Apply transitive closure over power relation that has cycles.
PowerRelation object with no cycles.
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
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.
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)