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Stable preference aggregation with infinite population

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Abstract

In this paper, we explore the stability of the aggregation procedure of individual preferences. In particular, we propose the stability under the addition of social preference, which is a normative property of democratic collective decision making. We establish impossibility and possibility theorems for non-dictatorial aggregation procedures.

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Notes

  1. Barberà (2010) surveys works on strategy-proofness since the works of Gibbard (1973) and Satterthwaite (1975).

  2. See, for example, Reny (2001). Blair and Muller (1983) clarify the relationship between the independence of irrelevant alternatives and strategy-proofness.

  3. Fey (2004) examines the majority rule under an infinite population setting. Cato et al. (2021) axiomatize approval voting with an infinite population.

  4. Gibbard (1969) and Mas-Colell and Sonnenschein (1972) show a difficulty of quasi-transitive social choice.

  5. Their results are extended by many authors, such as Brown (1974, 1975), Armstrong (1980), and Cato (2013a, 2013b).

  6. See also Cato (2020).

  7. Fleurbaey and Michel (2003) consider an equitable social ordering over the infinite-dimensional utility space and demonstrate that the concept of an ultrafilter can be unitized effectively in their framework of intergenerational equity.

  8. Sato (2013) proposes a weaker version of individual strategy-proofness and examines the relationship between the weaker version and the standard version.

  9. See also related arguments provided by Cato et al. (2021). See also Wilkinson (2021) for related frameworks with infinite utility streams.

  10. See Mihara (1997a) for a related argument on the infinite-population assumption.

  11. See Cato (2016) for operational characterizations of these properties.

  12. For example, see Willard (1970, p.80, Theorem 12.11).

  13. For a characterization of the Borda rule, see Young (1974).

  14. An ordering that is consistent with the original ordering is called an ordering extension. See Cato (2012) for a systematic argument on ordering extensions.

  15. The construction of this social welfare function is introduced by Cato (2017).

  16. Note that \(C(D(R))=D(C(R))\) for any binary relation R.

  17. See Sen (1970), Chapter 1\(^*\)) for detailed arguments on the set of greatest elements.

  18. Mihara (2001) examines an explicit construction of a social choice function of coalitional strategy-proofness, ontoness, and non-dictatorship under some restricted domain.

  19. Cato (2019) examines a strongly Paretian social welfare function under infinite population.

References

  • Aliprantis CD, Border KC (2006) Infinite dimensional analysis: a Hitchhiker’s guide, 3rd edn. Springer-Verlag, Berlin

    Google Scholar 

  • Armstrong TE (1980) Arrow’s theorem with restricted coalition algebras. J Math Econ 7(1):55–75

    Article  Google Scholar 

  • Arrow KJ (1951; 1963; 2012) Social choice and individual values. 1\(^{st}\) ed. (1951), Wiley, New York, NY; 2\(^{nd}\) ed. (1963), with ‘Notes on the theory of social choice’, Wiley, New York, NY; 3\(^{rd}\), (eds) (2012) with ‘Foreword to the third edition’ by Eric Maskin. Yale University Press, New Haven, CT

  • Barberà S (2010) Strategy proof social choice. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol 2. North-Holland, Amsterdam, pp 731–831 (chapter 25)

    Chapter  Google Scholar 

  • Barberà S, Dutta B, Sen A (2001) Strategy-proof social choice correspondences. J Econ Theory 101(2):374–394

    Article  Google Scholar 

  • Barberà S, Berga D, Moreno B (2010) Individual versus group strategy-proofness: when do they coincide? J Econ Theory 145(5):1648–1674

    Article  Google Scholar 

  • Batteau P, Blin JM, Monjardet B (1981) Stability of aggregation procedures, ultrafilters and simple games. Econometrica 49(2):527–534

    Article  Google Scholar 

  • Blair D, Muller E (1983) Essential aggregation procedures on restricted domains of preferences. J Econ Theory 30(1):34–53

    Article  Google Scholar 

  • Bossert W (1999) Intersection quasi-orderings: an alternative proof. Order 16(3):221–225

    Article  Google Scholar 

  • Bossert W, Cato S (2020) Acyclicity, anonymity, and prefilters. J Math Econ 87:134–141

    Article  Google Scholar 

  • Bossert W, Cato S (2021) Superset-robust collective choice rules. Math Soc Sci 109:126–136

    Article  Google Scholar 

  • Bossert W, Suzumura K (2010) Consistency, choice, and rationality. Harvard University Press, Cambridge

    Book  Google Scholar 

  • Bossert W, Suzumura K (2011) Multi-profile intergenerational social choice. Soc Choice Welf 37(3):493–509

    Article  Google Scholar 

  • Brown DJ (1974) An approximate solution to Arrow’s problem. J Econ Theory 9(4):375–383

    Article  Google Scholar 

  • Brown DJ (1975) Aggregation of preferences. Q J Econ 89(3):456–469

    Article  Google Scholar 

  • Cato S (2011) Maskin monotonicity and infinite individuals. Econ Lett 110(1):56–59

    Article  Google Scholar 

  • Cato S (2012) Szpilrajn, Arrow and Suzumura: concise proofs of extension theorems and an extension. Metroeconomica 63(2):235–249

    Article  Google Scholar 

  • Cato S (2013) Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings. Soc Choice Welf 41(1):169–202

    Article  Google Scholar 

  • Cato S (2013) Social choice, the strong Pareto principle, and conditional decisiveness. Theor Decis 75(4):563–579

    Article  Google Scholar 

  • Cato S (2016) Rationality and operators. Springer, Singapore

    Book  Google Scholar 

  • Cato S (2017) Unanimity, anonymity, and infinite population. J Math Econ 71:28–35

    Article  Google Scholar 

  • Cato S (2019) The possibility of Paretian anonymous decision-making with an infinite population. Soc Choice Welf 53(4):587–601

    Article  Google Scholar 

  • Cato S (2020) Quasi-stationary social welfare functions. Theor Decis 89:85–106

    Article  Google Scholar 

  • Cato S (2021) Preference aggregation and atoms in measures. J Math Econ 94:102446

    Article  Google Scholar 

  • Cato S, Rémila E, Solal P (2021) Infinite-population approval voting: A proposal. Synthese 199(3):10181–10209

    Article  Google Scholar 

  • Dasgupta PS, Hammond PJ, Maskin ES (1979) The implementation of social choice rules: some general results on incentive compatibility. Rev Econ Stud 46(2):185–216

    Article  Google Scholar 

  • Duggan J, Schwartz T (2000) Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized. Soc Choice Welf 17(1):85–93

    Article  Google Scholar 

  • Ferejohn J, Page T (1978) On the foundations of intertemporal choice. Am J Agr Econ 60(2):269–275

    Article  Google Scholar 

  • Fey M (2004) May’s theorem with an infinite population. Soc Choice Welf 23(2):275–293

    Article  Google Scholar 

  • Fishburn PC (1970) Arrow’s impossibility theorem: concise proof and infinite voters. J Econ Theory 2(1):103–106

    Article  Google Scholar 

  • Fleurbaey M, Michel P (2003) Intertemporal equity and the extension of the Ramsey criterion. J Math Econ 39(7):777–802

    Article  Google Scholar 

  • Gibbard A (1969) Social choice and the Arrow condition, unpublished paper

  • Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41(4):587–601

    Article  Google Scholar 

  • Hansson B (1976) The existence of group preference functions. Public Choice 28(1):89–98

    Article  Google Scholar 

  • Kirman AP, Sondermann D (1972) Arrow’s theorem, many agents, and invisible dictators. J Econ Theory 5(2):267–277

    Article  Google Scholar 

  • Le Breton M, Zaporozhets V (2009) On the equivalence of coalitional and individual strategy-proofness properties. Soc Choice Welf 33(2):287–309

    Article  Google Scholar 

  • Mas-Colell A, Sonnenschein H (1972) General possibility theorems for group decisions. Rev Econ Stud 39(2):185–192

    Article  Google Scholar 

  • Mihara HR (1997) Arrow’s theorem and Turing computability. Econ Theor 10(2):257–276

    Article  Google Scholar 

  • Mihara HR (1997) Anonymity and neutrality in Arrow’s Theorem with restricted coalition algebras. Soc Choice Welf 14(4):503–512

    Article  Google Scholar 

  • Mihara HR (2000) Coalitionally strategyproof functions depend only on the most-preferred alternatives. Soc Choice Welf 17(3):393–402

    Article  Google Scholar 

  • Mihara HR (2001) Existence of a coalitionally strategyproof social choice function: a constructive proof. Soc Choice Welf 18(3):543–553

    Article  Google Scholar 

  • Pazner EA, Wesley E (1977) Stability of social choices in infinitely large societies. J Econ Theory 14(2):252–262

    Article  Google Scholar 

  • Reny PJ (2001) Arrow’s theorem and the Gibbard-Satterthwaite theorem: a unified approach. Econ Lett 70(1):99–105

    Article  Google Scholar 

  • Sato S (2013) A sufficient condition for the equivalence of strategy-proofness and nonmanipulability by preferences adjacent to the sincere one. J Econ Theory 148(1):259–278

    Article  Google Scholar 

  • Satterthwaite MA (1975) Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J Econ Theory 10(2):187–217

    Article  Google Scholar 

  • Sen AK (1970) Collective choice and social welfare. Holden-Day, San Francisco

    Google Scholar 

  • Suzumura K (1976) Remarks on the theory of collective choice. Economica 43(172):381–390

    Article  Google Scholar 

  • Szpilrajn E (1930) Sur l’extension de l’ordre partiel. Fundam Math 1(16):386–389

    Article  Google Scholar 

  • Tanaka Y (2003) Oligarchy for social choice correspondences and strategy-proofness. Theor Decis 55(3):273–287

    Article  Google Scholar 

  • Wilkinson H (2021) Infinite aggregation: expanded addition. Philos Stud 178(6):1917–1949

    Article  Google Scholar 

  • Willard S (1970) General topology. Addison-Wesley Publishing Company, Reading

    Google Scholar 

  • Young HP (1974) An axiomatization of Borda’s rule. J Econ Theory 9(1):43–52

    Article  Google Scholar 

Download references

Acknowledgements

I am grateful to Marc Fleurbaey, Johannes Hörner, Hannu Salonen, Philippe Solal, Kotaro Suzumura, and two anonymous referees for their helpful suggestions. I also thank GATE L-SE for hospitality. The financial support from the Mitsubishi Foundation through Grants No. ID201920011 and from KAKENHI through Grants Nos. JP18K01501 and JP20H01446 is gratefully acknowledged. This paper is prepared with support from the Institute of Social Science, University of Tokyo, and its institute-wide joint research project, “Methodology of Social Sciences: How to Measure Phenomena and Values.”

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  1. Institute of Social Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, 113-0033, Japan

    Susumu Cato

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Cato, S. Stable preference aggregation with infinite population. Soc Choice Welf 59, 287–304 (2022). https://doi.org/10.1007/s00355-022-01389-4

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