Gyula Maksa

Functional equations involving means and their Gauss composition

In this paper the equivalence of the two functional equations f(M 1 (x,y))+f(M 2 (x,y))=f(x)+f(y) (x,y ∈ I) and 2f(M 1 ⊗ M 2 (x,y)) = f(x) + f(y) (x,y ∈ I) is studied, where M 1 and M 2 are two variable strict means on an open real interval I, and M 1 ⊗ M 2 denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown function f: I → R) for the cases when M 1 and M 2 are the arithmetic and geometric means, respectively, and also in the case when M 1 , M 2 , and M 1 ⊗ M 2 are quasi-arithmetic means. If M 1 and M 2 are weighted arithmetic means, then, depending on the algebraic character of the weight, the above equations can be equivalent and also non-equivalent to each other.

Collective judgement: combining individual value judgements

Abstract This paper addresses a fundamental problem of collective decision making: how to derive a collective value judgement from the individual value judgements of the members of a committee. Three structural conditions will be introduced, which correspond to certain consistency requirements for the collective judgement. It will be shown that the formula for the collective value judgement, based on these consistency conditions, is a quasilinear mean of the individual judgements, and moreover, that the generating function of the corresponding quasilinear mean is independent of the number of people in the committee. Some uniqueness properties are considered and, finally, it is shown that the quasilinear mean is suitable as a social choice function satisfying six Arrowian conditions.

A functional equation arising from ranked additive and separable utility

All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rank-dependent expected utility, all nonnegative solutions are obtained without any regularity conditions. An unexpected possibility of reduction to convexity makes the present proof possible.

Solution of a functional equation arising in an axiomatization of the utility of binary gambles

For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation (z/p)γ−1[zγ(p)] = φ−1[φ(z)ψ(p)] (z, p ∈]0, 1[) is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation h(u)[g(u+ v) − g(v)] = f(v)g(u + v) (u ∈ R+, v ∈ R).

Consistent aggregation of scale families of selection probabilities

Abstract There are many situations where an individual or group of individuals has to select the best option or options from some available set of options according to various criteria. Such choices or decisions are often probabilistic. It is also frequently necessary to combine or aggregate the selection data obtained in different contexts or from different individuals. We suppose that the selection probabilities are functions of the (ratio) scale values of the options. On one hand the aggregated (“overall”) selection probability of each option should be a function of all componentwise (“individual”) selection probabilities of all options; on the other hand it should be a function of the aggregated scale values. Using functional equation techniques, we derive from these suppositions, under quite weak technical conditions, that the scale values are aggregated by products of powers, in particular by weighted geometric means, and both the individual and the overall selection probabilities are relatively simple functions of the scale values of the options. Finally, as a special case, we characterize Luces choice model.

Stability of the parametric fundamental equation of information for nonpositive parameters

Summary.In this note we prove that the parametric fundamental equation of information is stable in the sense of Hyers and Ulam provided that the parameter is nonpositive. We also prove, as a corollary, that the system of equations that defines the recursive and semi-symmetric information measures depending on a nonpositive parameter is stable in a certain sense.

On the Stability of a Sum Form Equation

In this paper we prove that an equation of sum form is stable and almost superstable.

General and continuous solutions of the entropy equation

We present the general and continuous solutions of Shannon’s functional equation on the positive octant. We show that extensions to the positive octant yield more general, non‐separable, solutions than the entropy expression. We also show that strict monotonicity of the entropy measure is not a required axiom to derive the entropy solution, rather a milder condition of it being an increasing non‐constant function is sufficient.

Convexity with respect to families of means

AbstractIn this paper we investigate continuity properties of functions