# Distance rationalization of voting rules

## Abstract

The concept of *distance rationalizability* allows one to define new voting rules or rationalize existing ones via a consensus, i.e., a class of elections that have a unique, indisputable winner, and a distance over elections: A candidate is declared an election winner if she is the consensus candidate in one of the nearest consensus elections. Many classic voting rules are defined or can be represented in this way. In this paper, we focus on the power and the limitations of the distance rationalizability approach. Lerer and Nitzan (J Econ Theory 37(1):191–201, 1985) and Campbell and Nitzan (Soc Choice Welf 3(1):1–16, 1986) show that if we do not place any restrictions on the notions of distance and consensus then essentially all voting rules can be distance-rationalized. We identify a natural class of distances on elections—votewise distances—which depend on the submitted votes in a simple and transparent manner, and investigate which voting rules can be rationalized via distances of this type. We also study axiomatic properties of rules that can be defined via votewise distances.

## References

- Aleskerov F, Chistyakov V, Kalyagin V (2010) The threshold aggregation. Econ Lett 107(2):261–262CrossRefGoogle Scholar
- Arrow K (1951; revised edition, 1963) Social choice and individual values, Wiley, New YorkGoogle Scholar
- Baigent N (1987a) Metric rationalisation of social choice functions according to principles of social choice. Math Soc Sci 13(1):59–65CrossRefGoogle Scholar
- Baigent N (1987b) Preference proximity and anonymous social choice. Q J Econ 102(1):161–169CrossRefGoogle Scholar
- Baigent N, Klamler C (2004) Transitive closure, proximity and intransitivities. Econ Theory 23(1):175–181CrossRefGoogle Scholar
- Bartholdi J III, Tovey C, Trick M (1989) Voting schemes for which it can be difficult to tell who won the election. Soc Choice Welf 6(2):157–165CrossRefGoogle Scholar
- Bauer F, Stoer J, Witzgall C (1961) Absolute and monotonic norms. Numerische Matematic 3:257–264CrossRefGoogle Scholar
- Bogard K (1973) Preference structures I: distances between transitive preference relations. J Math Sociol 3:49–67CrossRefGoogle Scholar
- Bogard K (1975) Preference structures II: distances between transitive preference relations. SIAM J Appl Math 29:254–262CrossRefGoogle Scholar
- Boutilier C, Procaccia A (2012, July) A dynamic rationalization of distance rationalizability. In: Proceedings of the 26th AAAI conference on artificial intelligence. AAAI Press, pp 1278–1284Google Scholar
- Brams S, Fishburn P (2002) Voting procedures. In: Arrow K, Sen A, Suzumura K (eds) Handbook of social choice and welfare, vol 1. Elsevier, Amsterdam, pp 173–236CrossRefGoogle Scholar
- Brandt F (2009) Some remarks on Dodgson’s voting rule. Math Logic Q 55(4):460–463CrossRefGoogle Scholar
- Campbell D, Nitzan S (1986) Social compromise and social metrics. Soc Choice Welf 3(1):1–16CrossRefGoogle Scholar
- Caragiannis I, Procaccia A, Shah N (2013) When do noisy votes reveal the truth?. In: Proceedings of the 13th ACM Conference on electronic commerce, pp 143–160Google Scholar
- Chebotarev PY, Shamis E (1998) Characterizations of scoring methods for preference aggregation. Ann Oper Res 80:299–332CrossRefGoogle Scholar
- Condorcet J (1785) Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. (Facsimile reprint of original published in Paris, 1972, by the Imprimerie Royale)Google Scholar
- Conitzer V, Rognlie M, Xia L (2009, July) Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of the 21st international joint conference on artificial intelligence. AAAI Press, pp 109–115Google Scholar
- Conitzer V, Sandholm T (2005, July) Common voting rules as maximum likelihood estimators. In: Proceedings of the 21st conference on uncertainty in artificial intelligence. AUAI Press, pp 145–152Google Scholar
- Cook W, Seiford L (1978) Priority ranking and consensus information. Manag Sci 24:1721–1732CrossRefGoogle Scholar
- Cook W, Seiford L (1982) On the Borda-Kendall consensus method for priority ranking problems. Manag Sci 28:621–637CrossRefGoogle Scholar
- Deza MM, Deza E (2009) Encyclopedia of distances. Springer, BerlinCrossRefGoogle Scholar
- Eckert D, Klamler C (2011) Distance-based aggregation theory. In: Herrera-Viedma E, Garca-Lapresta JL, Kacprzyk J, Fedrizzi M, Nurmi H, Zadrozny S (eds) Consensual processes. Springer, Berlin, pp 3–22CrossRefGoogle Scholar
- Elkind E, Faliszewski P, Slinko A (2010, May) On the role of distances in defining voting rules. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, pp 375–382Google Scholar
- Elkind E, Faliszewski P, Slinko A (2011) Homogeneity and monotonicity of distance-rationalizable voting rules. In: Proceedings of the 10th international conference on autonomous agents and multiagent systems, pp 821–828Google Scholar
- Elkind E, Faliszewski P, Slinko A (2012) Rationalizations of Condorcet consistent rules via distances of hamming type. Soc Choice Welf 4(39):891–905CrossRefGoogle Scholar
- Elkind E, Shah N (2014, July) Electing the most probable without eliminating the irrational: Voting over intransitive domains. In: Proceedings of the 30th conference on uncertainty in artificial intelligenceGoogle Scholar
- Elkind E, Slinko A (2015) Rationalizations of voting rules. In: Brandt F, Conitzer V, Endriss U, Lang J, Procaccia AD (eds) Handbook of computational social choice, Chapt 8. Cambridge University Press, CambridgeGoogle Scholar
- Farkas D, Nitzan S (1979) The Borda rule and Pareto stability : a comment. Econometrica 47:1305–1306CrossRefGoogle Scholar
- Goldsmith J, Lang J, Mattei N, Perny P (2014) Voting with rank dependent scoring rules. In: Proceedings of the 28th AAAI conference on artificial intelligence. AAAI Press, pp 698–704Google Scholar
- Hemaspaandra E, Hemaspaandra L, Rothe J (1997) Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. J ACM 44(6):806–825CrossRefGoogle Scholar
- Hudry O, Monjardet B (2010) Consensus theories. An oriented survey. Math Soc Sci 190:139–167Google Scholar
- Kendall M, Gibbons J (1990) Rank correlation methods. Oxford University Press, OxfordGoogle Scholar
- Lerer E, Nitzan S (1985) Some general results on the metric rationalization for social decision rules. J Econ Theory 37(1):191–201CrossRefGoogle Scholar
- Litvak B (1982) Information given by the experts. Methods of acquisition and analysis. Radio and Communication, MoscowGoogle Scholar
- Litvak B (1983) Distances and consensus rankings. Cybernetics and systems analysis 19(1):71–81. (Translated from Kibernetika, No. 1, pp 57–63, January-February 1983)Google Scholar
- Meskanen T, Nurmi H (2008) Closeness counts in social choice. In: Braham M, Steffen F (eds) Power, freedom, and voting. Springer, BerlinGoogle Scholar
- Miller M, Osherson D (2009) Methods for distance-based judgment aggregation. Soc Choice Welf 4(32):575–601CrossRefGoogle Scholar
- Moulin H (1991) Axioms of cooperative decision making. Cambridge University Press, CambridgeGoogle Scholar
- Nitzan S (1981) Some measures of closeness to unanimity and their implications. Theory Decis 13(2):129–138CrossRefGoogle Scholar
- Nitzan S (1989) More on preservation of preference proximity and anonymous social choice. Q J Econ 104(1):187–190CrossRefGoogle Scholar
- Pfingsten A, Wagener A (2003) Bargaining solutions as social compromises. Theory Decis 55(4):359–389CrossRefGoogle Scholar
- Pivato M (2013) Voting rules as statistical estimators. Soc Choice Welf 40(2):581–630CrossRefGoogle Scholar
- Schechter E (1997) Handbook of analysis and its foundations. Academic Press, New YorkGoogle Scholar
- Xia L, Conitzer V (2011, July) A maximum likelihood approach towards aggregating partial orders. In: Proceedings of the 22nd international joint conference on artificial intelligence, pp 446–451Google Scholar
- Xia L, Conitzer V, Lang J (2010) Aggregating preferences in multiissue domains by using maximum likelihood estimators. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, pp 399–406Google Scholar
- Young H (1975) Social choice scoring functions. SIAM J Appl Math 28(4):824–838CrossRefGoogle Scholar
- Young H (1977) Extending Condorcet’s rule. J Econ Theory 16(2):335–353CrossRefGoogle Scholar
- Young H, Levenglick A (1978) A consistent extension of Condorcet’s election principle. SIAM J Appl Math 35(2):285–300CrossRefGoogle Scholar