Maximal domains for strategy-proof rules in the candidates selection problem

• Autores: Dolors Berga , Shigehiro Serizawa
• Localización: Abstracts of the Fifth Spanish Meeting on Game Theory and Applications / coord. por Jesús Mario Bilbao Arrese , Francisco Ramón Fernández García , 2002, ISBN 84-472-0733-1, pág. 35
• Idioma: inglés
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• Resumen

  • We consider the problem where a set of voters have to elect candidates among a set of two alternatives. This is the same framework as in Barberà, Sonnenschein, and Zhou (1991), where they obtain �voting by committees� as the characterization class of all onto and strategy-proof voting schemes on the domain of separable preferences. Furthermore, separable preferences turn out to be the maximal rich domain where voting by committees are strategy-proof.

    In this paper we study maximal domains of preferences for strategy-proofness of any rule and not only voting by committees. We obtain that the set of separable preferences is a maximal domain for the existence of strategy-proof and onto social choice functions satisfying the additional requirement of no vetoerness. For more than two alternatives this result is still a conjecture.

    Barberà, Sonnenschein, and Zhou (1991) (henceforth, BSZ) considered the problem where a finite set of voters N must choose, from a finite set K, which objects will be adopted. These objects can be bills considered by a legislature, candidates to enter a club, or they can also be interpreted as public goods with two feasible levels 0, 1. This last interpretation was given in Barberà, Gul, and Sttacchetti (1993) (henceforth, BGS) where the more general setting of any number of feasible levels for public goods is considered. BSZ obtained two important results concerning strategy-proofness. On the one hand, they characterized voting by committees (which we also call: generalized median voter schemes) as the only onto social choice functions satisfying strategy-proofness when agents� preferences are separable. On the other hand, they obtain the set of separable preferences as a maximal rich domain for which voting by committees with neither veto nor dummy voters are strategy-proof.

    Recently, other results improving this necessary condition on preferences for which voting by committees are strategy-proof have been obtained. See Serizawa (1995) and Barberà, Massó, and Neme (1999) for discrete sets of alternatives, and Barberà, Massó, and Serizawa (1996) and Berga (2000) for the case where social alternatives are continuously measured. Although any strategy-proof rule on separable preferences must be voting by committees, a strategy-proof social choice function may not belong to this class when the rule is defined outside that domain. Thus, this literature may exclude interesting rules.

    From their two results mentioned above, two natural questions arise. We initially concentrate in BSZ�s framework with two objects.

    The first one is related to the existence of other domains of preferences preserving voting by committees as the unique class of strategy-proof social choice functions. By BSZ, we know that the subdomain of additive preferences preserves these results. In our paper, we consider �rich domains� and we conjecture that �a social choice function on a rich domain is strategy-proof if and only if it is a generalized median voter scheme�.

    The second question is how large the domain of preferences can be and still preserve the existence of strategy-proof rules (not necessarily generalized median voter schemes)? We qualify this question in two different ways. First, we employ the no vetoer condition to rule out trivial rules such as dictators.

    Second, we require domains to be rich; that is, to contain a minimal variety of preferences. Then, we conjecture that �the unique maximal rich domain for strategy-proofness and the no vetoer condition is the domain of separable preferences�.

    Additionally, in this paper we also show how relevant is the rich domain condition for our results. In Theorem 2 we state our main result (unique for the moment) which says that �the domain of separable preferences is a maximal domain for strategy-proofness and the no vetoer condition�. We provide and example of an onto and strategy-proof social choice function satisfying the no vetoer condition which is not a generalized median voter scheme and which is defined on a non-rich domain. Moreover, we think we can obtain (we conjecture) that a slightly variation of this domain is maximal for our properties.

    Thus, the domain of separable preferences is not unique in Theorem 2.

    These two questions are in the same line as the ones already answered in Berga and Serizawa (2000) where the authors do not restrict a priori the class of rules to be considered and they study the problem of the provision of a single public good. There, they obtain the domain of convex preferences as the unique maximal domain including a minimally rich one allowing for strategy-proof and onto rules satisfying the no vetoer condition. In this paper we restrict to the case of two public goods (or two objects), however, we believe that our result can be generalized to any number of goods. This is still part of current research