Space of alternatives as a foundation of a mathematical model concerning decision-making under conditions of uncertainty

Abstract

We show a mathematical model based on “a priori” possible data and coherent subjective probabilities. A set of possible alternatives is viewed as a set of all possible samples whose size is equal to 1 selected from a finite population. Such a finite population coincides with those coherent previsions of a univariate random quantity representing all possible alternatives considered “a priori”. We consider a discrete probability distribution of all possible samples. We approximately get the standardized normal distribution from this probability distribution. Within this context an event is not a measurable set so we do not consider random variables viewed as measurable functions into a probability space characterized by a σ-algebra. Anyway, a parameter space is always provided with a metric structure that we introduce after studying the range of possibility. This metric structure is useful in order to obtain different quantitative measures that allow us of considering meaningful relationships between random quantities. When we study multivariate random quantities we introduce antisymmetric tensors satisfying simplification and compression reasons with respect to these random quantities into this metric structure.