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Stable preference aggregation with infinite population

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.

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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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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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