Socially desirable approximations for Dodgson's voting rule

  • Authors:

  • Ioannis Caragiannis;Christos Kaklamanis;Nikos Karanikolas;Ariel D. Procaccia

  • Affiliations:

  • University of Patras and RACTI, Patras, Greece;University of Patras and RACTI, Patras, Greece;University of Patras and RACTI, Patras, Greece;Harvard University, Cambridge, MA, USA

  • Venue:
  • Proceedings of the 11th ACM conference on Electronic commerce
  • Year:
  • 2010

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Abstract

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. In a previous paper [Caragiannis et al., SODA 2009] we have asked whether there are approximation algorithms for Dodgson's rule that are monotonic or homogeneous. In this paper 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 of a known voting rule yields a monotonic, homogeneous, polynomial-time O(m log 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.