Asking Infinite Voters ‘Who is a J?’: Group Identification Problems in \(\mathbb {N}\)

  • Federico FioravantiEmail author
  • Fernando Tohmé


We analyze the problem of classifying individuals in a group N taking into account their opinions about which of them should belong to a specific subgroup NN, in the case that |N| > . We show that this problem is relevant in cases in which the group changes in time and/or is subject to uncertainty. The approach followed here to find the ensuing classification is by means of a Collective Identity Function (CIF) that maps the set of opinions into a subset of N. Kasher and Rubinstein (Logique & Analyse, 160, 385–395 1997) characterized different CIFs axiomatically when |N| < , in particular, the Liberal and Oligarchic aggregators. We show that in the infinite setting, the liberal result is still valid but the result no longer holds for the oligarchic case and give a characterization of all the aggregators satisfying the same axioms as the Oligarchic CIF. In our motivating examples, the solution obtained according to the alternative CIF is most cogent.


Social choice Aggregation group identification problem Infinite voters 



  1. Barthelemy, J.R., Leclerc, B., Monjardet, B. (1986). On the use of ordered sets in problems of comparison and consensus of classifications. Journal of Classification, 3, 187–224.MathSciNetCrossRefGoogle Scholar
  2. Biais, B., Bisiére, C., Bouvard, M., Casamatta, C. (2018). The blockchain folk theorem. Toulousse School of Economics Working Paper [17-817].Google Scholar
  3. Cho, W.J., & Ju, B.-G. (2017). Multinary group identification. Theoretical Economics, 12, 513–531.MathSciNetCrossRefGoogle Scholar
  4. Cho, W.J., & Saporiti, A. (2015). Incentives, fairness, and efficiency in group identification, The School of Economics Discussion Paper Series 1117 Economics, The University of Manchester.Google Scholar
  5. Fishburn, P.C. (1970). Arrow’s impossibility theorem: concise proof and infinite voters. Journal of Economic Theory, 2, 103–106.MathSciNetCrossRefGoogle Scholar
  6. Fishburn, P.C., & Rubinstein, A. (1986). Aggregation of equivalence relations. Journal of Classfication, 3, 61–65.MathSciNetCrossRefGoogle Scholar
  7. Kasher, A. (1993). Jewish collective identity. In Goldberg, D.T., & Kraus, M. (Eds.) Jewish Identity. Philadelphia: Temple University Press.Google Scholar
  8. Kasher, A., & Rubinstein, A. (1997). On the question “Who is a J?”: a social choice approach. Logique & Analyse, 160, 385–395.MathSciNetzbMATHGoogle Scholar
  9. McMorris, F.R., & Powers, R.C. (2008). A characterization of majority rule for hierarchies. Journal of Classification, 25, 153–158.MathSciNetCrossRefGoogle Scholar
  10. Mirkin, B. (1975). On the problem of reconciling partitions. In Blalock, H.M., Aganbegian, A., Borodkin, F.M., Boudon, R., Capecchi, V. (Eds.) Quantitative sociology, international perspectives on mathematical and statistical modelling (pp. 441–449). Academic Press: New York.Google Scholar
  11. Saporiti, A. (2012). A proof for ‘Who is a J’ impossibility theorem. Economics Bulletin, 32, 494–501.Google Scholar
  12. Shorish, J. (2018). Blockchain state machine representation, open science framework. (
  13. Sung, S.C., & Dimitrov, D. (2005). On the axiomatic characterization of ‘Who is a J?’. Logique & Analyse, 48, 101–112.MathSciNetzbMATHGoogle Scholar
  14. Suppes, P. (1972). Axiomatic set theory. Toronto: Dover Publications.zbMATHGoogle Scholar

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© The Classification Society 2019

Authors and Affiliations

  1. 1.INMABBUniversidad Nacional del Sur, CONICETBahía BlancaArgentina

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