions()

Examples

pr <- as.PowerRelation("ab > ac > abc > b > a > {} > c > bc")
makePowerRelationMonotonic(pr)
#> (abc ~ ab) > ac > (bc ~ b) > a > (c ~ {})
# (abc ~ ab) > ac > (bc ~ b) > a > (c ~ {})

# notice that missing coalitions are automatically added,
# except for the empty set
pr <- as.PowerRelation("a > b > c")
makePowerRelationMonotonic(pr)
#> (abc ~ ab ~ ac ~ a) > (bc ~ b) > c
# (abc ~ ab ~ ac ~ a) > (bc ~ b) > c

# setting addMissingCoalitions to FALSE changes this behavior
pr <- as.PowerRelation("a > ab > c ~ {} > b")
makePowerRelationMonotonic(pr, addMissingCoalitions = FALSE)
#> (ab ~ a) > (b ~ c ~ {})
# (ab ~ a) > (b ~ c ~ {})

# notice that an equivalence class containing an empty coalition
# automatically moves all remaining coalitions to that equivalence class.
pr <- as.PowerRelation("a > {} > b > c")
makePowerRelationMonotonic(pr)
#> (abc ~ ab ~ ac ~ a) > (bc ~ b ~ c ~ {})
# (abc ~ ab ~ ac ~ a) > (bc ~ b ~ c ~ {})