Book in progress
In March 2018, I started working on a new textbook called "From Decision Theory to Game Theory: Reasoning about Decisions of Others".
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.
The book will be published by Cambridge University Press.
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.
Chapter 7 investigates games with unawareness, where a player may be unaware of some the choices that other players, or he himself, can make. Or a player may believe that others are unaware of some of the choices that he himself can make. Important is that a player cannot reason about choices that he is unaware of, and believes that others cannot reason about choices they are unaware of, and so on. This is called the unawareness principle, which plays a key role in this chapter. The choices that a player is aware of establishes his view of the game. We provide a definition of common belief in rationality for this setting, and propose a recursive elimination procedure that characterizes, for every view, precisely those choices the player can rationally make under common belief in rationality. We also explore the setting where the players' beliefs about the opponents' views are fixed. We finally show that in the framework of unawareness, imposing simple or symmetric belief hierarchies leads to trivial cases of unawareness, where all players are believed to hold the same view of the game.
Chapter 8 focuses on psychological games, where the preferences about your own choices may depend on the beliefs about the beliefs of others (higher-order beliefs). Such games may be useful for modelling psychological phenomena like surprise, meeting the other person's expectations, or disappointment aversion. We restrict to psychological games where player i's utility only depends on his second-order belief, that is, his belief about the opponent's belief about player i's choice, but not on higher-order beliefs. More precisely, we assume that the player's utility depends on his second-order expectation, which is a summary statistic of his second-order belief, and that his utility depends linearly on his second-order expectation. Such a psychological game induces a collection of one-person decision problems, where the states of player i's decision problem involves the (extreme) second-order expectations, and where player i's conditional preference relation has an expected utility representation. It is shown that common belief in rationality can be formalized in essentially the same way as for standard games. However, the recursive elimination procedure that characterizes those choices that are possible under common belief in rationality is fundamentally different from the procedures in the other chapters: It proceeds by recursively eliminating choices and second-order expectations, rather than eliminating choices and states.
Chapter 9 explores simple and symmetric belief hierarchies in psychological games. It is shown that common belief in rationality in combination with a simple belief hierarchy leads to psychological Nash equilibrium, and that common belief in rationality in combination with a symmetric belief hierarchy yields psychological correlated equilibrium. Several examples demonstrate that common belief in rationality with a simple, or symmetric, belief hierarchy may severely limit the ability to surprise the other person.
The online appendix offers for Chapter 2 an axiomatic characterization of those conditional preference relation that admit an expected utility representation. Moreover, it provides some economic applications for each of the Chapters 2 until 9.
These chapters and the online appendix can be downloaded below.
Acknowledgments and introduction
Part I: Decision Problems
2.1 Decision making under certainty
2.2 Decision making under uncertainty
2.3 Expected utility representation
2.4 Utility design procedure
2.5 Unique relative preference intensities
2.6 Strict dominance
2.7 Proofs
Solutions to in-chapter questions
Problems
Literature
Part II: Standard Static Games
Chapter 3: Common Belief in Rationality in Standard Games
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 in Standard Games
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 with Incomplete Information
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 with Incomplete Information
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
Part IV: Unawareness
Chapter 7: Common Belief in Rationality with Unawareness
7.1 Unawareness
7.2 Belief hierarchies, beliefs diagrams and types
7.3 Common belief in rationality
7.4 Recursive procedure
7.5 Bottom-up procedure
7.6 Fixed beliefs on views
7.7 Correct and symmetric beliefs
7.8 Proofs
Solutions to in-chapter questions
Problems
Literature
Part V: Psychological Games
Chapter 8: Common Belief in Rationality in Psychological Games
8.1 Example
8.2 Psychologogical games
8.3 Common belief in rationality
8.4 Recursive procedure
8.5 States-first procedure
8.6 When elimination of choices and states is enough
8.7 Proofs
Solutions to in-chapter questions
Problems
Literature
Chapter 9: Correct and Symmetric Beliefs in Psychological Games
9.1 Correct beliefs
9.2 Symmetric beliefs
9.3 Comparison of the concepts
9.4 Proofs
Solutions to in-chapter questions
Problems
Literature
Chapter 2: Decision Problems
2.8 Case of two choices
2.9 Case of preference reversals
2.10 General case
2.11 Economic applications
2.12 Proofs
Chapter 3: Common Belief in Rationality in Standard Games
3.7 Economic applications
Chapter 4: Correct and Symmetric Beliefs in Standard Games
4.6 Economic applications
Chapter 5: Common Belief in Rationality with Incomplete Information
5.7 Economic applications
Chapter 6: Correct and Symmetric Beliefs with Incomplete Information
6.6 Economic Applications
Chapter 7: Common Belief in Rationality with Unawareness
7.9 Economic Applications
Chapter 8: Common Belief in Rationality in Psychological Games
8.8 Economic Applications
Chapter 9: Correct and Symmetric Beliefs in Psychological Games
9.5 Economic Applications