Completeness and transitivity of preferences on mixture sets

Abstract

In this paper, we show that the presence of the Archimedean and the mixturecontinuity properties of a binary relation, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without decisiveness (completeness), or decisiveness without consistency, or in the presence of a weak consistency condition, neither. The basic result can be sharpened when specialized from the context of a generalized mixture set to that of a mixture set in the sense of Herstein-Milnor (1953). We relate the results to the antecedent literature, and view them as part of an investigation into the interplay of the structure of the choice space and the behavioral assumptions on the binary relation defined on it; the ES research program due to Eilenberg (1941) and Sonnenschein (1965), and one to which Schmeidler (1971) is an especially influential contribution.

Topological connectedness and behavioral assumptions on preferences: a two-way relationship

“…Uzawa (1960) pioneered this line of research by showing that convexity of a complete preference relation defined on a convex subset of a topological vector space with closed upper sections and a transitive asymmetric part implies that the relation itself is transitive. This result is reproduced and generalized by Sonnenschein (1965, Theorems 5 and 6) and picked up by Galaabaatar, Khan, and Uyanık (2018).…”

“…Uzawa (1960) pioneered this line of research by showing that convexity of a complete preference relation defined on a convex subset of a topological vector space with closed upper sections and a transitive asymmetric part implies that the relation itself is transitive. This result is reproduced and generalized by Sonnenschein (1965, Theorems 5 and 6) and picked up by Galaabaatar, Khan, and Uyanık (2018). 33 Note that to say that the symmetric part of a relation is complete renders the relation trivial, whereas to say that the asymmetric part is complete furnishes the contradiction that an element of the space is preferred to itself.…”

“…46 This being said, it is important to emphasize that applications that work with strict preference relations as their primitive, such as Peleg (1970) and Majumdar and Sen (1976), 47 or those that simply drop both completeness and transitivity such as Nishimura and Ok (2016), also do not fall under the ambit of this paper. The same is true for applications relying on linear structures: they do not fall under the exclusively topological rubric of this paper, and require different mathematical techniques and tools, and we investigate such structures elsewhere; see Galaabaatar, Khan, and Uyanık (2018). 48 But again, just because the setting is one of uncertainty, it does not mean that the results are necessarily out of the bounds of our treatment elaborated in this paper.…”

The Intermediate Value Theorem and Decision-Making in Psychology and Economics: An Expositional Consolidation

“…23 In addition to the notion of the up-and-down function abstracted into a binary relation, Section 3 can also be viewed as an abstraction of the notion of equality into an equivalence relation, and it is this that facilitates the formulation of the solvability and restricted solvability properties. This section then turns to the discipline that gave rise to measurement and decision theory: 24 Section 4 turns to the new material as categorized under extensions and use of the IVP for functions separately from binary relations, and summarized in four portmanteau theorems, and two main Propositions. 25 Section 5 is devoted to the proofs.…”

On an extension of a theorem of Eilenberg and a characterization of topological connectedness

“…This is to say, they focused their attention on the mixing operation, rather than on the objects of choice itself. Wold (1943-44) used a similar scalar-continuity property in his work, but with an additional monotonicity assumption; also see Fishburn (1982), Wakker (1989), Bridges and Mehta (1995), Herden and Pallack (2001) and Candeal, Induráin and Molina (2012), Galaabaatar, Khan and Uyanık (2019) on numerical representation of preferences. In addition to this parametrized topological setting, we also mention for the record, a third setting, that involves no topology at all, but constrains itself to a purely algebraic structure.…”