Set and revealed preference axioms for multi-valued choice
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Set and revealed preference axioms for multi-valued choice Hans Peters1
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Panos Protopapas2
Ó The Author(s) 2020
Abstract We consider choice correspondences that assign a subset to every choice set of alternatives, where the total set of alternatives is an arbitrary finite or infinite set. We focus on the relations between several extensions of the condition of independence of irrelevant alternatives on one hand, and conditions on the revealed preference relation on sets, notably the weak axiom of revealed preference, on the other hand. We also establish the connection between the condition of independence of irrelevant alternatives and so-called strong sets; the latter characterize a social choice correspondence satisfying independence of irrelevant alternatives. Keywords Revealed preference axioms Multi-valued choice Independence of irrelevant alternatives Strong sets
1 Introduction 1.1 Background This paper contributes to a question with a long history. A (single-valued) choice function assigns to each choice set (i.e., subset of the set of all alternatives) an element of that choice set. The condition of independence of irrelevant alternatives (IIA) requires that if the alternative chosen from a choice set is still available in a subset of the choice set, then it should also be chosen from that subset. This condition already occurs as condition no. 7 in Nash’s axiomatic bargaining model (1950).1 Closely related is the Weak Axiom of Revealed 1
IIA is not to be confused with the condition with the same name in social choice theory, see Arrow (1963). It became standard to refer to Nash’s condition as IIA, e.g., Roth (1979).
The authors thank Bettina Klaus, Flip Klijn, Jordi Masso´, Luis Pedro Santos-Pinto, Dries Vermeulen, and two reviewers for helpful suggestions. & Hans Peters [email protected] Extended author information available on the last page of the article
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H. Peters P. Protopapas
Preference (WARP), which says that if x is chosen from some choice set where also y is available, then y should never be chosen from any choice set where also x is available.2 Another way of phrasing this, is that the preference relation revealed by the choice function has no cycles of length two. As is well-known and easy to show, for collections of choice sets that are closed under nonempty intersection, IIA and WARP are equivalent.3 1.2 This paper: main outlook In this paper, we consider choice correspondences instead of choice functions: these assign to each choice set a nonempty subset, rather than a unique element. In the classical choice literature, often the expression ‘choice function’ is used where we use ‘choice correspondence’. The main interpretation that we have in mind is that, indeed, a set is chosen, rather than a short-list from which later a singleton has to be chosen—for instance, a set of courses (starter, main, dessert) and a selection of wines in a restaurant; or a committee of fixed or variable size in a society or university.
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