skip to main content
10.1145/3033274.3085138acmconferencesArticle/Chapter ViewAbstractPublication PagesecConference Proceedingsconference-collections
research-article
Public Access

Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties

Published:20 June 2017Publication History

ABSTRACT

We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative chosen by an omniscient algorithm. The worst-case distortion of a social choice rule is, therefore, a measure of how close it always gets to the optimal alternative without any knowledge of the underlying costs. Under this model, it has been conjectured that Ranked Pairs, the well-known weighted-tournament rule, achieves a distortion of at most 3 (Anshelevich et al. 2015). We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst-case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the worst case, the simpler Copeland rule is at least as good as Ranked Pairs. And as long as we are limited to (weighted or unweighted) tournament rules, we demonstrate that randomization cannot help achieve an expected worst-case distortion of less than 3. Using the concept of approximate majorization within the distortion framework, we prove that Copeland and Randomized Dictatorship achieve low constant factor fairness-ratios (5 and 3 respectively), which is a considerable generalization of similar results for the sum of costs and single largest cost objectives. In addition to all of the above, we outline several interesting directions for further research in this space.

Skip Supplemental Material Section

Supplemental Material

04b_03goel.mp4

References

  1. Elliot Anshelevich, Onkar Bhardwaj, and John Postl. 2015. Approximating Optimal Social Choice under Metric Preferences. Association for the Advancement of Artificial Intelligence, 15th Conference of the (2015). Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Elliot Anshelevich and John Postl. 2016. Randomized Social Choice Functions Under Metric Preferences. 25th International Joint Conference on Artificial Intelligence (2016). Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Elliot Anshelevich and Shreyas Sekar. 2016. Blind, Greedy, and Random: Algorithms for Matching and Clustering Using Only Ordinal Information. Association for the Advancement of Artificial Intelligence, 15th Conference of the (2016). Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. 2004. Local search heuristics for k-median and facility location problems. SIAM Journal on computing (2004). Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Salvador Barbara and Matthew Jackson. 1988. Maximin, leximin, and the protective criterion: characterizations and ' comparisons. Journal of Economic Theory (1988).Google ScholarGoogle Scholar
  6. Salvador Barberà. 2001. An introduction to strategy-proof social choice functions. Social Choice and Welfare (2001).Google ScholarGoogle Scholar
  7. Craig Boutilier, Ioannis Caragiannis, Simi Haber, Tyler Lu, Ariel D Procaccia, and Or Sheffet. 2015. Optimal social choice functions: A utilitarian view. Artificial Intelligence (2015). Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Ioannis Caragiannis, Christos Kaklamanis, Panagiotis Kanellopoulos, and Maria Kyropoulou. 2009. On low-envy truthful allocations. International Conference on Algorithmic Decision Theory (2009). Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Ioannis Caragiannis and Ariel D Procaccia. 2011. Voting almost maximizes social welfare despite limited communication. Artificial Intelligence (2011). Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Yiling Chen, John K Lai, David C Parkes, and Ariel D Procaccia. 2013. Truth, justice, and cake cutting. Games and Economic Behavior (2013).Google ScholarGoogle Scholar
  11. Zvi Drezner and Horst W Hamacher. 1995. Facility location. Springer-Verlag New York, NY.Google ScholarGoogle Scholar
  12. James M Enelow and Melvin J Hinich. 1984. The spatial theory of voting: An introduction. CUP Archive.Google ScholarGoogle Scholar
  13. Michal Feldman, Amos Fiat, and Iddan Golomb. 2016. On voting and facility location. Proceedings of the 2016 ACM Conference on Economics and Computation (2016). Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Aris Filos-Ratsikas and Peter Bro Miltersen. 2014. Truthful approximations to range voting. International Conference on Web and Internet Economics (2014).Google ScholarGoogle ScholarCross RefCross Ref
  15. Peter C Fishburn. 1977. Condorcet social choice functions. SIAM Journal on applied Mathematics (1977).Google ScholarGoogle Scholar
  16. Allan Gibbard. 1973. Manipulation of voting schemes: a general result. Econometrica: journal of the Econometric Society (1973).Google ScholarGoogle Scholar
  17. Ashish Goel, Anilesh Kollagunta Krishnaswamy, and Kamesh Munagala. 2016. Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties. arXiv preprint arXiv:1612.02912 (2016).Google ScholarGoogle Scholar
  18. Ashish Goel and Adam Meyerson. 2006. Simultaneous optimization via approximate majorization for concave profits or convex costs. Algorithmica (2006).Google ScholarGoogle Scholar
  19. Ashish Goel, Adam Meyerson, and Serge Plotkin. 2001. Approximate majorization and fair online load balancing. Symposium on Discrete Algorithms: Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms (2001). Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Jon Kleinberg, Yuval Rabani, and Éva Tardos. 1999. Fairness in routing and load balancing. Foundations of Computer Science, 1999. 40th Annual Symposium on (1999). Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Amit Kumar and Jon Kleinberg. 2000. Fairness measures for resource allocation. Foundations of Computer Science, 41st Annual Symposium on (2000). Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Richard J Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. 2004. On approximately fair allocations of indivisible goods. Proceedings of the 5th ACM conference on Electronic commerce (2004). Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Hervé Moulin. 1980. On strategy-proofness and single peakedness. Public Choice (1980).Google ScholarGoogle Scholar
  24. Hervé Moulin. 1986. Choosing from a tournament. Social Choice and Welfare (1986).Google ScholarGoogle Scholar
  25. Hervé Moulin, Felix Brandt, Vincent Conitzer, Ulle Endriss, Ariel D Procaccia, and Jérôme Lang. 2016. Handbook of Computational Social Choice. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Ariel D Procaccia and Jeffrey S Rosenschein. 2006. The distortion of cardinal preferences in voting. International Workshop on Cooperative Information Agents (2006). Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Ariel D Procaccia and Junxing Wang. 2014. Fair enough: Guaranteeing approximate maximin shares. Proceedings of the fifteenth ACM conference on Economics and computation (2014). Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. John Rawls. 2009. A theory of justice. Harvard University Press.Google ScholarGoogle Scholar
  29. Mark Allen Satterthwaite. 1975. Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of economic theory (1975).Google ScholarGoogle Scholar
  30. Markus Schulze. 2003. A new monotonic and clone-independent single-winner election method. Voting matters (2003).Google ScholarGoogle Scholar
  31. Piotr Skowron and Edith Elkind. 2017. Social Choice Under Metric Preferences: Scoring Rules and STV. Association for the Advancement of Artificial Intelligence, 16th Conference of the (2017).Google ScholarGoogle Scholar

Index Terms

  1. Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        EC '17: Proceedings of the 2017 ACM Conference on Economics and Computation
        June 2017
        740 pages
        ISBN:9781450345279
        DOI:10.1145/3033274

        Copyright © 2017 ACM

        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 20 June 2017

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article

        Acceptance Rates

        EC '17 Paper Acceptance Rate75of257submissions,29%Overall Acceptance Rate664of2,389submissions,28%

        Upcoming Conference

        EC '24
        The 25th ACM Conference on Economics and Computation
        July 8 - 11, 2024
        New Haven , CT , USA

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader