This chapter serves as a building block for modeling and solving planning problems that involve more than one decision maker. The focus is on making a single decision in the presence of other decision makers that may interfere with the outcome. The planning problems in Chapters 10 to 12 will be viewed as a sequence of decision-making problems. The ideas presented in this chapter can be viewed as making a one-stage plan. With respect to Chapter 2, the present chapter reduces the number of stages down to one and then introduces more sophisticated ways to model a single stage. Upon returning to multiple stages in Chapter 10, it will quickly be seen that many algorithms from Chapter 2 extend nicely to incorporate the decision-theoretic concepts of this chapter.
Since there is no information to carry across stages, there will be no need for a state space. Instead of designing a plan for a robot, in this chapter we will refer to designing a strategy for a decision maker (DM). The planning problem reduces down to a decision-making problem. In later chapters, which describe sequential decision making, planning terminology will once again be used. It does not seem appropriate yet in this chapter because making a single decision appears too degenerate to be referred to as planning.
A consistent theme throughout Part III will be the interaction of multiple DMs. In addition to the primary DM, which has been referred to as the robot, there will be one or more other DMs that cannot be predicted or controlled by the robot. A special DM called nature will be used as a universal way to model uncertainties. Nature will usually be fictitious in the sense that it is not a true entity that makes intelligent, rational decisions for its own benefit. The introduction of nature merely serves as a convenient modeling tool to express many different forms of uncertainty. In some settings, however, the DMs may actually be intelligent opponents who make decisions out of their own self-interest. This leads to game theory, in which all decision makers (including the robot) can be called players.
Section 9.1 provides some basic review and perspective that will help in understanding and relating later concepts in the chapter. Section 9.2 covers making a single decision under uncertainty, which is typically referred to as decision theory. Sections 9.3 and 9.4 address game theory, in which two or more DMs make their decisions simultaneously and have conflicting interests. In zero-sum game theory, which is covered in Section 9.3, there are two DMs that have diametrically opposed interests. In nonzero-sum game theory, covered in Section 9.4, any number of DMs come together to form a noncooperative game, in which any degree of conflict or competition is allowable among them. Section 9.5 concludes the chapter by covering justifications and criticisms of the general models formulated in this chapter. It useful when trying to apply decision-theoretic models to planning problems in general.
This chapter was written without any strong dependencies on Part II. In fact, even the concepts from Chapter 2 are not needed because there are no stages or state spaces. Occasional references to Part II will be given, but these are not vital to the understanding. Most of the focus in this chapter is on discrete spaces.