Choice by lexicographic semiorders

Abstract

In Tversky's (1969) model of a lexicographic semiorder, a preference is generated via the sequential application of numerical criteria by declaring an alternative x better than an alternative y if the first criterion that distinguishes between x and y ranks x higher than y by an amount exceeding a fixed threshold. We generalize this idea to a fully fledged model of boundedly rational choice. We explore the connection with sequential rationalizability of choice (Apesteguia and Ballester 2010, Apesteguia and Ballester 2010) and we provide axiomatic characterizations of both models in terms of observable choice data.

Partial knowledge restrictions on the two-stage threshold model of choice

“…(The same is true, mutatis mutandis, for Proposition 3.14.) 17 For example, under single-valued choice we have that E is empty, W f is the transitive closure of H f , and the condition that xW f y ⇒ ¬[yH f x] (used in Theorem 4.3) amounts to acyclicity of H f (used in Proposition 3.3).…”

“…Moreover, it is not difficult to confirm that each single-valued characterization in Section 3 is a corollary of the corresponding multi-valued result. 17 …”

“…To give this enterprise some concreteness, we shall commit to a particular model of how choices are determined by the Cherepanov et al [8], and Masatlioglu and Nakajima [20] on search and consideration sets; and Mandler et al [15], Manzini and Mariotti [17], and Bajraj andÜlkü [4] on procedural models. 4 The observer might be able to determine salience levels, say, using knowledge of the physiology of vision and the spatial arrangement of the choice alternatives.…”

State dependent choice

“…For example, in a variation of our model in which each property can be satisfied in various degrees represented by a partial order (i.e. a 'state dependent' version of our [42] model of choice by lexicographic semiorder) the correspondence would be lost.…”

A simple model of two-stage choice

“…The intuition comes from observing the connection between indirect and weak reversals: if c is RSM-representable, then an indirect x, y reversal amounts to a chain of three connected weak reversals y , z , z, w , and w , x . 23 As such, Strong Exclusivity rules out particular four-cycles of weak reversals: if c displays a chain of weak reversals (arising from an indirect x, y reversal), then it cannot display a weak x, y reversal. In a similar fashion, the incompatibility between direct and weak reversals implied by Strong Exclusivity rules out particular three-cycles of weak reversals.…”