Are the duplication cost and Robinson-Foulds distance equivalent?

In the tree reconciliation approach for species tree inference, a tree that has the minimum reconciliation score for given gene trees is taken as an estimate of the species tree. The scoring models used in existing tree reconciliation methods include the duplication, mutation, and deep coalescence costs. Since existing inference methods all are heuristic, their performances are often evaluated by using the Robinson-Foulds (RF) distance between the true species trees and the estimates output on simulated multi-locus datasets. To better understand these methods, we study the relationships between the duplication cost and the RF distance. We prove that the gap between the duplication cost and the RF distance is unbounded, but the symmetric duplication cost is logarithmically equivalent to the RF distance. The relationships between other reconciliation costs and the RF distance are also investigated.