Abstract
A well-studied approach to the design of voting rules views them as maximum likelihood estimators; given votes that are seen as noisy estimates of a true ranking of the alternatives, the rule must reconstruct the most likely true ranking. We argue that this is too stringent a requirement and instead ask: how many votes does a voting rule need to reconstruct the true ranking? We define the family of pairwise-majority consistent rules and show that for all rules in this family, the number of samples required from Mallows’s noise model is logarithmic in the number of alternatives, and that no rule can do asymptotically better (while some rules like plurality do much worse). Taking a more normative point of view, we consider voting rules that surely return the true ranking as the number of samples tends to infinity (we call this property accuracy in the limit); this allows us to move to a higher level of abstraction. We study families of noise models that are parameterized by distance functions and find voting rules that are accurate in the limit for all noise models in such general families. We characterize the distance functions that induce noise models for which pairwise-majority consistent rules are accurate in the limit and provide a similar result for another novel family of position-dominance consistent rules. These characterizations capture three well-known distance functions.
- K. Arrow. 1951. Social Choice and Individual Values. John Wiley and Sons.Google Scholar
- H. Azari, D. Parks, and L. Xia. 2012. Random utility theory for social choice. In Proceedings of the 26th Annual Conference on Neural Information Processing Systems (NIPS’12). 126--134. Google ScholarDigital Library
- J. Bartholdi, C. A. Tovey, and M. A. Trick. 1989. Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare 6 (1989), 157--165.Google ScholarCross Ref
- C. Boutilier, I. Caragiannis, S. Haber, T. Lu, A. D. Procaccia, and O. Sheffet. 2012. Optimal social choice functions: A utilitarian view. In Proceedings of the 13th ACM Conference on Electronic Commerce (EC’12). 197--214. Google ScholarDigital Library
- C. Boutilier and A. D. Procaccia. 2012. A dynamic rationalization of distance rationalizability. In Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI’12). 1278--1284. Google ScholarDigital Library
- M. Braverman and E. Mossel. 2008. Noisy sorting without resampling. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’08). 268--276. Google ScholarDigital Library
- I. Caragiannis, A. D. Procaccia, and N. Shah. 2014. Modal ranking: A uniquely robust voting rule. In Proceedings of the 28th AAAI Conference on Artificial Intelligence (AAAI’14). 616--622. Google ScholarDigital Library
- F. Chierichetti and J. Kleinberg. 2012. Voting with limited information and many alternatives. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12). 1036--1055. Google ScholarDigital Library
- V. Conitzer and T. Sandholm. 2005. Common voting rules as maximum likelihood estimators. In Proceedings of the 21st Annual Conference on Uncertainty in Artificial Intelligence (UAI’05). 145--152. Google ScholarDigital Library
- D. E. Critchlow, M. A. Fligner, and J. S. Verducci. 1991. Probability models on rankings. Journal of Mathematical Psychology 35, 3 (1991), 294--318.Google ScholarCross Ref
- E. Elkind and G. Erdélyi. 2012. Manipulation under voting rule uncertainty. In Proceedings of the 11th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS’12). 627--634. Google ScholarDigital Library
- E. Elkind, P. Faliszewski, and A. Slinko. 2009. On distance rationalizability of some voting rules. In Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge (TARK’09). 108--117. Google ScholarDigital Library
- E. Elkind, P. Faliszewski, and A. Slinko. 2010. On the role of distances in defining voting rules. In Proceedings of the 9th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS’10). 375--382. Google ScholarDigital Library
- E. Elkind and A. Slinko. 2015. Rationalizations of voting rules. In Handbook of Computational Social Choice, F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A. D. Procaccia (Eds.). Cambridge University Press, Chapter 8.Google Scholar
- P. C. Fishburn. 1974. Paradoxes of voting. American Political Science Review 68, 2 (1974), 537--546.Google ScholarCross Ref
- M. A. Fligner and J. S. Verducci. 1986. Distance based ranking models. Journal of the Royal Statistical Society B 48, 3 (1986), 359--369.Google Scholar
- G. Lebanon and J. Lafferty. 2002. Cranking: Combining rankings using conditional probability models on permutations. In Proceedings of the 9th International Conference on Machine Learning (ICML’02). 363--370. Google ScholarDigital Library
- T. Y. Liu. 2011. Learning to Rank for Information Retrieval. Springer-Verlag.Google ScholarDigital Library
- T. Lu and C. Boutilier. 2011. Learning Mallows models with pairwise preferences. In Proceedings of the 28th International Conference on Machine Learning (ICML’11). 145--152. Google ScholarDigital Library
- C. L. Mallows. 1957. Non-null ranking models. Biometrika 44 (1957), 114--130.Google ScholarCross Ref
- A. Mao, A. D. Procaccia, and Y. Chen. 2013. Better human computation through principled voting. In Proceedings of the 27th AAAI Conference on Artificial Intelligence (AAAI’13). 1142--1148. Google ScholarDigital Library
- T. Meskanen and H. Nurmi. 2008. Closeness counts in social choice. In Power, Freedom, and Voting, M. Braham and F. Steffen (Eds.). Springer-Verlag, 289--306.Google Scholar
- S. Obraztsova, E. Elkind, and N. Hazon. 2011. Ties matter: Complexity of voting manipulation revisited. In Proceedings of the 10th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS’11). 71--78. Google ScholarDigital Library
- T. Pfeiffer, X. A. Gao, A. Mao, Y. Chen, and D. G. Rand. 2012. Adaptive polling for information aggregation. In Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI’12). 122--128. Google ScholarDigital Library
- A. D. Procaccia, S. J. Reddi, and N. Shah. 2012. A maximum likelihood approach for selecting sets of alternatives. In Proceedings of the 28th Annual Conference on Uncertainty in Artificial Intelligence (UAI’12). 695--704. Google ScholarDigital Library
- L. Xia and V. Conitzer. 2008. Generalized scoring rules and the frequency of coalitional manipulability. In Proceedings of the 9th ACM Conference on Electronic Commerce (EC’08). 109--118. Google ScholarDigital Library
- L. Xia and V. Conitzer. 2011. A maximum likelihood approach towards aggregating partial orders. In Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11). 446--451. Google ScholarDigital Library
- H. P. Young. 1988. Condorcet’s theory of voting. American Political Science Review 82, 4 (1988), 1231--1244.Google ScholarCross Ref
Index Terms
- When Do Noisy Votes Reveal the Truth?
Recommendations
When do noisy votes reveal the truth?
EC '13: Proceedings of the fourteenth ACM conference on Electronic commerceA well-studied approach to the design of voting rules views them as maximum likelihood estimators; given votes that are seen as noisy estimates of a true ranking of the alternatives, the rule must reconstruct the most likely true ranking. We argue that ...
When do noisy votes reveal the truth?
EC '13: Proceedings of the fourteenth ACM conference on Electronic commerceA well-studied approach to the design of voting rules views them as maximum likelihood estimators; given votes that are seen as noisy estimates of a true ranking of the alternatives, the rule must reconstruct the most likely true ranking. We argue that ...
Computing the margin of victory for various voting rules
EC '12: Proceedings of the 13th ACM Conference on Electronic CommerceThe margin of victory of an election, defined as the smallest number k such that k voters can change the winner by voting differently, is an important measurement for robustness of the election outcome. It also plays an important role in implementing ...
Comments