Constant-Factor Distortion Mechanisms for k-Committee Election

In the k-committee election problem, we wish to aggregate the preferences of n agents over a set of alternatives and select a committee of k alternatives that minimizes the cost incurred by the agents. While we typically assume that agent preferences are captured by a cardinal utility function, in many contexts we only have access to ordinal information, namely the agents' rankings over the outcomes. As preference rankings are not as expressive as cardinal utilities, a loss of efficiency is inevitable, and is quantified by the notion of distortion. We study the problem of electing a k-committee that minimizes the sum of the \ell-largest costs incurred by the agents, when agents and candidates are embedded in a metric space. This problem is called the \ell-centrum problem and captures both the utilitarian and egalitarian objectives. When k >= 2, it is not possible to compute a bounded-distortion committee using purely ordinal information. We develop the first algorithms (that we call mechanisms) for the \ell-centrum problem (when k >= 2), which achieve O(1)-distortion while eliciting only a very limited amount of cardinal information via value queries. We obtain two types of query-complexity guarantees: O(log k log n) queries per agent, and O(k^2 log^2 n) queries in total (while achieving O(1)-distortion in both cases). En route, we give a simple adaptive-sampling algorithm for the \ell-centrum k-clustering problem.