Abstract
In the previous chapter we primarily discussed six choice mechanisms: naive, resolute, dogmatic, sagacious, lenient and cautious choice.1 These choice mechanisms specify the rule according to which preferences are transformed into choice sets. The present chapter is concerned with the choice sets themselves, and not with the mechanism they are generated from. Specifically, we describe in which way the choice sets vary as we change the opportunity sets, that is we analyse a choice function (see Definition 6.2). Depending on the way the choice sets vary, we say that the choice function exhibits a certain property.
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Notes
In the context of subgame perfection (section 7.4), we also introduced the perfect choice mechanism. We do not consider this last choice mechanism any further.
This may become clearer from the following static example: Suppose a binary relation with aRb, aRc, bRa, cRa, cPb. Then static choice (that is trivial choice as specified in Definition 7.2) generates the following choice function: C(a, b, c) = a, c, C(a, b) = a, b, C(a, c) = a, c, C(b, c) = c. So despite intransitivity (bRa, aRc, and notbRc) choice satisfies property α.
Moreover, Sen’s (1971, p.310) Theorem 3 shows that a choice function satisfying properties α and β generates an ordering R, which in turn, generates a choice function C such that C = C. If also R = R, then a perfect correspondence exists between choice and preference.
This is Arrow’s (1959) property C4, p. 123 stated in a dynamic terminology.
The property in its dynamic application goes back to Hammond (1977), though he appears unaware of its far-reaching implications. In Hammond’s study, H-ordinality is labelled as ordinality.
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© 1998 Springer-Verlag Berlin Heidelberg
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von Auer, L. (1998). Properties of Dynamic Choice Functions. In: Dynamic Preferences, Choice Mechanisms, and Welfare. Lecture Notes in Economics and Mathematical Systems, vol 462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58879-2_8
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DOI: https://doi.org/10.1007/978-3-642-58879-2_8
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