Book Epistemic Game Theory II

Book in progress


In March 2018, I started working on a new textbook called "Epistemic Game Theory II: Games as Decision Problems".

It will be a follow-up to my book called "Epistemic Game Theory: Reasoning and Choice". The purpose of the new book will be to explore non-standard games, such as games with incomplete information, games with unawareness and psychological games, in a unified way from an decision-theoretic and epistemic perspective.


In games with incomplete information the players may have uncertainty about the opponents' utility functions. In games with unawareness the players may be unaware of some choices for the opponents, or even some of the choices for themselves. In psychological games, the utility of a player may not only depend on his choice and and his first-order belief about the opponents' choices, but also on what he believes about the opponents' beliefs.


Despite the difference between these various classes of games, this book will show that these games can be explored in a unified way from a decision-theoretic and epistemic perspective. For each of these game classes, we will investigate the central concept of common belief in rationality and an associated recursive procedure, together with some version of Nash equilibrium and correlated equilibrium.


The version of Nash equilibrium will always be characterized by the notion of a simple belief hierarchy, reflecting the idea that a player believes that his opponents are correct about his beliefs. The idea of a simple belief hierarchy, in combination with common belief in rationality, will lead to a Nash equilibrium in a standard game, to a generalized Nash equilibrium in games with incomplete information, and to a psychological Nash equilibrium in psychological games.


The version of correlated equilibrium will always be characterized by the new notion of a symmetric belief hierarchy. The idea of a symmetric belief hierarchy, in combination with common belief in rationality, will give rise to a correlated equilibrium in standard games, to a Bayesian equilibrium in games with incomplete information, and to a psychological correlated equilibrium in psychological games.


It will be shown that in games with unawareness, symmetric and simple belief hierarchies will lead to trivial situations of unawareness, and will therefore not be explored separately for this class of games. 


In Chapter 2 of the book we start by studying one-person decision problems under uncertainty. 

The primitive object is that of a conditional preference relation, which assigns to every probabilistic belief over states a preference relation over the available choices. In the chapter we impose axioms on a conditional preference relation which are both necessary and sufficient for it having an expected utility representation. We do so for three different scenarios: The case of the two choices, the case where no choice is weakly dominated by another choice, and the general case. For every scenario we also show how to compute an expected utility representation if the axioms are satisfied.


Chapter 3 investigates the concept of common belief in rationality in standard static games. We show how to formalize this notion, and how it can be characterized by means of a recursive elimination procedure. 


Chapter 4 is about the ideas of a simple belief hierarchy and a symmetric belief hierarchy. We show how common belief in rationality, together with a simple belief hierarchy, leads to the concept of Nash equilibrium. Similarly, common belief in rationality in combination with a symmetric belief hierarchy yields correlated equilibrium.


Chapter 5 investigates games with incomplete information, where players may be uncertain about the precise conditional preference relations, or utility functions, that the opponents have. For this setting, it first gives a formal definition of common belief in rationality, and then explores how the resulting choices can be characterized by a recursive elimination procedure. Towards the end of the chapter, it concentrates on fixed beliefs on the opponents' utility functions. 


Chapter 6 applies the ideas of a simple belief hierarchy and a symmetric belief hierarchy to games with incomplete information. It is shown that common belief in rationality together with a simple belief hierarchy leads to a new concept called generalized Nash equilibrium. Moreover, if we combine common belief in rationality with a symmetric belief hierarchy, we obtain the concept of Bayesian equilibrium. As such, generalized Nash equilibrium is the incomplete information counterpart to Nash equilibrium, whereas Bayesian equilibrium is the incomplete information counterpart to correlated equilibrium. At the end, we look at scenarios where there are fixed beliefs on the players' utility functions. 


These five chapters can be downloaded below.


Part I: Decision Problems


Chapter 2: Decision Problems

2.1 Decision making under certainty

2.2 Decision making under uncertainty

2.3 Expected utility representation

2.4 Case of two choices

2.5 Case of no weak dominance

2.6 General case

2.7 Strict dominance

2.8 Proofs

Solutions to in-chapter questions

Problems

Literature


Part II: Standard Static Games


Chapter 3: Common Belief in Rationality

3.1 Games as decision problems

3.2 Belief hierarchies, beliefs diagrams and types

3.3 Common belief in rationality

3.4 Recursive procedure

3.5 Order of elimination

3.6 Proofs

Solutions to in-chapter questions

Problems

Literature


Chapter 4: Correct and Symmetric Beliefs

4.1 Correct beliefs

4.2 Symmetric beliefs

4.3 One theory per choice

4.4 Comparison of the concepts

4.5. Proofs

Solutions to in-chapter questions

Problems

Literature


Part III: Incomplete Information


Chapter 5: Common Belief in Rationality

5.1 Incomplete information

5.2 Belief hierarchies, beliefs diagrams, and types

5.3 Common belief in rationality

5.4 Recursive procedure

5.5 Fixed beliefs on utilities

5.6 Proofs

Solutions to in-chapter questions

Problems

Literature


Chapter 6: Correct and symmetric beliefs

6.1 Correct beliefs

6.2 Symmetric beliefs

6.3 Fixed beliefs on utilities

6.4 Comparison of the concepts

6.5 Proofs

Solutions to in-chapter questions

Problems

Literature


Bibliography