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 approximation 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. Finally, for any constant e > 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 - e.
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Coppersmith et al. (2006) studied this question.