Majority rule in the absence of a majority

Nehring, Klaus and Pivato, Marcus (2018): Majority rule in the absence of a majority.

Abstract

Which is the best, impartially most plausible consensus view to serve as the basis of democratic group decision when voters disagree? Assuming that the judgment aggregation problem can be framed as a matter of judging a set of binary propositions (“issues”), we develop a multi-issue majoritarian approach based on the criterion of supermajority efficiency (SME). SME reflects the idea that smaller supermajorities must yield to larger supermajorities so as to obtain better supported, more plausible group judgments. As it is based on a partial ordering, SME delivers unique outcomes only in special cases. In general, one needs to make cardinal, not just ordinal, trade- offs between different supermajorities. Hence we axiomatically characterize the class of additive majority rules, whose (generically unique) outcome can be interpreted as the “on balance most plausible” consensus judgment.

• Item Type: MPRA Paper
• Language: English
• Keywords: judgement aggregation; majority rule; majoritarian; hyperreal; Condorcet
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations

References

Balinski, M., Laraki, R., 2010. Majority Judgment: Measuring, Ranking, and Electing. MIT Press, Boston, MA.

Bandelt, H., Barthelemy, J., 1984. Medians in median graphs. Discrete Applied Mathematics, 131–142.

Barberà, S., Masso, J., Neme, A., 1997. Voting under constraints. Journal of Economic Theory 76, 298–321.

Barthélémy, J.-P., Janowitz, M. F., 1991. A formal theory of consensus. SIAM J. Discrete Math. 4 (3), 305–322.

Barthélémy, J.-P., Monjardet, B., 1981. The median procedure in cluster analysis and social choice theory. Math. Social Sci. 1 (3), 235–267.

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, Am- sterdam, pp. 309–316.

Bossert, W., Sprumont, Y., 2014. Strategyproof preference aggregation. Games and Economic Behavior 85, 109–126.

Cedroni, L., Garzia, D. (Eds.), 2010. Voting Advice Applications - The State of the Art. Scriptaweb, Napoli.

Christiano, T., 2008. Democracy. In: Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy, fall 2008 Edition. Metaphysics Research Lab, Stanford University. URL http://plato.stanford.edu/archives/fall2008/entries/democracy/

Clifford, A. H., 1954. Note on Hahn’s theorem on ordered abelian groups. Proc. Amer. Math. Soc. 5, 860–863.

Condorcet, M. d., 1785. An essay on the application of analysis to the probability of decisions rendered by the plurality of votes. In: McLean and Urken (1995), pp. 91–112.

Condorcet, M. d., 1788. On the constitution and the functions of provincial assemblies. In: McLean and Urken (1995), pp. 113–144.

Cusanus, N., 1434. On Catholic harmony, Book III. In: McLean and Urken (1995), Ch. 37, pp. 77–80.

Dietrich, F., Spiekermann, K., March 2013a. Epistemic democracy with defensible premises. Economics and Philosophy 29 (01), 87–120.

Dietrich, F., Spiekermann, K., 2013b. Independent opinions? on the causal foundations of belief formation and jury theorems. Mind 122 (487), 655–685.

Estlund, D., 2008. Democratic Authority. Princeton UP, Princeton and Oxford.

Galton, F., March 1907. Vox populi. Nature 75, 450–451.

Genest, C., Zidek, J. V., 1986. Combining probability distributions: a critique and an annotated bibliog- raphy. Statist. Sci. 1, 114–148.

Goldblatt, R., 1998. Lectures on the hyperreals. Vol. 188 of Graduate Texts in Mathematics. Springer- Verlag, New York, an introduction to nonstandard analysis.

Goodin, R., Spiekermann, K., 2018. An Epistemic Theory of Democracy. Oxford University Press, Oxford, UK.

Gravett, K. A. H., 1956. Ordered abelian groups. Quart. J. Math. Oxford Ser. (2) 7, 57–63.

Guilbaud, G.-T., Octobre-Décembre 1952. Les théories de l’intérêt général et le problème logique de l’aggrégation. Economie Appliquée V (4), 501–551.

Halpern, J. Y., 2010. Lexicographic probability, conditional probability, and nonstandard probability. Games Econom. Behav. 68 (1), 155–179.

Hausner, M., Wendel, J. G., 1952. Ordered vector spaces. Proc. Amer. Math. Soc. 3, 977–982.

Kemeny, J. G., Fall 1959. Math without numbers. Daedalus 88, 571–591.

Kornhauser, L., Sager, L., 1986. Unpacking the court. Yale Law Journal 96, 82–117.

List, C., Pettit, P., 2002. Aggregating sets of judgements: an impossibility result. Economics and Philos- ophy 18, 89–110.

List, C., Pettit, P., 2011. Group Agency. Oxford University Press, Oxford, UK.

Lull, R., 1299. The art of elections. In: McLean and Urken (1995), pp. 73–76.

Manin, B., 1995. The Principles of Representative Government. Cambridge University Press, Cambridge, UK.

McKelvey, R., 1986. Covering, dominance, and institution-free properties of social choice. Amer. J. Polit. Sci. 30, 283–314.

McLean, I., Urken, A. (Eds.), 1995. Classics of Social Choice. Michigan University Press.

McMorris, F. R., Mulder, H. M., Powers, R. C., 2000. The median function on median graphs and semi- lattices. Discrete Appl. Math. 101 (1-3), 221–230.

Myerson, R. B., 1995. Axiomatic derivation of scoring rules without the ordering assumption. Soc. Choice Welf. 12 (1), 59–74.

Nehring, K., Pivato, M., 2011. Incoherent majorities: the McGarvey problem in judgement aggregation. Discrete Applied Mathematics 159, 1488–1507.

Nehring, K., Pivato, M., 2014. How indeterminate is sequential majority voting? A judgement aggregation perspective. In: The mathematics of decisions, elections, and games. Vol. 624 of Contemp. Math. Amer. Math. Soc., Providence, RI, pp. 55–88.

Nehring, K., Pivato, M., 2018. The median rule in judgement aggregation. (preprint).

Nehring, K., Pivato, M., Puppe, C., 2014. The Condorcet set: Majority voting over interconnected propo- sitions. J.Econ.Theory 151, 268–303.

Nehring, K., Pivato, M., Puppe, C., 2016. Unanimity overruled: Majority voting and the burden of history. Journal of Theoretical Politics 28 (4), 552–597.

Nehring, K., Puppe, C., 2007. The structure of strategy-proof social choice I: General characterization and possibility results on median spaces. J.Econ.Theory 135, 269–305.

Nehring, K., Puppe, C., 2010. Abstract Arrowian aggregation. J. Econom. Theory 145 (2), 467–494.

Peleg, B., Zamir, S., 2012. Extending the Condorcet jury theorem to a general dependent jury. Soc. Choice Welf. 39 (1), 91–125.

Pivato, M., 2013a. Additive representation of separable preferences over infinite products. to appear in Theory and Decision.

Pivato, M., 2013b. Voting rules as statistical estimators. Soc. Choice Welf. 40 (2), 581–630.

Pivato, M., 2017. Epistemic democracy with correlated voters. J. Math. Econom. 72, 51–69.

Schwartzberg, M., 2015. Epistemic democracy and its challenges. Annual Review of Political Science 18, 187–203.

Slater, P., 1961. Inconsistencies in a schedule of paired comparisons. Biometrica 48, 303–312.

Smith, J. H., 1973. Aggregation of preferences with variable electorate. Econometrica 41, 1027–1041.

Tangian, A., 2014. Mathematical Theory of Democracy. Springer, Heidelberg.

Tideman, T. N., 1987. Independence of clones as a criterion for voting rules. Soc. Choice Welf. 4 (3), 185–206.

Waldron, J., 1999. The Dignity of Legislation. Cambridge University Press.

Young, H. P., 1975. Social choice scoring functions. SIAM J. Appl. Math. 28, 824–838.

Young, H. P., 1988. Condorcet’s theory of voting. American Political Science Review 82 (4), 1231–1244.

Young, H. P., Levenglick, A., 1978. A consistent extension of Condorcet’s election principle. SIAM J. Appl. Math. 35 (2), 285–300.

Zavist, T. M., Tideman, T. N., 1989. Complete independence of clones in the ranked pairs rule. Soc. Choice Welf. 6 (2), 167–173.