Language Of Game Theory, The: Putting Epistemics Into The Mathematics Of Games

Putting Epistemics into the Mathematics of Games

Adam Brandenburger

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814513456
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This volume contains eight papers written by Adam Brandenburger and his co-authors over a period of 25 years. These papers are part of a program to reconstruct game theory in order to make how players reason about a game a central feature of the theory. The program — now called epistemic game theory — extends the classical definition of a game model to include not only the game matrix or game tree, but also a description of how the players reason about one another (including their reasoning about other players' reasoning). With this richer mathematical framework, it becomes possible to determine the implications of how players reason for how a game is played. Epistemic game theory includes traditional equilibrium-based theory as a special case, but allows for a wide range of non-equilibrium behavior.

Contents:

• An Impossibility Theorem on Beliefs in Games (Adam Brandenburger and H Jerome Keisler)
• Hierarchies of Beliefs and Common Knowledge (Adam Brandenburger and Eddie Dekel)
• Rationalizability and Correlated Equilibria (Adam Brandenburger and Eddie Dekel)
• Intrinsic Correlation in Games (Adam Brandenburger and Amanda Friedenberg)
• Epistemic Conditions for Nash Equilibrium (Robert Aumann and Adam Brandenburger)
• Lexicographic Probabilities and Choice Under Uncertainty (Lawrence Blume, Adam Brandenburger, and Eddie Dekel)
• Admissibility in Games (Adam Brandenburger, Amanda Friedenberg and H Jerome Keisler)
• Self-Admissible Sets (Adam Brandenburger and Amanda Friedenberg)

Readership: Graduate students and researchers in the fields of game theory, theoretical computer science, mathematical logic and social neuroscience.
Key Features:

• Focuses on epistemic game theory — an emerging approach to game theory
• Likely strong interest in these tools from other disciplines, includingtheoretical computer science, mathematical logic, and social neuroscience
• Prominent co-author team: Robert Aumann (Hebrew University, Nobel Laureate 2005); Lawrence Blume (Cornell University); Eddie Dekel (Northwestern University and Tel Aviv University); Amanda Freedeneurg (Arizona State University); H Jerome Keisler (University of Wisconsin Madison)

Chapter 1

An Impossibility Theorem on Beliefs in Games

Adam Brandenburger and H. Jerome Keisler

Originally published in Studia Logica, 84, 211–240.

Keywords: belief model; complete belief model; game; first order logic; modal logic; paradox.

Financial support: Harvard Business School, Stern School of Business, National Science Foundation, and Vilas Trust Fund.

Acknowledgments: We are indebted to Amanda Friedenberg and Gus Stuart for many valuable discussions bearing on this chapter. Samson Abramsky, Ken Arrow, Susan Athey, Bob Aumann, Joe Halpern, Christopher Harris, Aviad Heifetz, Jon Levin, Martin Meier, Mike Moldoveanu, Eric Pacuit, Rohit Parikh, Martin Rechenauer, Hannu Salonen, Dov Samet, Johan van Benthem, Daniel Yamins, and Noson Yanofsky provided important input. Our thanks, too, to participants in the XIII Convegno di Teoria dei Giochi ed Applicazioni (University of Bologna, June 1999), the Tenth International Conference on Game Theory (State University of New York at Stony Brook, July 1999), the 2004 Association for Symbolic Logic Annual Meeting (Carnegie Mellon University, May 2004), and seminars at Harvard University, New York University, and Northwestern University, and to referees.

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:

Ann believes that Bob assumes that
Ann believes that Bob’s assumption is wrong

This is formalized to show that any belief model of a certain kind must have a “hole.” An interpretation of the result is that if the analyst’s tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.

1. Introduction

In game theory, the notion of a player’s beliefs about the game — even a player’s beliefs about other players’ beliefs, and so on — arises naturally. Take the basic game-theoretic question: Are Ann and Bob rational, does each believe the other to be rational, and so on? To address this, we need to write down what Ann believes about Bob’s choice of strategy — to decide whether she chooses her strategy optimally given her beliefs (i.e., whether she is rational). We also have to write down what Ann believes Bob believes about her strategy choice — to decide whether Ann believes Bob chooses optimally given his beliefs (i.e., whether Ann believes Bob is rational). Etc. Beliefs about beliefs about. . . in games are basic.

In this chapter we ask: Doesn’t such talk of what Ann believes Bob believes about her, and so on, suggest that some kind of self-reference arises in games, similar to the well-known examples of self-reference in mathematical logic. If so, then is there some kind of impossibility result on beliefs in games, that exploits this self-reference?

There is such a result, a game-theoretic version of Russell’s paradox.¹ We give it first in words. By an assumption (or strongest belief) we mean a belief that implies all other beliefs. Consider the follow...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Foreword
• About the Author
• Acknowledgments
• Introduction
• Chapter 1. An Impossibility Theorem on Beliefs in Games
• Chapter 2. Hierarchies of Beliefs and Common Knowledge
• Chapter 3. Rationalizability and Correlated Equilibria
• Chapter 4. Intrinsic Correlation in Games
• Chapter 5. Epistemic Conditions for Nash Equilibrium
• Chapter 6. Lexicographic Probabilities and Choice Under Uncertainty
• Chapter 7. Admissibility in Games
• Chapter 8. Self-Admissible Sets
• Subject Index
• Author Index