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Infinite-population approval voting: A proposal

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Abstract

In this study, we propose a new direction of research on the axiomatic analysis of approval voting, which is a common democratic decision method. Its novelty is to examine an infinite population setting, which includes an application to intergenerational problems. In particular, we assume that the set of the population is countably infinite. We provide several extensions of the method of approval voting for this setting. As our main result, axiomatic characterizations of the extensions are offered by revealing a direct link between approval voting and the Borda rule. The characterized methods are natural extensions of the standard approval voting method for the finite-population case and are regarded as minimum requirements for other possible infinite-population extensions, which are reasonably democratic.

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Notes

  1. Rawls (1971) examines the problem of just saving in his theory of justice, employing a non-utilitarian approach. Rawls’s argument is formalized by Arrow (1973) and Dasgputa (1974) by using a model with an infinite population.

  2. This interpretation is related to Mihara’s (1997) argument, which provides a rationale for Arrovian social choice with an infinite population.

  3. McCarthy et al. (2019, p 2) provide similar rationales of an infinite-population framework of moral judgments.

  4. One may question whether our results are valid for the case where the set of individuals is uncountably infinite. The results in the second step are valid, but our results in the third step do not hold because our continuity axioms are constructed for the countable-set structure. If we impose an uncountable set of generations, we need related but other types of axioms.

  5. See also Smith (1973), Goertz and Maniquet (2011), and Pivato (2013) .

  6. An exception is a work by Fey (2004), who examines infinite-population versions of the simple majority rule.

  7. This axiom indicates that if the electorate is composed of two voters approving disjoint sets of outcomes, then the ballot aggregation function selects the union of these two ballots.

  8. Ramsey (1928) follows Sidgwick’s proposal of time neutrality by taking suggestions of J.M. Keynes.

  9. See Section 4 for the definition of the operator \(+\) on \(\Sigma \).

  10. It is noteworthy that, in a sense, Theorem 2 is a type of generalization of a characterization result for a variable, finite-population setting. We can identify a finite-population set of concerned generations as active voters in each profile in \(\Omega \), and, by regarding unconcerned generations in the distant future to be non-existent, we can induce the finite-population structure. Our result suggests that our aggregation rule(s) must choose the same outcome as the approval voting method under this finite-population model.

  11. In the literature on intergenerational equity, the “lim inf” criterion is employed by Fleurbaey and Michel (2003) and Chambers (2009). In their works, it is used to compare infinite streams of outcomes/utilities. Here, we follow this strategy to compare infinite streams of ballots.

  12. The overtaking criterion is defined as follows: given two infinite utility streams \((u_i)_{\in {\mathbb {N}}}, (u'_i)_{i \in {\mathbb {N}}} \in {\mathbb {R}}_+^{{\mathbb {N}}}\), \((u_i)_{\in {\mathbb {N}}}\) is at least as good as \((u'_i)_{\in {\mathbb {N}}}\) if there exists \(m \in {\mathbb {N}}\) such that \(\sum _{i =1}^n u_i \ge \sum _{i =1}^n u'_i \text{ for } \text{ each } n \ge m\).

  13. See Fleurbaey and Michel (2003), who propose the “lim inf” criterion as follows: \((u_i)_{\in {\mathbb {N}}}\) is at least as good as \((u'_i)_{\in {\mathbb {N}}}\) if \(\lim _m \inf _{n \ge m} \sum _{i =1}^n (u_i-u_i') \ge 0\). They argue the difference between the “lim inf” criterion and the overtaking criterion in their framework.

  14. See Asheim and Tungodden (2004) and Basu and Mitra (2007). They introduce variants of Brock’s axiom. In particular, the formulation of axioms by Asheim and Tungodden (2004) is closely related to our axioms because they employ expressions with the lim inf and lim sup. A substantial difference is that they define their continuity axioms for binary relations, while we define our continuity axioms for choice functions.

  15. A natural interpretation of strong choice continuity is associated with tentative incompleteness. Choice continuity does not exclude unnecessary tentative incompleteness shown in Example 5. Strong choice continuity works as an axiom for resolving such incompleteness.

  16. Roughly speaking, odd generations appear earlier than even generations. Notably, the rule respects the interests of the earlier generations in a sense. This non-invariant nature is associated with what we may call the present bias. However, this bias occurs in limited cases. If one tries to avoid this bias and impose full anonymity, decision-making is either limited to profiles in \(\Omega \) or another voting rule must be considered.

  17. See Litak (2018) for the use of the Axiom of Choice in a voting model with an infinite population.

  18. We now mention a slightly technical point on the relationship between finite-population and infinite-population settings. Our characterization results extend the existing works on a finite but “variable-population” case. A newly introduced operator \(\oplus \) is needed for extensions to the infinite-population setting because of the variable nature of consistency. For approval voting methods or scoring methods, variable-population settings are considered to be quite natural; see Fishburn (1978a,1978b) and Young (1974). Further, plausible features of these voting methods are captured with variable population axioms. However, by using the result in a fixed-population setting, our extensions can be simpler. Indeed, Baigent and Xu (1991) propose a set of axioms for characterizing approval voting under a finite- and fixed-population setting. If we use variations of their axioms, our extensions can be done without the operator \(\oplus \).

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Acknowledgements

We would like to thank an editor and two anonymous reviewers of this journal for their helpful comments and suggestions on an earlier version of the paper. Financial support from JSPS KAKENHI (18K01501, 20H01446), Mutsubishi Foundation (ID201920011), research programs IDEX Lyon from University of Lyon (project IN-DEPTH) within the programme Investissements d’Avenir” (ANR-16-IDEX-0005), and “Mathématiques de la décision pour l’ingénierie physique et sociale” (MODMAD) is gratefully acknowledged. Cato acknowledges an institute-wide joint research project, “Methodology of Social Sciences” in the Institute of Social Science, University of Tokyo.

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

    Susumu Cato

  2. Université Jean Monnet and GATE LSE, 10, rue Trefilerie, 42 023, Saint-Etienne Cedex 2, France

    Eric Rémila & Philippe Solal

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  1. Susumu Cato
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Cato, S., Rémila, E. & Solal, P. Infinite-population approval voting: A proposal. Synthese 199, 10181–10209 (2021). https://doi.org/10.1007/s11229-021-03242-0

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