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Public Choice

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07-04-2018

Weak rationalizability and Arrovian impossibility theorems for responsive social choice

Author: John Duggan

Published in: Public Choice | Issue 1-2/2019

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Abstract

This paper provides representation theorems for choice functions satisfying weak rationality conditions: a choice function satisfies \(\alpha\) if and only if it can be expressed as the union of intersections of maximal sets of a fixed collection of acyclic relations, and a choice function satisfies \(\gamma\) if and only if it consists of the maximal elements of a relation that can depend on the feasible set in a particular, well-behaved way. Other rationality conditions are investigated, and these results are applied to deduce impossibility theorems for social choice functions satisfying weak rationality conditions along with positive responsiveness conditions. For example, under standard assumptions on the set of alternatives and domain of preferences, if a social choice function satisfies Pareto optimality, independence of irrelevant alternatives, a positive responsiveness condition for revealed social preferences, and a new rationality condition \(\delta ^{*}\) (a strengthening of \(\gamma\)), then some individual must have near dictatorial power.

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Equivalently, B is a linear order if \(R_{B}\) is transitive and anti-symmetric; it is a weak order if \(R_{B}\) is transitive; it is a quasiorder if \(R_{B}\) is negatively transitive; and it is a suborder if \(R_{B}\) is negatively acyclic, in the sense that there do not exist a natural number k and alternatives \(x_{1},\ldots ,x_{k} \in X\) such that \(\lnot x_{j} R_{B} x_{j+1}\) for all \(j <k\) and \(\lnot x_{k}R_{B}x_{1}\).

See Duggan (2013) for a review of the related literature and for general results on the uncovered set.

The result is also shown in Theorem 1 of Schwartz (1976). He assumes the collection of feasible sets consists of the nonempty, finite subsets of alternatives, and he defines a condition W1 that is equivalent to the conjunction of \(\alpha\) and \(\gamma\).

The axiom of Aizerman and Malishevski (1981) requires the opposite inclusion as well, so that \(C(Y)=C(Z)\), but this holds in the presence of \(\alpha\). Condition W5 of Schwartz (1976) suffers from a typo: \(\not \subseteq\) should be \(\subseteq\). See Weymark (1983) for a discussion of why condition \(\phi\) must be applied with some restriction on the size of feasible sets.

To verify that \(\beta '\) implies \(\beta\), consider \(Y,Z \in {\mathcal {X}}\) and assume \(Y \subseteq Z\) and \(C(Y) \cap C(Z) \ne \emptyset\). In particular, \(C(Z) \cap Y \ne \emptyset\), so \(\beta '\) implies \(C(Y) \subseteq C(Y \cup Z)=C(Z)\), as required.

Sen’s (1971) Theorem T.8 shows that the conjunction of \(\alpha\) and \(\beta\) is equivalent to the weak congruence axiom, which is equivalent to rationalizability by a weak order, which is equivalent to ACA; see Proposition 3.

The term “pseudo-relation” is not my own: it appears in the overhead slides of a seminar at Caltech, collected in Aizerman et al. (1991). The term “pseudo-rationalizability” is used differently than by Moulin (1985).

Mas-Colell and Sonnenschein (1972) assume that if some individual is indifferent between x and y and then this becomes a strict preference, then this is sufficient to break a social tie, and if some individual has a strict preference and then this becomes indifference, again a social tie is broken. In contrast, the antecedent of r-Tie break condition requires a strict preference reversal, making the axiom considerably weaker.

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Title: Weak rationalizability and Arrovian impossibility theorems for responsive social choice
Author: John Duggan
Publication date: 07-04-2018
Publisher: Springer US
Published in: Public Choice / Issue 1-2/2019
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-018-0528-2

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Introduction to a special issue in honor of Kenneth Arrow

Kenneth Arrow’s impossibility theorem stretching to other fields

Reflections on Arrow’s theorem and voting rules

Arrow, and unexpected consequences of his theorem

Social welfare with net utilities

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