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

TL;DR

This paper analyzes the effectiveness of Borda counting in identifying top-$k$ items from $m$-wise partial rankings, providing theoretical bounds and demonstrating its advantages over spectral MLE methods in non-parametric settings.

Contribution

The paper offers new theoretical performance bounds for Borda counting in $m$-wise ranking scenarios, extending previous pairwise comparison results and demonstrating its robustness in non-parametric models.

Findings

Borda counting accurately identifies top-$k$ items when score separation exceeds a threshold.

The bounds are tighter for pairwise comparisons ($m=2$) than previous results.

Numerical experiments show Borda counting outperforms spectral MLE in non-parametric data.

Abstract

We analyze the performance of the Borda counting algorithm in a non-parametric model. The algorithm needs to utilize probabilistic rankings of the items within $m$-sized subsets to accurately determine which items are the overall top-$k$ items in a total of $n$ items. The Borda counting algorithm simply counts the cumulative scores for each item from these partial ranking observations. This generalizes a previous work of a similar nature by Shah et al. using probabilistic pairwise comparison data. The performance of the Borda counting algorithm critically depends on the associated score separation $\Delta_k$ between the $k$-th item and the $(k+1)$-th item. Specifically, we show that if $\Delta_k$ is greater than certain value, then the top-$k$ items selected by the algorithm is asymptotically accurate almost surely; if $\Delta_k$ is below certain value, then the result will be inaccurate with a constant probability. In the special case of $m=2$, i.e., pairwise comparison, the resultant bound is tighter than that given by Shah et al., leading to a reduced gap between the error probability upper and lower bounds. These results are further extended to the approximate top-$k$ selection setting. Numerical experiments demonstrate the effectiveness and accuracy of the Borda counting algorithm, compared with the spectral MLE-based algorithm, particularly when the data does not necessarily follow an assumed parametric model.