Towards the full information chain theory: question difficulty

Abstract: A general problem of optimal information acquisition for its use in decision making problems is considered. This motivates the need for developing quantitative measures of information sources' capabilities for supplying accurate information depending on the particular content of the latter. In this article, the notion of a real valued difficulty functional for questions identified with partitions of problem parameter space is introduced and the overall form of this functional is derived that satisfies a particular system of reasonable postulates. It is found that, in an isotropic case, the resulting difficulty functional depends on a single scalar function on the parameter space that can be interpreted -- using parallels with classical thermodynamics -- as a temperature-like quantity, with the question difficulty itself playing the role of thermal energy. Quantitative relationships between difficulty functionals of different questions are also explored.