Abstract
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 N′⊆ N, 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.
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Notes
The problem of finding a consensus among classifications has been analyzed for many different structures. The second issue of volume 3 of the Journal of Classification was devoted to different aspects of this problem. The most relevant article, for our purposes, in that issue is Barthelemy et al. (1986) while other important contributions, in the case that the classifications constitute equivalence relations, are Mirkin (1975) and Fishburn and Rubinstein (1986). The specific case in which classifications have a hierarchical structure has been analyzed in McMorris and Powers (2008).
Similar problems arise even if the boundaries of the country do not change. Think about the problem of determining if Americans are pro or against immigration. Events like 9/11 and the election of Donald Trump as president of the USA indicate that the opinions of the citizens must be dated.
Fishburn shows that Arrow’s Impossibility Theorem no longer holds when the number of agents is infinite.
If the partition is \((S, \emptyset , \emptyset , \mathbb {N}\setminus S)\), for a \(S \subseteq \mathbb {N}\), Lemma 2 yields Lemma 1.
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Fioravanti, F., Tohmé, F. Asking Infinite Voters ‘Who is a J?’: Group Identification Problems in \(\mathbb {N}\). J Classif 37, 58–65 (2020). https://doi.org/10.1007/s00357-018-9295-5
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DOI: https://doi.org/10.1007/s00357-018-9295-5