Mathematical Decision Making

Since both, decision- and game theory vitally employ optimization at their core, this chapter will provide the basic ideas, concepts and modeling aspects of optimization. It is intended to provide the mathematical basics for the further chapters. The presentation is to the point of a simple, compact and self-contained description of: (i) what is decision- and game-theory about, (ii) how do the two areas differ, and (iii) how does the practical work with these models look like when we strive for solutions. Specifically, we discuss preference relations, real and stochastic ordering relations and optimization as the most general covering framework, including single- and multi-goal optimization, with applications in being decision theory and game theory. Numeric examples accompany each section and concept. The opening of the chapter will specifically set the notation for all upcoming (mathematical) descriptions, to be consistent throughout the entire presentation (and book).