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We consider the problem of ranking $N$ objects starting from a set of noisypairwise comparisons provided by a crowd of equal workers. We assume thatobjects are endowed with intrinsic qualities and that the probability withwhich an object is preferred to another depends only on the difference betweenthe qualities of the two competitors. We propose a class of non-adaptiveranking algorithms that rely on a least-squares optimization criterion for theestimation of qualities. Such algorithms are shown to be asymptotically optimal(i.e., they require $O(\frac{N}{\epsilon^2}\log \frac{N}{\delta})$ comparisonsto be $(\epsilon, \delta)$-PAC). Numerical results show that our schemes arevery efficient also in many non-asymptotic scenarios exhibiting a performancesimilar to the maximum-likelihood algorithm. Moreover, we show how they can beextended to adaptive schemes and test them on real-world datasets.

Ranking a set of objects: a graph based least-square approach