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Anonymous, neutral, and resolute social choice revisited

A Correction to this article was published on 12 February 2021

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Abstract

We revisit the incompatibility of anonymity and neutrality in single-valued social choice. We first analyze the irresoluteness outlook these two axioms together with Pareto efficiency impose on social choice rules and deliver a method to refine irresolute rules without violating anonymity, neutrality, and efficiency. Next, we propose a weakening of neutrality called consequential neutrality that requires resolute social choice rules to assign each alternative to the same number of profiles. We explore social choice problems in which consequential neutrality resolves impossibilities that stem from the fundamental tension between anonymity, neutrality, and resoluteness.

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Notes

  1. 1.

    Zwicker (2016) delivers an introduction to the theory of voting where major results regarding anonymity and neutrality are included.

  2. 2.

    They propose a weaker version of anonymity in a similar way by assuming that individuals are divided into sub-committees and requiring that, within each sub-committee, individuals have equal influence on the collective decision, while people in different sub-committees may enjoy different levels of influence.

  3. 3.

    So, given any distinct \(x,y\in A\) and \(P_i\in {\mathcal {L}}(A)\), precisely one of \(xP_iy\) and \(yP_ix\) holds. Moreover, \(xP_iy\) and \(yP_iz\) implies \(xP_iz\) for all \(x,y,z\in A\) and \(P_i\in {\mathcal {L}}(A)\). Finally, \(xP_ix\) does not hold for any \(x\in A\).

  4. 4.

    Having already noted \({\mathcal {D}}_m^*(n)\subseteq {\mathcal {D}}_m(n)\), we now remark that \({\mathcal {D}}_m^*(n)=\{1\}\implies {\mathcal {D}}_m(n)=\{1\}\) for all \(m,n\in {\mathbb {N}}\). To prove this by contradiction, we first let \(k\in {\mathcal {D}}_m(n)\backslash \{1\}\). Thus, we have \(k\in {\mathcal {D}}(n)\) and \(k\le m\). Due to the fundamental theorem of arithmetic, k has a prime divisor \(k^*\), which divides n as well, hence \(k^*\in {\mathcal {D}}_m^*(n)\), implying \({\mathcal {D}}_m^*(n)\ne \{1\}\).

  5. 5.

    This condition is equivalent to asking the greatest common divisor of m! and n to be 1, as mentioned by Doğan and Giritligil (2015), who reconsider the problem through a group theoretic approach. Interestingly, as Doğan and Giritligil (2015) as well as Bubboloni and Gori (2014) show, \(gcd(m!, n) = 1\) turns out to be necessary and sufficient for the existence of anonymous and neutral social welfare functions (i.e., functions which assign to every preference profile a strict ranking of alternatives).

  6. 6.

    Note that the irresoluteness outlook of an SCR f does not specify to how many profiles f assigns k alternatives when \(k\in K_f\).

  7. 7.

    Note that the iterative plurality rule presents a generalization of our concept of an SCR in the sense that it is defined on the domain of preference profiles over every \(B\in {{\mathcal {A}}}\).

  8. 8.

    Part (ii) of Theorem 2 can be proven with rules other than the iterative plurality rule, such as Coombs rule, as in Moulin (1983) p. 24. We are thankful to a referee for pointing out this fact.

  9. 9.

    Define, for any \(P_N\in {\mathcal {L}}(A)^N\), \(\alpha (P_N,x,k)=\#\{i\in N: \#\{y\in A\backslash \{x\}:x P_i y\}\ge k\}\), which gives the number of individuals that rank x higher than at least k alternatives. Now, define the fallback bargaining rule \(\varphi :{\mathcal {L}}(A)^N\rightarrow {\mathcal {A}}\) so that, \(\forall x\in A\), \(x\in \varphi (P_N)\) iff

    $$\begin{aligned} {\text {*}}{max}_{k\in \{0,\ldots ,m-1\}} \left\{ k\in {\mathbb {N}}: \alpha (P_N,x,k)=n\right\} \ge {\text {*}}{max}_{k\in \{0,\dots ,m-1\}} \left\{ k\in {\mathbb {N}}: \alpha (P_N,y,k)=n\right\} \end{aligned}$$

    for all \(y\in A \backslash \{x\}.\)

  10. 10.

    Perry and Powers (2008) propose a method to count, in the case of two alternatives, the number of aggregation rules that satisfy anonymity and neutrality. Bubboloni and Gori (2016) give a method to perform these counts for any number of alternatives.

  11. 11.

    We are providing computational results for only some small values of m and n because as m and n increase, these values grow dramatically (see Tables 2 and 3 in the Appendix). As diminution in the ratios are also fast, these values appear to be sufficient for this conclusion. Since the aim of this counting exercise is to assess the comparison of numbers of functions that satisfy the two versions of neutrality, we leave out other axioms (such as anonymity and efficiency), although it certainly is an interesting question.

  12. 12.

    This holds for any \(x_i\in A\).

  13. 13.

    Note that \(\sigma ^{r+m}(x_i)=\sigma ^{r}(x_i)\) for all \(r\in {\mathbb {N}}\) and \(i\in \{1,\ldots ,m\}\).

  14. 14.

    To see this, note that the numerator in \(\left( {\begin{array}{c}m\\ k\end{array}}\right) =\frac{m\cdot ... \cdot (m-k+1)}{k!}\) is divisible by m whereas none of \(\{2,\ldots ,k\}\) divides m.

  15. 15.

    To see this, consider \(g:{\mathcal {L}}(A)^{N}\rightarrow {\mathcal {A}}\) such that \(g(P_{N})=g(P_{N}^{\prime })=x\) and \(g(P_{N}^{\prime \prime })=g(P_{N}^{\prime \prime \prime })=y\), which is both anonymous and CN while not efficient.

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Correspondence to Ali I. Ozkes.

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We are grateful to Walter Bossert, Denis Cornaz, Fatih Demirkale, Lars Ehlers, Ayça Ebru Giritligil, Jeffrey Hatley, Sean Horan, Hervé Moulin, Jean Lainé, Clemens Puppe, Yves Sprumont, and William Zwicker for helpful discussions. The paper extensively benefited from the thoughtful comments of three anonymous referees to whom we are grateful. Our work is partly supported by the Projects ANR-14-CE24-0007-01, CoCoRICo-CoDec, and IDEX ANR-10-IDEX-0001-02 PSL* MIFID.

The original online version of this article was revised: Due to production team missed to incorporate the author corrections namely placement of Table 1, typo in footnote of Tables 2 and 3 and affiliation order of the first author. Now, they have been corrected.

Appendix: Observations on the numbers of CN, neutral, and resolute SCRs

Appendix: Observations on the numbers of CN, neutral, and resolute SCRs

Tables 2 and 3 below show ratios the observations made in Sect. 4 are based on.

Table 2 The ratio of \({\#{\mathcal {F}}_{m,n}^{NEUTRAL}}/{\#{\mathcal {F}}_{m,n}^{CN}}\) for different values of (mn)
Table 3 The ratio of \({\#{\mathcal {F}}_{m,n}^{CN}}/{\#{\mathcal {F}}^{RESOLUTE}_{m,n}}\) for different values of (mn)

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Ozkes, A.I., Sanver, M.R. Anonymous, neutral, and resolute social choice revisited. Soc Choice Welf 57, 97–113 (2021). https://doi.org/10.1007/s00355-020-01308-5

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