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Collective rationality and decisiveness coherence

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Abstract

Arrow’s impossibility theorem states that if an aggregation rule satisfies unrestricted domain, weak Pareto, independence of irrelevant alternatives, and collective rationality, then there exists a dictator. Among others, Arrow’s postulate of collective rationality is controversial. We propose a new axiom for an aggregation rule, decisiveness coherence, which is weaker than collective rationality. It is shown that given the Arrovian axioms other than collective rationality, a dictatorship arises if and only if decisiveness coherence is satisfied. Moreover, we introduce weak versions of decisive coherence and examine these implications.

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Notes

  1. Suzumura (1983) and Campbell and Kelly (2002) survey various Arrovian impossibility results under modified postulates.

  2. Sen (1995) examines the notion of rationality in social choice through Buchanan’s criticism.

  3. Ferejohn and Grether (1977) formulate a weakening of path independence. Bandyopadhyay (1988) provides a version of path independence that guarantees full rationality. Bandyopadhyay (1990) examines its implication for social choice theory.

  4. Cato (2010) provides a systematic approach to proving their impossibility theorems. Sen (1977) provides a classic survey on this subject.

  5. Quesada (2002) and Ozdemir and Sanver (2007) examine domain conditions that yield Arrow’s theorem.

  6. Cato (2016b) provides a comprehensive argument on fundamental properties of binary relations.

  7. See Cato (2016b) for more detail on the operational expressions of the rationality properties.

  8. Blair et al. (1976) provide a comprehensive analysis for path-independent social choice functions.

  9. See Samuelson (1938) and Houthakker (1950). Suzumura (1983) provides quite general formulations of both axioms of revealed preference.

  10. Suppose that f satisfies decisive congruence. By way of contradiction, assume that \(x \notin C_{R^*}(S \cup T)\) but \(x \in C_R(C_R(S) \cup C_R(T))\) for some \(x \in X\). Then, there exists \(y \in S \cup T\) such that \((y,x) \in \bigcap _{i \in M}P(R_i)\) for some \(M \in {\mathcal {D}}_f\). By definition, \((y,x) \in P(f(\mathbf {R}))\). Without loss of generality, we can assume that \(y \in S\). Since \((y,x) \in P(f(\mathbf {R}))\), it follows that \(x \in C_R(T)\) and \(y \notin C_R(S)\). Since \(y \notin C_R(S)\), finiteness implies that \((z,y) \in tc(P(f(\mathbf {R})))\) for some \(z \in C_R(S)\). Since \(tc(P(f(\mathbf {R}))) \subseteq tc(f(\mathbf {R}))\), \((z,y) \in tc(f(\mathbf {R}))\). If \((x,z) \in f(\mathbf {R})\), then \((x,y) \in tc(f(\mathbf {R}))\) and \((y,x) \in \bigcap _{i \in M}P(R_i)\). This contradicts (3). Thus, \((z,x) \in P(f(\mathbf {R}))\), by completeness. Therefore, we have \(x \notin C_R(C_R(S) \cup C_R(T))\), which is a contradiction. Thus, quasi path independence is satisfied.

  11. For example, the following rule is not serially dictatorial, but satisfies all axioms: \(f(\mathbf{R})=R_1\) for all profiles.

  12. Takayama and Yokotani (2017) carefully examine the structure of the set of conditionally decisive coalitions.

  13. Consider the following axiom, which is an extension of decisive coherence:

    Conditional decisiveness coherence: For all \(\mathbf {R} \in \mathcal {A}\), all \(M \subseteq N\), and all \(M' \in {\mathcal {D}}_f(M)\),

    $$\begin{aligned} \left( f(\mathbf {R})\circ \left( \left( \bigcap _{i \in M} I(R_i) \right) \cap \left( \bigcap _{i \in M'} P(R_i) \right) \right) \right) \cup \left( \left( \left( \bigcap _{i \in M} I(R_i) \!\right) \cap \left( \bigcap _{i \in M'} P(R_i) \right) \right) \circ f(\mathbf {R}) \right) \subseteq P(f(\mathbf {R})). \end{aligned}$$
  14. Weak decisiveness coherence\(^{\star }\) is not necessary for (5).

  15. Sen (1977, 1993) discusses the meanings and significance of this approach.

  16. See, for example, Kalai et al. (1979), Border (1983) and Fleurbaey et al. (2005), and Tadenuma (2005) Le Breton and Weymark (2011) provide a comprehensive survey of Arrovian results in economic domains.

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Acknowledgements

I thank Maggie Penn, a managing editor of this journal, and three anonymous referees for their helpful comments. I thank Marc Fleurbaey, Kohei Kamaga, Koichi Tadenuma, Maurice Salles, Yohei Sekiguchi, and Kotaro Suzumura for discussions and comments. This work was supported by JSPS KAKENHI Grant number 26870477 and Postdoctoral Fellowship for Research Abroad of JSPS. I also thank the hospitality of Princeton University.

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Cato, S. Collective rationality and decisiveness coherence. Soc Choice Welf 50, 305–328 (2018). https://doi.org/10.1007/s00355-017-1085-1

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