Ludwig - Maximilians - Universität München

In this work we recall and discuss optimality criteria for decision making under uncertainty with respect to different assumptions concerning the structure of the information available. First, an overview of the basic concepts of classical decision theory is given. Here, particular emphasis is placed on explaining classical decision criteria (i.e. criteria for the case that the uncertainty can be characterized by using classical probabilistic models) from literature and discussing the assumptions underlying them (specifically Bernoulli-, Maximin and Hodges & Lehmann-criterion are discussed). Afterwards, in order to establish the mathematical basis necessary, a brief introduction to the theory of linear optimization is provided. Thereby, particular interest lies in recalling theoretical results concerning the resolvability of linear optimization problems. Subsequently, we demonstrate how linear optimization theory can be used to construct algorithms for determining optimal decisions (with respect to the classical criteria) in finite decision problems. Examples of concrete decision problems are given. Next, we give some (theoretical and practical) examples that support the idea of introducing imprecise probabilistic models to decision theory. Accordingly, two common generalizations of classical probability theory are explained: Credal sets and interval probability. Connections of the two concepts are shown up. Finally, we recall criteria for optimal decision making, if the uncertainty is characterized by a credal set or an interval probability respectively (specifically interval dominance, E-admissibility, maximality, Γ-Maximin-criterion, Γ-Maximax-criterion and a criterion combining the two latter). Again, for all criteria discussed, we illustrate and explain algorithms for determining optimal decisions.