Abhay Parekh

Sinho Chewi, Forest Yang, Avishek Ghosh et al.

Suppose we are given observations, where each observation is drawn independently from one of $k$ known distributions. The goal is to match each observation to the distribution from which it was drawn. We observe that the maximum likelihood estimator (MLE) for this problem can be computed using weighted bipartite matching, even when $n$, the number of observations per distribution, exceeds one. This is achieved by instantiating $n$ duplicates of each distribution node. However, in the regime where the number of observations per distribution is much larger than the number of distributions, the Hungarian matching algorithm for computing the weighted bipartite matching requires $\mathcal O(n^3)$ time. We introduce a novel randomized matching algorithm that reduces the runtime to $\tilde{\mathcal O}(n^2)$ by sparsifying the original graph, returning the exact MLE with high probability. Next, we give statistical justification for using the MLE by bounding the excess risk of the MLE, where the loss is defined as the negative log-likelihood. We test these bounds for the case of isotropic Gaussians with equal covariances and whose means are separated by a distance $η$, and find (1) that $\gg \log k$ separation suffices to drive the proportion of mismatches of the MLE to 0, and (2) that the expected fraction of mismatched observations goes to zero at rate $\mathcal O({(\log k)}^2/η^2)$.

Vijay Kamble, Nihar Shah, David Marn et al.

A major challenge in obtaining evaluations of products or services on e-commerce platforms is eliciting informative responses in the absence of verifiability. This paper proposes the Square Root Agreement Rule (SRA): a simple reward mechanism that incentivizes truthful responses to objective evaluations on such platforms. In this mechanism, an agent gets a reward for an evaluation only if her answer matches that of her peer, where this reward is inversely proportional to a popularity index of the answer. This index is defined to be the square root of the empirical frequency at which any two agents performing the same evaluation agree on the particular answer across evaluations of similar entities operating on the platform. Rarely agreed-upon answers thus earn a higher reward than answers for which agreements are relatively more common. We show that in the many tasks regime, the truthful equilibrium under SRA is strictly payoff-dominant across large classes of natural equilibria that could arise in these settings, thus increasing the likelihood of its adoption. While there exist other mechanisms achieving such guarantees, they either impose additional assumptions on the response distribution that are not generally satisfied for objective evaluations or they incentivize truthful behavior only if each agent performs a prohibitively large number of evaluations and commits to using the same strategy for each evaluation. SRA is the first known incentive mechanism satisfying such guarantees without imposing any such requirements. Moreover, our empirical findings demonstrate the robustness of the incentive properties of SRA in the presence of mild subjectivity or observational biases in the responses. These properties make SRA uniquely attractive for administering reward-based incentive schemes (e.g., rebates, discounts, reputation scores, etc.) on online platforms.

Nihar B. Shah, Sivaraman Balakrishnan, Joseph Bradley et al.

Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector $w^* \in \mathbb{R}^d$ that represents the "qualities" of the $d$ items being compared; this class of models includes the two most widely used parametric models--the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector $w^*$ under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via its Laplacian spectrum. Thus, in settings where the subset of pairs may be chosen, our results provide principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors.

Nihar B. Shah, Sivaraman Balakrishnan, Joseph Bradley et al.

When eliciting judgements from humans for an unknown quantity, one often has the choice of making direct-scoring (cardinal) or comparative (ordinal) measurements. In this paper we study the relative merits of either choice, providing empirical and theoretical guidelines for the selection of a measurement scheme. We provide empirical evidence based on experiments on Amazon Mechanical Turk that in a variety of tasks, (pairwise-comparative) ordinal measurements have lower per sample noise and are typically faster to elicit than cardinal ones. Ordinal measurements however typically provide less information. We then consider the popular Thurstone and Bradley-Terry-Luce (BTL) models for ordinal measurements and characterize the minimax error rates for estimating the unknown quantity. We compare these minimax error rates to those under cardinal measurement models and quantify for what noise levels ordinal measurements are better. Finally, we revisit the data collected from our experiments and show that fitting these models confirms this prediction: for tasks where the noise in ordinal measurements is sufficiently low, the ordinal approach results in smaller errors in the estimation.