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.
Supplemental Material
- 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 ScholarDigital Library
- Elliot Anshelevich and John Postl. 2016. Randomized Social Choice Functions Under Metric Preferences. 25th International Joint Conference on Artificial Intelligence (2016). Google ScholarDigital Library
- 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 ScholarDigital Library
- 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 ScholarDigital Library
- Salvador Barbara and Matthew Jackson. 1988. Maximin, leximin, and the protective criterion: characterizations and ' comparisons. Journal of Economic Theory (1988).Google Scholar
- Salvador Barberà. 2001. An introduction to strategy-proof social choice functions. Social Choice and Welfare (2001).Google Scholar
- 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 ScholarDigital Library
- Ioannis Caragiannis, Christos Kaklamanis, Panagiotis Kanellopoulos, and Maria Kyropoulou. 2009. On low-envy truthful allocations. International Conference on Algorithmic Decision Theory (2009). Google ScholarDigital Library
- Ioannis Caragiannis and Ariel D Procaccia. 2011. Voting almost maximizes social welfare despite limited communication. Artificial Intelligence (2011). Google ScholarDigital Library
- Yiling Chen, John K Lai, David C Parkes, and Ariel D Procaccia. 2013. Truth, justice, and cake cutting. Games and Economic Behavior (2013).Google Scholar
- Zvi Drezner and Horst W Hamacher. 1995. Facility location. Springer-Verlag New York, NY.Google Scholar
- James M Enelow and Melvin J Hinich. 1984. The spatial theory of voting: An introduction. CUP Archive.Google Scholar
- 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 ScholarDigital Library
- Aris Filos-Ratsikas and Peter Bro Miltersen. 2014. Truthful approximations to range voting. International Conference on Web and Internet Economics (2014).Google ScholarCross Ref
- Peter C Fishburn. 1977. Condorcet social choice functions. SIAM Journal on applied Mathematics (1977).Google Scholar
- Allan Gibbard. 1973. Manipulation of voting schemes: a general result. Econometrica: journal of the Econometric Society (1973).Google Scholar
- 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 Scholar
- Ashish Goel and Adam Meyerson. 2006. Simultaneous optimization via approximate majorization for concave profits or convex costs. Algorithmica (2006).Google Scholar
- 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 ScholarDigital Library
- 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 ScholarDigital Library
- Amit Kumar and Jon Kleinberg. 2000. Fairness measures for resource allocation. Foundations of Computer Science, 41st Annual Symposium on (2000). Google ScholarDigital Library
- 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 ScholarDigital Library
- Hervé Moulin. 1980. On strategy-proofness and single peakedness. Public Choice (1980).Google Scholar
- Hervé Moulin. 1986. Choosing from a tournament. Social Choice and Welfare (1986).Google Scholar
- Hervé Moulin, Felix Brandt, Vincent Conitzer, Ulle Endriss, Ariel D Procaccia, and Jérôme Lang. 2016. Handbook of Computational Social Choice. Cambridge University Press. Google ScholarDigital Library
- Ariel D Procaccia and Jeffrey S Rosenschein. 2006. The distortion of cardinal preferences in voting. International Workshop on Cooperative Information Agents (2006). Google ScholarDigital Library
- 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 ScholarDigital Library
- John Rawls. 2009. A theory of justice. Harvard University Press.Google Scholar
- 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 Scholar
- Markus Schulze. 2003. A new monotonic and clone-independent single-winner election method. Voting matters (2003).Google Scholar
- 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 Scholar
Index Terms
Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties
Recommendations
Relating Metric Distortion and Fairness of Social Choice Rules
NetEcon '18: Proceedings of the 13th Workshop on Economics of Networks, Systems and ComputationImproved Metric Distortion for Deterministic Social Choice Rules
EC '19: Proceedings of the 2019 ACM Conference on Economics and ComputationIn this paper, we study the metric distortion of deterministic social choice rules that choose a winning candidate from a set of candidates based on voter preferences. Voters and candidates are located in an underlying metric space. A voter has cost ...
Comments