  Suppose that we wish to estimate a vector $\mathbf{x}$ from a set of binary
paired comparisons of the form "$\mathbf{x}$ is closer to $\mathbf{p}$ than to
$\mathbf{q}$" for various choices of vectors $\mathbf{p}$ and $\mathbf{q}$. The
problem of estimating $\mathbf{x}$ from this type of observation arises in a
variety of contexts, including nonmetric multidimensional scaling, "unfolding,"
and ranking problems, often because it provides a powerful and flexible model
of preference. We describe theoretical bounds for how well we can expect to
estimate $\mathbf{x}$ under a randomized model for $\mathbf{p}$ and
$\mathbf{q}$. We also present results for the case where the comparisons are
noisy and subject to some degree of error. Additionally, we show that under a
randomized model for $\mathbf{p}$ and $\mathbf{q}$, a suitable number of binary
paired comparisons yield a stable embedding of the space of target vectors.
Finally, we also show that we can achieve significant gains by adaptively
changing the distribution for choosing $\mathbf{p}$ and $\mathbf{q}$.
