Book in progress

Epistemic Game Theory II: Non-Standard Games

In March 2018, I started working on a new textbook called "Epistemic Game Theory II: Non-Standard Games".

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 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 an 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 the book we will start with standard games as a reference point. Chapter 2 of the book, on common belief in rationality in standard games, and Chapter 3, on Nash equilibrium and correlated equilibrium in standard games, are now ready. The chapters can be downloaded below.

Part 1: Standard Static Games

Chapter 2: Common Belief in Rationality

2.1 Idea of common belief in rationality

2.2 Games and decision problems

2.3 Belief hierarchies, beliefs diagrams and types

2.4 Definition of common belief in rationality

2.5 Recursive procedure

2.6 Why the procedure works

2.7 Order of elimination

2.8 Proofs

Solutions to in-chapter questions

Problems

Literature

Chapter 3: Correct and Symmetric Beliefs

3.1 Correct beliefs

3.2 Symmetric beliefs

3.3 One theory per choice

3.4 Comparison of the concepts

3.5. Proof

Solutions to in-chapter questions

Problems

Literature