János Aczél

Procedures for Synthesizing Ratio Judgements

Requirements which seem reasonable for functions synthesizing judgements (quantities or their ratios), in particular separability, associativity or bisymmetry, cancellativity, consensus, reciprocal or homogeneity properties are investigated and all functions satisfying them are determined.

On the possible merging functions

Abstract In this paper, we study merging functions, functions which combine individual judgements into a merged or aggregate or consensus judgement. In particular, we study such functions under several simple axioms, symmetry, linear homogeneity, and agreement (which says that if all individuals agree, the merged judgement agrees with those of all of the individuals). We show that under one or more of these assumptions, the possible merging procedures are very few if we want certain statements involving the merged functions to be meaningful in the precise sense used in the theory of measurement, and that in many cases the arithmetic mean or the geometric mean are the only possible merging functions. The results are applied to group consensus problems, to performance analysis of alternative new technologies or of students or job applicants, and to the development of measures of price level.

On scientific laws without dimensional constants

A foundational paper of Lute [S] shows that the general form of a “scientific law” is greatly restricted by knowledge of the “admissible transformations” of the dependent and independent variables, transformations such as that from grams to pounds or inches to meters. The restrictions are discovered by formulating a functional equation from knowledge of the admissible transformations. Lute’s basic approach has been clarified and extended by Lute [6, 71, Rozeboom [15, 161, Osborne [12], and Roberts and Rosenbaum [ 141. A fundamental assumption in all the results which have been obtained so far is that the admissible transformations can be applied independently to all of the independent variables. In this paper we modify this assumption, and discover that in this situation, knowledge of the admissible transformations does not always restrict the form of the scientific law as greatly as in the cases previously studied. Specifically, suppose x1, x2 ,..., x, + 1 are n + 1 variables, z is the set of admissible transformations for the ith variable, i= 1, 2,..., n + 1, and x,+ , is some unknown function u(x,, x2,..., x,). The problem is to find the general form of the function u knowing the sets 5, i.e., to find the general form of the “scientific law”

Determining merged relative scores

The starting point in this paper is the following type of situation. Performances of people or technologies are measured by their scores with respect to different benchmarks relative to those of some base performance. In order to arrive at a decision one wants to merge (average, aggregate) the scores of each individual or technology into a single number which can then be compared with that of another individual or technology. The situation is similar, at least technically, to price indices where present prices of different products are considered relative to prices of these products in a base year and we again want to merge these into one number. In this paper one solves most of these problems. One also states a few problems which are still open

A mixed theory of information. III. Inset entropies of degree β

In generalization of a purely probabilistic result ( Daroczy, 1970 , Information and Control 16 , 36–41), the general form of. symmetric, β -recursive entropies of randomized systems of events is determined in the framework of the mixed theory of information.

Measuring information beyond communication theory—Why some generalized information measures may be useful, others not

Summary (and Keywords)Non-communication models for information theory: games and experiments. Measures of uncertainty and information: entropies, divergences, information improvements.Some useful properties of information measures, symmetry, bounds, behaviour under composition, branching, conditional measures, sources. Rényi measures, measures of higher degree.Promising and not so promising generalizations. Measures which depend not just upon the probabilities but (also) upon the subject matters.

The Role of Some Functional Equations in Decision Analysis

This paper presents some functional equations that have played an essential role in the characterization of utility and probability functions in decision analysis. We survey some previous results with improvements and derive several new results. We also discuss some simple methods for the solution of these equations and highlight some subtle points about their use. We show that one functional equation can determine several unknown functions within it, and that relaxing differentiability and requiring only continuity of the functions leads to generalizations of many well-known results in utility and Bayesian probability theory.

Extension of a generalized Pexider equation

The equations k(s+t) = (s)+n(t) and k(s+t) = m(s)n(t), called Pexider equations, have been completely solved on R 2 . If they are assumed to hold only on an open region, they can be extended to R 2 (the second when k is nowhere 0) and solved that way. In this paper their common generalization k(s + t) = (s) + m(s)n(t) is extended from an open region to R 2 and then completely solved if k is not constant on any proper interval. This equation has further interesting particular cases, such as k(s + t) = (s) + m(s)k(t) and k(s+t) = k(s) +m(s)n(t), that arose in characterization of geometric and power means and in a problem of equivalence of certain utility representations, respectively, where the equations may hold only on an open region in R 2 . Thus these problems are solved too.

Notes on generalized information functions

Solutions of functional equations, connected with deviations (divergences, inaccuracies) and with inset entropies of degree α, are supplemented and completed. No regularity assumptions are made.

Measuring information beyond communication theory: Some probably useful and some almost certainly useless generalizations

Abstract Non-communication models for information theory: games and experiments. Measures of uncertainty and information: entropies, divergences, information improvements. Some useful properties of information measures, symmetry, bounds, behaviour under composition, branching, conditional measures, sources. Renyi measures, measures of higher degree. Promising and not so promising generalizations. Measures which depend not just upon the probabilities but (also) upon the object matters.