Socially desirable approximations for dodgson’s voting rule

In 1876, Charles Lutwidge Dodgson suggested the intriguing voting rule that today bears his name. Although Dodgson’s rule is one of the most well-studied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is NP-hard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice point of view—it fails basic social choice desiderata such as monotonicity and homogeneity. However, this does not preclude the existence of approximation algorithms for Dodgson that are monotonic or homogeneous, and indeed it is natural to ask whether such algorithms exist. In this article, we give definitive answers to these questions. We design a monotonic exponential-time algorithm that yields a 2-approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomial-time O(log m)-approximation algorithm (where m is the number of alternatives); this result is tight as well due to a complexity-theoretic lower bound. Furthermore, we show that a slight variation on a known voting rule yields a monotonic, homogeneous, polynomial-time O(mlog m)-approximation algorithm and establish that it is impossible to achieve a better approximation ratio even if one just asks for homogeneity. We complete the picture by studying several additional social choice properties; for these properties, we prove that algorithms with an approximation ratio that depends only on m do not exist.