The geometry of random paired comparisons

Suppose that we are able to obtain binary paired comparisons of the form “x is closer to p than to q” for various choices of vectors p and q. Such observations arise in a variety of contexts, including nonmetric multidimensional scaling, unfolding, and ranking problems, often because they provide a powerful and flexible model of preference. In this paper we give a theoretical bound for how well we can expect to estimate x under a randomized model for p and q. We also show that we can achieve significant gains by adaptively changing the distribution for choosing p and q.

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