Computational aspects of distortion

The distortion framework in social choice theory allows quantifying the efficiency of (randomized) selection of an alternative based on the preferences of a set of agents. We make two fundamental contributions to this framework. First, we develop a linear-programming-based algorithm for computing the optimal randomized decision on a given instance, which is simpler and faster than the state-of-the-art solutions. For practitioners who may prefer to deploy a classical decision-making rule over the aforementioned optimal rule, we develop an algorithm based on non-convex quadratic programming for computing the exact distortion of any (and the best) randomized positional scoring rule. For a small number of alternatives, we find that the exact distortion bounds are significantly better than the asymptotic bounds established in prior literature and lead to different recommendations on which rules to use. These results rely on a novel characterization of the instances yielding the worst distortion, which may be of independent interest.