Abstract
This paper considers the trade-off between unanimity and anonymity in collective decision-making with an infinite population. This efficiency-equity trade-off is afundamental difficulty in making a normative judgment in a conflict betweengenerations. In particular, it is known that this trade-off is quite sensitive in the formulation of unanimity axioms. In this study, we consider the trade-off in a preference-aggregation framework instead of the standard utility-aggregationframework. We show that there exists a social welfare function that satisfies I-strong Pareto, independence of irrelevant alternatives, and finite anonymity. This contrasts with an impossibility result in the standard utility-aggregation framework, and this means that the trade-off is also sensitive for background frameworks of aggregations.
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Notes
This is proved by Cato (2017).
In our main result (Theorem 1), we show that the social welfare function also satisfies finite insensitivity, which is stronger than finite anonymity. The axiom requires that social preferences are invariant for changes of preferences in any finite subset of individuals.
R is complete if \(\forall x,y \in X\), \((x,y) \in R\) or \((y,x) \in R\), and R is transitive if \(\forall x,y,z \in X\), if \((x,y) \in R\) and \((y,z) \in R\), then \((x,z) \in R\). Cato (2016) examines the implications of these conditions.
Non-dictatorship requires that there exists no \(i \in N\) such that \(P(R_i) \subseteq P(f(\mathbf {R}))\) for all \(\mathbf {R} \in \mathcal {R}^N\).
For other types of anonymity in this setting, see Cato (2017).
For a set A, \((A)^c\) (or \(A^c\)) denotes the complement of A.
See Aliprantis and Border (2006).
A coalition G is decisive if \(\bigcap _{i \in G} P(R_i) \subseteq P(f(\mathbf {R}))\) for all \(\mathbf {R} \in \mathcal {R}^N\).
Note that the social welfare function proposed here satisfies finite insensitivity.
Hansson (1976) shows that if a quasi-social welfare function satisfies weak Pareto and IIA, then the family of decisive coalitions forms a filter on N.
Reflexivity is obvious. Transitivity is shown as follows. For all \(x,y,z \in X\), if \((x,y) \in f_{\hat{\Omega }}(\mathbf{R})\) and \((y,z) \in f_{\hat{\Omega }}(\mathbf{R})\), then both \(\{ i \in N: (y,x) \in P(R_i)\}\) and \(\{ i \in N: (z,y) \in P(R_i)\}\) are finite. Since \(\{ i \in N: (z,x) \in P(R_i)\} \subseteq \{ i \in N: (y,x) \in P(R_i)\} \cup \{ i \in N: (z,y) \in P(R_i)\}\), \(\{ i \in N: (z,x) \in P(R_i)\}\) is finite. We obtain \((x,z) \in f_{\hat{\Omega }}(\mathbf{R})\).
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I thank an associate editor of this journal, two anonymous reviewers, Marc Fleurbaey, and Hannu Salonen for their helpful suggestions. This study was financially supported by JSPS KAKENHI (18K01501), and was also supported by the Postdoctoral Fellowship for Research Abroad of the JSPS.
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Cato, S. The possibility of Paretian anonymous decision-making with an infinite population. Soc Choice Welf 53, 587–601 (2019). https://doi.org/10.1007/s00355-019-01199-1
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DOI: https://doi.org/10.1007/s00355-019-01199-1