On Top-k Selection from m-wise Partial Rankings via Borda Counting

Wenjing Chen, Ruida Zhou, Chao Tian, Cong Shen

Abstract: We analyze the performance of Borda counting algorithm on noisy m-wise ranking data to accurately select the top-k items from a total of n items. This generalizes a previous result of a similar nature reported by Shah et al. on the noisy pairwise comparison data. We show that the associated score separation Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> between the k-th item and the (k+1)-th item plays an important role: if Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> is greater than a threshold depending on (n, k) and the scoring system in Borda counting, then the top-k selection is accurate asymptotically almost surely; if Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> is below a threshold, then the top-k selection will not be accurate with at least a constant probability. This separation between the two thresholds depends on m and the scoring systems in the Borda counting procedure.