Ordering by weighted number of wins gives a good ranking for weighted tournaments

We consider the following simple algorithm for feedback arc set problem in weighted tournaments — order the vertices by their weighted indegrees. We show that this algorithm has an approxima- tion guarantee of 5 if the weights satisfy probability constraints (for any pair of vertices u and v, wuv + wvu = 1). Special cases of feedback arc set problem in such weighted tournaments include feedback arc set problem in unweighted tournaments and rank aggregation. To complement the upper bound, for any constant ǫ > 0, we exhibit an infinite family of (unweighted) tournaments for which the above algorithm (irrespective of how ties are broken) has an approximation ratio of 5 − ǫ.