Continuous Choice Functions Research Articles

Smooth approximations by continuous choice-functions

We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake, we prove that the theory of densely ordered sets with generic predicates is \(\aleph _0\)-categorical. A model of the theory and a particular continuous choice-function is constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a by-product, other functions with surprising properties can be constructed.

Non-existence of continuous choice functions

Let X be a compact, or path-connected, metric space whose topological dimension is at least 2. We show that there does not exist a continuous choice function (i.e., single-valued choice correspondence) defined on the collection of all finite feasible sets in X. Not to be void of content, therefore, a revealed preference theory in the context of most infinite consumption spaces must either relinquish the fundamental continuity property or allow for multi-valued choice correspondences.

Collective choice functions on non-convex problems

Collective choice problems on sets in \({\frak R}_+^n\) arise naturally in economics. Such problems have been extensively studied both in the theory of revealed preferences (Peters and Wakker, 1991) and in axiomatic bargaining theory under the assumption of convexity. However, our knowledge of collective choice functions on non-convex problems is still patchy. In this paper I study the existence and characterisation of continuous choice functions on the domain \(\Gamma\) of comprehensive problems. The main result completely characterises rational choice functions that are continuous and satisfy Weak Pareto Optimality: they form the class of Monotone Path Choice Functions on \(\Gamma\). I also show that any discontinuous rational and weakly Pareto optimal choice function must be non- anonymous.

Revealed group preferences on non-convex choice problems

This paper studies the conditions under which the basic results of the revealed preference theory can be established on the domain of choice problems which include non-convex feasible sets; the exercise is closely related to the works of Peters and Wakker (1991) and Bossert (1994). We show that while no continuous choice function can satisfy strong Pareto optimality over the class of all compact and comprehensive choice problems, strong Pareto optimality, Arrow's choice axiom, upper hemicontinuity and a weak compromisation postulate turn out to be necessary and sufficient to represent choice correspondences by continuous, strictly increasing and quasiconcave real-valued functions. Some applications of our main findings to axiomatic bargaining theory are also studied.

Continuous choice functions and the strong axiom of revealed preference

In this note, it is shown that a result due to Uzawa showing that the weak axiom of revealed preference together with a regularity condition implies the strong axiom of revealed preference can be derived from the regularity conditionalone, if the choice function is continuous in prices and in income. The observation that the weak axiom of revealed preference is not needed suggests that this regularity condition is remarkably strong. The main result of the paper is quite general in the sense that the only additional assumption on the choice function is that the whole budget be spent.

Local selections and local dendrites

Let X be a Peano continuum, C( X) its space of subcontinua, and C( X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C( X) is a continuous choice function; the selection σ is rigid if σ( A) ϵ B ⊂ A implies σ( A) = σ( B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C( X, ε) admits a selection (rigid selection). Further, C( X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C( X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.

A generalization of a continuous choice function theorem.

A generalization of a continuous choice function theorem.

Continuous choice functions and convexity

Continuous choice functions and convexity