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Unanimity in attribute-based preference domains

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Abstract

We provide several characterizations of unanimity decision rules, in a public choice model where preferences are constrained by attributes possessed by the alternatives (Nehring and Puppe, Games Econ Behavior 59:132–153, 2007a; Nehring and Puppe, J Econ Theory 135:269–305, 2007b). Solidarity conditions require that when some parameters of the economy change, the agents whose parameters are kept fixed either all weakly lose or they all weakly win. Population-monotonicity (Thomson, Math Oper Res 8:319–326, 1983a; Thomson, J Econ Theory 31:211–226, 1983b) applies to the arrival and departure of agents, while replacement-domination (Moulin, Q J Econ 102:769–783, 1987) applies to changes in preferences. We show that either solidarity property is compatible with voter-sovereignty and strategy-proofness if and only if the attribute space is quasi-median (Nehring, Social aggregation without veto, Mimeo, 2004), and with Pareto-efficiency if and only if the attribute space is a tree. Each of these combinations characterizes unanimity.

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Notes

  1. For surveys on these two conditions, see Thomson (1995, 1999).

  2. Solidarity conditions were also studied in the problem of locating multiple public goods by Miyagawa (1998, 2001), Ehlers (2002, 2003), and Umezawa (2012) and in the problem of selecting a probabilitic location by Ehlers and Klaus (2001). These models differ from ours in that they include some alternatives which are not considered best by any preference in the domain.

  3. In a companion paper, Gordon (2007a) studies Pareto-efficiency and solidarity in a general public choice problem and establishes several of these conditions’ general implications. We defer the discussion of the relation between the two papers to the end of Sect. 5.

  4. Note that the set \(A\) can be recovered from an attribute space \(\mathcal {H}\) since \(A=\bigcup \limits _{H\in \mathcal {H}}H\)

  5. Nehring and Puppe work with linear orderings (2007a, b). Here we consider weak orderings. Either domain is rich enough for our results to hold.

  6. Nehring and Puppe (2007b) provide a necessary and sufficient condition on the structure of winning coalitions under which voting by issues is well-defined: the “intersection property.” Barberà, Massó and Neme (1997) provide a different necessary and sufficient condition, also labelled “intersection property” under which separable voting under constraints is well-defined.

  7. Other properties are studied on these domains by Demange (1982), Danilov (1994) and Schummer and Vohra (2002).

  8. Buneman (1971) and Bandelt and Dress 1986 provide related constructions, without assuming a median space. Their construction, however, allows introducing “latent alternatives”, i.e. additional alternatives outisde of the set \(A\) in order to construct a tree consistent with an attribute structure that satisfies \(\left( T\right) \). Our result shows that when the space is median, latent alternatives are not needed. In addition, our construction relies on elementary arguments.

  9. Steps 3 and 4 in the proof can be deduced from the main result in Nehring and Puppe (2007a), more precisely from Claim (a) in their Theorem. For the sake of completeness, we provide a direct proof that exploits the special structure of this model to avoid the complexities of their analysis

  10. The proofs in Vohra (1999) and Klaus (1999, 2001) require infinitely many alternatives. This assumption is implicitly used, for example, in Vohra’s Lemma 3.

  11. In a similarly general framework, Bu (2013) establishes a general equivalence between false-name-proofness, which requires non-manipulability via the creation of fictitious identities, and strategy-proofness, anonymity and population-monotonicity. It would be interesting to study the implications of false-name-proofness in the class of attribute-based preference domains.

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  1. Departement of Economics, Sciences Po, 28 rue des Saints-Pères, 75007 , Paris, France

    Sidartha Gordon

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  1. Sidartha Gordon
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Correspondence to Sidartha Gordon.

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I am thankful to Hans Bandelt, Biung-Ghi Ju, Jim Schummer and William Thomson for helpful conversations and comments and to two anonymous referees for comments.

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Gordon, S. Unanimity in attribute-based preference domains. Soc Choice Welf 44, 13–29 (2015). https://doi.org/10.1007/s00355-014-0809-8

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