Subjective Probability and Expected Utility, A Stochastic Approximation Evaluation

The topic of this article is stochastic algorithms for evaluation of the utility and subjective probability based on the decision maker’s preferences. The main direction of the presentation is toward development of mathematically grounded algorithms for subjective probability and expected utility evaluation as a function of both the probability and the rank of the alternative. The stochastic assessment is based on mathematically formulated axiomatic principles and stochastic procedures and on the utility theory without additivity. The uncertainty of the human preferences is eliminated as is typical for the stochastic programming. Numerical presentations are shown and discussed. Indian Journ l of Applied Re earch Website: www.theglobaljournals.com (ISSN 2249-555X) INTRODUCTION The representation of complex systems including human decisions as objective function needs mathematical tools for evaluation of qualitative human knowledge. In decision making theory the primitive are preferences relations as description of people's strategies, guided both by internal expectations about their own capabilities of getting results, and by external feedback of this result (Keeney & Raiffa, 1993). Such modeling addresses theory of measurement (scaling), utility theory and Bayesian approach in decision making. The Bayesian statistical technique in decision making is applicable when the information and uncertainty in respect of problems, hypothesis and parameters can be expressed by probability distribution and functional representation of human preferences (Griffiths & Tenenbaum, 2006). Such an approach needs careful analysis of the terms measurement, formalization and admissible mathematical operations in the modeling. This is a fundamental level that requires the use of basic mathematical terms like sets, relations and operations over them, and their gradual elaboration to more complex and specific terms like value and utility functions, operators on mathematically structured sets and harmonization of these descriptions with set of axioms. In this aspect we enter the theory of measurements and the expected utility theory (Fishburn, 1970). The evaluation of qualitative human knowledge and the mathematical inclusion of the subjective probabilities and utility posed many difficulties and needs a special attention. Generally the human notions and preferences have qualitative or verbal expression. The wisely merge of the qualitative and verbal expression as human preferences and quantitative mathematical description causes many efforts. The violations of transitivity of the preferences lead to declinations in utility and subjective probability assessment (Cohen & al., 1988; Kahneman & Tversky, 1979; Fishburn, 1988; Machina, 2009). Such declinations explain the DM behavior observed in the Allais Paradox (Allais, 1953). A long discussion for the role of the mathematic and the Bayesian theory in the human decision making reality has been started yet. New extensions of axiomatic bases of the developed mathematical theories are considered for further wide developments of von Neumann’s theory. Fruitful directions of researches are development of a nonadditive subjective utility theory. The mathematical results of Schmeidler in respect of subjective probability and utility description make a great impression on this development (Shmeidler, 1989). The paper suggests a reasonable wellfounded mathematical approach and methods for subjective probability and utility evaluation based on the von Neumann’s utility theory and the Kahneman’s and Schmeidler’s findings. We propose and discuss a stochastic programming for subjective probability and utility polynomial evaluation as machine learning based on the human preferences. Numerical presentations are shown and discussed. MATHEMATICAL FORMULATIONS AND BACKGROUND The difficulties that come from the mathematical approach are due to the probability and subjective uncertainty of the DM expression and the cardinal character of the expressed human’s preferences. The mathematical description is the following. Let X be the set of alternatives (X⊆R). From practical point of view the empirical system of human preferences relations is a algebraic system with relations SR (X,(≈),()), where (≈) can be considered as the relation “indifferent or equivalent”, and () is the relation “prefer”. We look for equivalency of the empirical system with the numbered system of relations SR (R-real numbers, (=), (>)). The “indifference” relation (≈) is based on () and is defined by ((x≈y) ¬((xy)∨(xy))). We introduce a set S, which elements are named state of nature, following Schmaidler’s exposition (Shmeidler, 1989). Let Ω be algebra of subset of S. Denote by Do the set of all measurable finite step valued functions from S to P and denote by Dc the constant functions in Do. Let D be a convex subset of P which includes Dc, (Dc ⊆ Do ⊆ D). In the neo-Bayesian nomenclature elements of X are deterministic finite outcome (alternatives), elements of P are random outcomes or lotteries connected with the objective probabilities, and elements of D are acts connected with the uncertainty of human operations described with subjective probabilities. Elements of S