Ranking with pairwise comparisons

Ranking with pairwise comparisons arise in decision making and internet applications. Here we study the connections between three popular methods: HodgeRank, Principal Eigenvector (PEV), and Tropical Eigenvector. We show that their comparison can be embedded into one single optimization problem. As an example of this approach, we show that under iid noise, HodgeRank has asymptotically better probability of rank recovery than PEV. However, comparison remains open under the non-asymptotic regime, and this is one of the open problems in this area.