Abstract
We adopt an ‘epistemic’ interpretation of social decisions: there is an objectively correct choice, each voter receives a ‘noisy signal’ of the correct choice, and the social objective is to aggregate these signals to make the best possible guess about the correct choice. One epistemic method is to fix a probability model and compute the maximum likelihood estimator (MLE), maximum a posteriori (MAP) estimator or expected utility maximizer (EUM), given the data provided by the voters. We first show that an abstract voting rule can be interpreted as MLE or MAP if and only if it is a scoring rule. We then specialize to the case of distance-based voting rules, in particular, the use of the median rule in judgement aggregation. Finally, we show how several common ‘quasiutilitarian’ voting rules can be interpreted as EUM.
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References
Astié A: Comparaisons par paires et problèmes de classement: estimation et tests statistiques. Math Sci Humaines 32, 17–44 (1970)
Astié A (1971) Comparaisons par paires estimation de relations d’ordre et tests. Publ Inst Statist Univ Paris 20(1–2): 1–50 (1974)
Austen-Smith D, Banks J: Information aggregation, rationality, and the Condorcet Jury Theorem. Am Political Sci Rev 90, 34–45 (1996)
Balinski M, Laraki R: A theory of measuring, electing, and ranking. Proc Natl Acad Sci 104(21), 8720–8725 (2007)
Balinski M, Laraki R: Majority judgment: measuring, ranking, and electing. MIT Press, Boston (2011)
Barthélémy J-P, Monjardet B: The median procedure in cluster analysis and social choice theory. Math Soc Sci 1(3), 235–267 (1981)
Barthélémy J-P, Monjardet B (1988) The median procedure in data analysis: new results and open problems. In: Classification and related methods of data analysis (Aachen, 1987). North-Holland, Amsterdam, pp 309–316
Ben-Yashar R, Paroush J: Optimal decision rules for fixed-size committees in polychotomous choice situations. Soc Choice Welf 18(4), 737–746 (2001)
Black DS: On the rationale of group decision-making. J Political Econ 56, 23–34 (1948)
Bovens L, Rabinowicz W: Democratic answers to complex questions—an epistemic perspective. Synthese 150(1), 131–153 (2006)
Bradley RA, Terry ME: Rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39, 324–345 (1952)
Brams SJ, Fishburn PC: Approval voting. Birkhäuser, Boston (1983)
Bühlmann H, Huber PJ: Pairwise comparison and ranking in tournaments. Ann Math Stat 34, 501–510 (1963)
Chambers CP, Hayashi T: Preference aggregation under uncertainty: Savage vs. Pareto. Games Econ Behav 54(2), 430–440 (2006)
Cohen J: An epistemic conception of democracy. Ethics 97(1), 26–38 (1986)
Cohen A, Feigin P: On a model of concordance between judges. J Royal Stat Soc B 40(2), 203–213 (1978)
Condorcet Md (1785) Essai sur l’application de l’analyse à à la probabilité des décisions rendues à à la pluralité des voix. Paris
Conitzer V, Sandholm T (2005) Common voting rules as maximum likelihood estimators. In: 21st Annual conference on uncertainty in artificial intelligence (UAI-05), Edinburgh, pp 145–152
Conitzer V, Rognlie M, Xia L (2009) Preference functions that score rankings and maximum likelihood estimation. In: 21st International joint conference on artificial intelligence (IJCAI-09), Pasadena, CA, pp 109–115
Dietrich F: General representation of epistemically optimal procedures. Soc Choice Welf 26(2), 263–283 (2006)
Dietrich F, Spiekermann K (2011) Epistemic democracy with defensible premises (preprint)
Drissi-Bakhkhat M (2002) A statistical approach to the aggregation of votes. Ph.D. thesis, Université Laval, Québec
Drissi-Bakhkhat M, Truchon M: Maximum likelihood approach to vote aggregation with variable probabilities. Soc Choice Welf 23(2), 161–185 (2004)
Elkind E, Faliszewski P, Slinko A (2009) On distance-rationalizability of some voting rules. In: TARK ’09: proceedings of the 12th conference on theoretical aspects of rationality and knowledge
Elkind E, Faliszewski P, Slinko A (2010) Good rationalizations of voting rules. In: Proceedings of AAAI’10
Estlund D: Beyond fairness and deliberation: the epistemic dimension of democratic authority. In: Bohman, J, Rehg, W (eds) Deliberative democracy: essays on reason and politics, pp. 173–204. MIT Press, Cambridge (1997)
Fallis D: Epistemic value theory and judgment aggregation. Episteme 2(1), 39–55 (2005)
Fligner MA, Verducci JS: Distance based ranking models. J R Stat Soc Ser B 48(3), 359–369 (1986)
Fligner MA, Verducci JS: Multistage ranking models. J Am Stat Assoc 83(403), 892–901 (1988)
Fligner MA, Verducci JS: Posterior probabilities for a consensus ordering. Psychometrika 55(1), 53–63 (1990)
Fligner, MA, Verducci, JS (eds): Probability models and statistical analysis for ranking data. Springer, New York (1993)
Folland GB: Real analysis. Wiley, New York (1984)
Gilboa I, Samet D, Schmeidler D: Utilitarian aggregation of beliefs and tastes. J Political Econ 112, 932–938 (2004)
Hartmann S, Pigozzi G, Sprenger J: Reliable methods of judgement aggregation. J Log Comput 20, 603–617 (2010)
Hays WL: A note on average tau as a measure of concordance. J Am Stat Assoc 55, 331–341 (1960)
Hubert L, Schultz J: Quadratic assignment as a general data analysis strategy. Br J Math Stat Psychol 29(2), 190–241 (1976)
Hummel P: Jury theorems with multiple alternatives. Soc Choice Welf 34(1), 65–103 (2010)
Hylland A, Zeckhauser R: The impossibility of Bayesian group decision making with separate aggregation of beliefs and values. Econometrica 47(6), 1321–1336 (1979)
Kaniovski S: Aggregation of correlated votes and Condorcet’s Jury Theorem. Theory Decis 69(3), 453–468 (2010)
Kemeny JG: Math without numbers. Daedalus 88, 571–591 (1959)
Kendall M: A new measure of rank correlation. Biometrika 30, 81–93 (1938)
Kendall MG: Rank correlation methods. Griffin, London (1970)
Lebanon G, Lafferty J (2002) Cranking: combining rankings using conditional models on permutations. In: International conference on machine learning, pp 363–370
List C: Group knowledge and group rationality: a judgment aggregation perspective. Episteme 2(1), 25–38 (2005)
List C, Goodin RE: Epistemic democracy: generalizing the Condorcet Jury Theorem. J Political Philos 9(3), 277–306 (2001)
List C, Pettit P: Aggregating sets of judgements: an impossibility result. Econ Philos 18, 89–110 (2002)
List C, Puppe C (2009) Judgement aggregation: a survey. In: Oxford handbook of rational and social choice. Oxford University Press, Oxford, pp 457–482
Lorenz J, Rauhut H, Schweitzer F, Helbing D (2011) How social influence can undermine the wisdom of crowd effect. Proc Natl Acad Sci. http://www.pnas.org/content/early/2011/05/10/1008636108.abstract. Accessed May 16 2011
Mallows CL: Non-null ranking models. I. Biometrika 44, 114–130 (1957)
Meskanen T, Nurmi H: Closeness counts in social choice. In: Braham, M, Steffen, F (eds) Power, freedom, and voting, Springer, Berlin (2008)
Miller MK, Osherson D: Methods for distance-based judgment aggregation. Soc Choice Welf 32(4), 575–601 (2009)
Mongin P: Consistent Bayesian aggregation. J Econ Theory 66(2), 313–351 (1995)
Mongin P: The paradox of the Bayesian experts and state-dependent utility theory. J Math Econ 29(3), 331–361 (1998)
Myerson RB: Axiomatic derivation of scoring rules without the ordering assumption. Soc Choice Welf 12(1), 59–74 (1995)
Nehring K, Pivato M, Puppe C (2011) Condorcet admissibility: indeterminacy and path-dependence under majority voting on interconnected decisions (preprint)
Nitzan S: Collective preference and choice. Cambridge University Press, Cambridge (2010)
Pivato M (2011) Variable-population scoring rules (preprint)
Régnier S: Stailité d’un opérator de classification. Math Sci Humaines 60, 21–30 (1977)
Remage R Jr, Thompson WA Jr: Rankings from paired comparisons. Ann Math Stat 35, 739–747 (1964)
Remage R Jr, Thompson WA Jr: Maximum-likelihood paired comparison rankings. Biometrika 53, 143–149 (1966)
Smith JH: Aggregation of preferences with variable electorate. Econometrica 41, 1027–1041 (1973)
Spearman C: The proof and measurement of association between two things. Am J Psychol 15, 72–101 (1904)
Truchon M: Borda and the maximum likelihood approach to vote aggregation. Math Soc Sci 55(1), 96–102 (2008)
Truchon M, Gordon S: Social choice, optimal inference and figure skating. Soc Choice Welf 30(2), 265–284 (2008)
Truchon M, Gordon S: Statistical comparison of aggregation rules for votes. Math Soc Sci 57(2), 199–212 (2009)
Xia L, Conitzer V (2009) Finite local consistency characterizes generalized scoring rules. In: IJCAI’09, proceedings of the international joint conference on artificial intelligence
Xia L, Conitzer V (2011) A maximum likelihood approach towards aggregating partial orders. In: 23rd International joint conference on artificial intelligence (IJCAI-11), pp 446–451
Xia L, Conitzer V, Lang J (2010) Aggregating preferences in multi-issue domains by using maximum likelihood estimators. In: 9th International joint conference on autonomous agents and multi agent systems (AAMAS-10), Toronto, pp 399–406
Young HP: Social choice scoring functions. SIAM J Appl Math 28, 824–838 (1975)
Young HP: Optimal ranking and choice from pairwise decisions. In: Grofman, B, Owen, G (eds) Information pooling and group decision making, pp. 113–122. JAI Press, Greenwich (1986)
Young HP: Condorcet’s theory of voting. Am Political Sci Rev 82(4), 1231–1244 (1988)
Young HP: Optimal voting rules. J Econ Perspect 9(1), 51–64 (1995)
Young HP (1997) Group choice and individual judgments. In: Mueller DC (ed) Perspectives on public choice: a handbook. Cambridge University Press, Cambridge, Chap 9, pp 181–200
Young HP, Levenglick A: A consistent extension of Condorcet’s election principle. SIAM J Appl Math 35(2), 285–300 (1978)
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Pivato, M. Voting rules as statistical estimators. Soc Choice Welf 40, 581–630 (2013). https://doi.org/10.1007/s00355-011-0619-1
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DOI: https://doi.org/10.1007/s00355-011-0619-1