Abstract
All societies need to make collective decisions, and for this they need a voting rule. We ask how stable these rules are. Stable rules are more likely to persist over time. We consider a family of voting systems in which individuals declare intensities of preference through numbers in the unit interval. With these voting systems, an alternative defeats another whenever the amount of opinion obtained by the first alternative exceeds the amount of opinion obtained by the second alternative by a fixed threshold. The relevant question is what should this threshold be? We assume that each individual’s assessment of what the threshold should be is represented by a trapezoidal fuzzy number. From these trapezoidal fuzzy numbers we associate reciprocal preference relations on [0,m]. With these preferences over thresholds in place, we formalize the notion of a threshold being “self-selective”. We establish some mathematical properties of self-selective thresholds, and then describe a three stage procedure for selecting an appropriate threshold. Such a procedure will always select a threshold which can then be used for future decision making.
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References
Barberà, S., Jackson, M.O.: Choosing how to choose self-stable majority rules and constitutions. The Quarterly Journal of Economics 119, 1011–1048 (2004)
Copeland, A.H.: A ‘reasonable’ social welfare function. University of Michigan Seminar on Applications of Mathematics to the Social Sciences. University of Michigan, Ann Arbor (1951)
Delgado, M., Vila, M.A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets and Systems 93, 125–135 (1998)
Diss, M., Merlin, V.R.: On the stability of a triplet of scoring rules. Theory and Decision 69, 289–316 (2010)
Ferejohn, J.A., Grether, D.M.: On a class of rational social decisions procedures. Journal of Economic Theory 8, 471–482 (1974)
Fishburn, P.C.: The Theory of Social Choice. Princeton University Press, Princeton (1973)
García-Lapresta, J.L.: A general class of simple majority decision rules based on linguistic opinions. Information Sciences 176, 352–365 (2006)
García-Lapresta, J.L., Llamazares, B.: Aggregation of fuzzy preferences: Some rules of the mean. Social Choice and Welfare 17, 673–690 (2000)
García-Lapresta, J.L., Llamazares, B.: Majority decisions based on difference of votes. Journal of Mathematical Economics 35, 463–481 (2001)
García-Lapresta, J.L., Llamazares, B.: Preference intensities and majority decisions based on difference of support between alternatives. Group Decision and Negotiation 19, 527–542 (2010)
García-Lapresta, J.L., Rodríguez-Palmero, C.: Some algebraic characterizations of preference structures. Journal of Interdisciplinary Mathematics 7, 233–254 (2004)
Houy, N.: Some further characterizations for the forgotten voting rules. Mathematical Social Sciences 53, 111–121 (2007)
Houy, N.: Dinamics of stable sets of constitutions (2007), http://wakame.econ.hit-u.ac.jp/~riron/Workshop/2006/Houy.pdf
Houy, N.: On the (im)possibility of a set of constitutions stable at different levels (2009), ftp://mse.univ-paris1.fr/pub/mse/cahiers2004/V04039.pdf
Karni, E., Safra, Z.: Individual sense of justice: a utility representation. Econometrica 70, 263–284 (2002)
Koray, S.: Self-selective social choice functions verify Arrow and Gibbard-Satterthwaite theorems. Econometrica 68, 981–995 (2000)
Llamazares, B.: Simple and absolute special majorities generated by OWA operators. European Journal of Operational Research 158, 707–720 (2004)
Llamazares, B.: The forgotten decision rules: Majority rules based on difference of votes. Mathematical Social Sciences 51, 311–326 (2006)
Llamazares, B.: Choosing OWA operator weights in the field of Social Choice. Information Sciences 177, 4745–4756 (2007)
Llamazares, B., García-Lapresta, J.L.: Voting systems generated by quasiarithmetic means and OWA operators. In: Fodor, J., De Baets, B. (eds.) Principles of Fuzzy Preference Modelling and Decision Making, pp. 195–213. Academia Press, Ghent (2003)
Llamazares, B., García-Lapresta, J.L.: Extension of some voting systems to the field of gradual preferences. In: Bustince, H., Herrera, F., Montero, J. (eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, pp. 292–310. Springer, Berlin (2008)
McLean, I., Urken, A.B. (eds.): Classics of Social Choice. The University of Michigan Press, Ann Arbor (1995)
Morales, J.I.: Memoria Matemática sobre el Cálculo de la Opinion en las Elecciones. Imprenta Real, Madrid (1797); English version in McLean and Urken [22, pp. 197–235]: Mathematical Memoir on the Calculations of Opinions in Elections, pp. 197–236
Nurmi, H.: Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and Systems 6, 249–259 (1981)
Nurmi, H.: Fuzzy social choice: A selective retrospect. Soft Computing 12, 281–288 (2008)
Saari, D.G.: Consistency of decision processes. Annals of Operations Research 23, 103–137 (1990)
Segal, U.: Let’s agree that all dictatorships are equally bad. The Journal of Political Economy 108, 569–589 (2000)
Sen, A.K.: Collective Choice and Social Welfare. Holden-Day, San Francisco (1970)
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García-Lapresta, J.L., Piggins, A. (2012). Voting on How to Vote. In: Seising, R., Sanz González, V. (eds) Soft Computing in Humanities and Social Sciences. Studies in Fuzziness and Soft Computing, vol 273. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24672-2_17
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DOI: https://doi.org/10.1007/978-3-642-24672-2_17
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