Judgement aggregation functions and ultraproducts

Herzberg, Frederik S. (2008): Judgement aggregation functions and ultraproducts.

Abstract

The relationship between propositional model theory and social decision making via premise-based procedures is explored. A one-to-one correspondence between ultrafilters on the population set and weakly universal, unanimity-respecting, systematic judgment aggregation functions is established. The proof constructs an ultraproduct of profiles, viewed as propositional structures, with respect to the ultrafilter of decisive coalitions. This representation theorem can be used to prove other properties of such judgment aggregation functions, in particular sovereignty and monotonicity, as well as an impossibility theorem for judgment aggregation in finite populations. As a corollary, Lauwers and Van~Liedekerke's (1995) representation theorem for preference aggregation functions is derived.

• Item Type: MPRA Paper
• Language: English
• Keywords: Judgment aggregation function; ultraproduct; ultrafilter
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations

References

[Armstrong, 1980] Armstrong, T. (1980). Arrow’s theorem with restricted coalition algebras. Journal of Mathematical Economics, 7(1):55–75.

[Armstrong, 1985] Armstrong, T. (1985). Precisely dictatorial social welfare functions. Erratum and addendum to: “Arrow’s theorem with restricted coalition algebras”. Journal of Mathematical Economics, 14(1):57–59.

[Banaschewski, 1983] Banaschewski, B. (1983). The power of the ultrafilter theorem. Journal of the London Mathematical Society. Second Series, 27(2):193–202.

[Chang and Keisler, 1973] Chang, C. and Keisler, H. (1973). Model theory. Studies in Logic and the Foundations of Mathematics. 73. Amsterdam: North-Holland.

[Daniëls, 2006] Daniëls, T. (2006). Social choice and the logic of simple gemes. In Endriss, U. and Lang, J., editors, Proceedings of the 1st International Workshop on Computational Social Choice (COMSOC-2006), pages 125–138. Amsterdam: Institute for Logic, Language and Computation, Universiteit van Amsterdam.

[Dietrich and List, 2007] Dietrich, F. and List, C. (2007). Majority voting on restricted domains. Technical report, London School of Economics and Political Science.

[Dietrich and Mongin, 2007] Dietrich, F. and Mongin, P. (2007). The premiss-based approach to judgment aggregation. Technical report, Universiteit Maastricht.

[Eckert, 2008] Eckert, D. (2008). Guilbaud’s Theorem: An early contribution to judgment aggregation. Technical report, Institut für Finanzwissenschaft, Universität Graz.

[Eckert and Klamler, 2008] Eckert, D. and Klamler, C. (2008). A simple ultrafilter proof for an impossibility theorem in judgment aggregation. Technical report, Institut für Finanzwissenschaft, Universität Graz.

[Fishburn, 1970] Fishburn, P. (1970). Arrow’s impossibility theorem: concise proof and infinite voters. Journal of Economic Theory, 2:103–106.

[Gärdenfors, 2006] Gärdenfors, P. (2006). A representation theorem for voting with logical consequences. Economics and Philosophy, 22(2):181–190.

[Guilbaud, 1952] Guilbaud, G. (1952). Les théories de l’intérêt général et le problème logique de l’agrégation. Économie Appliquée, 5(4):501–584.

[Guilbaud, 2008] Guilbaud, G. (2008). Theories of the general interest and the logical problem of aggregation. Journal Electronique d’Histoire des Probabilités et de la Statistique, 4(1).

[Halpern, 1964] Halpern, J. (1964). The independence of the axiom of choice from the Boolean prime ideal theorem. Fundamenta Mathematicae, 55:57–66.

[Halpern and Levy, 1971] Halpern, J. and Levy, A. (1971). The Boolean prime ideal theorem does not imply the axiom of choice. In Axiomatic Set Theory. Proceedings of Symposia in Pure Mathematics. XIII. Part 1, pages 83–134. Providence, RI: American Mathematical Society.

[Hansson, 1976] Hansson, B. (1976). The existence of group preference functions. Public Choice, 38:89–98.

[Henkin, 1949] Henkin, L. (1949). The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14:159–166.

[Kirman and Sondermann, 1972] Kirman, A. and Sondermann, D. (1972). Arrow’s theorem, many agents, and invisible dictators. Journal of Economic Theory, 5(2):267–277.

[Lauwers and Van Liedekerke, 1995] Lauwers, L. and Van Liedekerke, L. (1995). Ultraproducts and aggregation. Journal of Mathematical Economics, 24(3):217–237.

[List and Puppe, 2007] List, C. and Puppe, C. (2007). Judgment aggregation: a survey. Technical report, Universität Karlsruhe.

[Ło´s, 1955] Ło´s, J. (1955). Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. In Skolem, T., Hasenjaeger, G., Kreisel, G., Robinson, A., Wang, H., Henkin, L., and Ło´s, J., editors, Mathematical Interpretation of Formal Systems, pages 98–113. Studies in Logic and the Foundation of Mathematics. Amsterdam: North- Holland.

[Monjardet, 1983] Monjardet, B. (1983). On the use of ultrafilters in social choice theory. In Social Choice and Welfare (Caen, 1980), Contributions to Economic Analysis. 145, pages 73–78. Amsterdam: North-Holland.

[Sen, 1966] Sen, A. (1966). A possibility theorem on majority decisions. Econometrica, 34(2):491–499.

[Tarski, 1930] Tarski, A. (1930). Une contribution à la théorie de la mesure. Fundamenta Mathematicae, 15:42–50.

[Tarski, 1933] Tarski, A. (1933). Poje¸cie prawdy w je¸zykach nauk dedukcyjnych. Warsaw: Nakładem Towarzystwa Naukowego Warszawskiego.

[Ulam, 1929] Ulam, S. (1929). Concerning functions of sets. Fundamenta Mathematicae, 14:231–233.