This chapter focuses on games with unawareness, where the players may be unaware of some of the choices that others can make. The player’s view specifies the choices in the game that he is aware of. The chapter starts by explaining how a game with unawareness can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices and views can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality. It is shown that the choices which are possible under common belief in rationality can be characterized by iterated strict dominance for unawareness. The chapter finally turns to the scenario of fixed beliefs on views, where the players hold some pre-specified beliefs about the opponents’ views.

This chapter investigates one-person decision problems under uncertainty. The main building block is that of a conditional preference relation: a mapping that assigns to every belief about the states a preference relation over the decision maker’s choices. Under certain conditions, such a conditional preference relation admits an expected utility representation, which allows us to summarize the conditional preference relation by a finite utility matrix. Throughout the book it is assumed that the conditional preference relation indeed has an expected utility representation.

This chapter focuses on standard games. It starts by explaining how a standard game can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality – the central line of reasoning which states that a player believes that others choose rationally, believes that others believe that others choose rationally, and so on. It is shown that the choices which are possible under common belief in rationality can be characterized by the iterated elimination of strictly dominated choices.

This chapter focuses on psychological games, where the players’ preferences may directly depend on higher-order beliefs. The chapter starts by explaining how a psychological game can be viewed as a collection of one-person decision problems. By using the same epistemic model as for standard games, it provides a formal definition of common belief in rationality. It is shown that the choices which are possible under common belief in rationality can be characterized by the iterated elimination of choices and second-order expectations. The chapter finally demonstrates when the easier procedure of iterated elimination of choices and states is sufficient for common belief in rationality.

This chapter starts by introducing the notion of a simple belief hierarchy, and shows that a simple belief hierarchy in combination with common belief in rationality leads to Nash equilibrium. It then turns to the weaker notion of symmetric belief hierarchies and shows, in a similar fashion, that a symmetric belief hierarchy in combination with common belief in rationality leads to correlated equilibrium. It finally investigates the one theory per choice condition, and demonstrates how it leads to canonical correlated equilibrium when combined with common belief in rationality and a symmetric belief hierarchy.

This chapter starts by introducing the notion of a simple belief hierarchy, and shows that a simple belief hierarchy in combination with common belief in rationality leads to generalized Nash equilibrium. It then turns to the weaker notion of symmetric belief hierarchies and shows, in a similar fashion, that a symmetric belief hierarchy in combination with common belief in rationality leads to Bayesian equilibrium. It subsequently investigates the one theory per choice-utility pair condition, and demonstrates how it leads to canonical Bayesian equilibrium when combined with common belief in rationality and a symmetric belief hierarchy. The chapter finally turns to the scenario of fixed beliefs on utilities, where the players hold some pre-specified beliefs about the opponents’ utility functions.

This chapter focuses on games with incomplete information, where the players may be uncertain about the utility functions of the other players. It starts by explaining how a game with incomplete information can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices and utility functions can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality. It is shown that the choices which are possible under common belief in rationality can be characterized by the generalized iterated strict dominance procedure. The chapter finally turns to the scenario of fixed beliefs on utilities, where the players hold some pre-specified beliefs about the opponents’ utility functions.