Mikhalevich V. The issue of uncertainty in decision problems and the guaranteed result principle

The doctoral thesis is dedicated to substantiating the approach towards decision making under uncertainty, as well the development of it and the corresponding mathematical apparatus, basing on formal logic principles. The introduced approach implements the idea of optimality principle axiomatization, which lies in foundation of formalizing the notion of (decision) "rationality". And in case of reliance on the guaranteed result principle in statistical form, the uncertainty of the states of the world appears as "randomness to a wide extent". The notions of decision-making systems and their situations and two corresponding forms of schemas and models were introduced for the purpose of interpreting the formal statement of the choice problem. The interconnection established between these two forms allows considering the situation to be parametric without loss of generality. The decision problem and the problem of preference choice on decisions are defined within the decision-making system. The notions of preference choice rule and schematic uncertainty with respect to the preference choice problem in wide rule classes are introduced for arbitrary situation schema classes. The existence criterion and classification were developed for the indicated uncertainties. Mutually dual statistical forms of guaranteed and best result principles are defined for choice problems on profits, as well as losses. The results of Ivanenko-Labkovsky presenting the uncertainty issue for preference choice problems on losses based on guaranteed result principle in statistical form and defining the multi-prior SEU model, are strengthened. An analogy to the Anscombe-Aumann theorem for choice problems in Bayesian form on losses as well as profits is obtained. The duality principle for choice problems on losses and profits based on guaranteed and best result principles in statistical forms was established and proved. The notions of model and non-reducible model for decision-making systems were introduced for unambiguous definition of preferences on decisions. The solution of the uncertainty issue for choice problems in Bayesian form requiring a utility function preserving decision and consequence preferences extends to decision problems in generalized neo-Bayesian form allowing randomness to a wide extent for consequences, on the assumption of the utility function's linearity. This solution is based on the transition to multiple choice problems. As a result for multiple decision-making systems, the following models were obtained: non-reducible multi-prior SEU models for choice problems in generalized neo-Bayesian form, which axiomatize the guaranteed and best result principles in statistical form, correspondingly; non-reducible SEU, CEU models and a multi-prior SEU model, all introduced by behaviorist traditions and generalizing the corresponding models by Anscombe-Aumann, Schmeidler and Gilboa-Schmeidler. The indicated models have proof of their corresponding necessary and sufficient criterion replacement conditions.