Transitive Closure

Apply transitive closure over power relation that has cycles.

Usage

transitiveClosure(powerRelation)

Arguments

powerRelation

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

Value

PowerRelation object with no cycles.

Details

A power relation is a binary relationship between coalitions that is transitive. For coalitions \(a, b, c \in 2^N\), this means that if \(a \succ b\) and \(b \succ c\), then \(a \succ c\).

A power relation with cycles is not transitive. A transitive closure over a power relation removes all cycles and turns it into a transitive relation, placing all coalitions within a cycle in the same equivalence class. If \(a \succ b \succ a\), from the symmetric definition in PowerRelation() we therefore assume that \(a \sim b\). Similarly, if \(a \succ b_1 \succ b_2 \succ \dots \succ b_n \succ a\), the transitive closure turns it into \(a \sim b_1 \sim b_2 \sim \dots \sim b_n\).

transitiveClosure() transforms a PowerRelation object with cycles into a PowerRelation object without cycles. As described above, all coalitions within a cycle then are put into the same equivalence class and all duplicate coalitions are removed.

Examples

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

# nothing changes
transitiveClosure(pr)
#> 1 > 2


pr <- suppressWarnings(as.PowerRelation("1 > 2 > 1"))

# 1 ~ 2
transitiveClosure(pr)
#> (1 ~ 2)


pr <- suppressWarnings(
  as.PowerRelation("1 > 3 > 1 > 2 > 23 > 2")
)

# 1 > 3 > 1 > 2 > 23 > 2 =>
# 1 ~ 3 > 2 ~ 23
transitiveClosure(pr)
#> (1 ~ 3) > (2 ~ 23)