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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Notes
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.
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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I’m grateful for helpful feedback from an anonymous referee and from Martin Osborne. All errors are my responsibility.
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Duggan, J. Weak rationalizability and Arrovian impossibility theorems for responsive social choice. Public Choice 179, 7–40 (2019). https://doi.org/10.1007/s11127-018-0528-2
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DOI: https://doi.org/10.1007/s11127-018-0528-2