**Description:**

Nash
equilibrium is the central solution concept in Game Theory. Since Nash’s
original paper in 1951, it has found countless applications in modeling
strategic behavior of traders in markets, (human) drivers and (electronic)
routers in congested networks, nations in nuclear disarmament negotiations, and
more. A decade ago, the relevance of this solution concept was called into
question by computer scientists, who proved (under appropriate complexity
assumptions) that computing a Nash equilibrium is an intractable problem. And
if centralized, specially designed algorithms cannot find Nash equilibria, why
should we expect distributed, selfish agents to converge to one? The remaining
hope was that at least approximate Nash equilibria can be efficiently computed.

Understanding
whether there is an efficient algorithm for approximate Nash equilibrium has
been the central open problem in this field for the past decade. In this book,
we provide strong evidence that even finding an approximate Nash equilibrium is
intractable. We prove several intractability theorems for different settings
(two-player games and many-player games) and models (computational complexity,
query complexity, and communication complexity). In particular, our main result
is that under a plausible and natural complexity assumption (“Exponential Time
Hypothesis for PPAD”), there is no polynomial-time algorithm for finding an
approximate Nash equilibrium in two-player games.

The
problem of approximate Nash equilibrium in a two-player game poses a unique
technical challenge: it is a member of the class PPAD, which captures the
complexity of several fundamental total problems, i.e., problems that always
have a solution; and it also admits a quasipolynomial time algorithm. Either
property alone is believed to place this problem far below NP-hard problems in
the complexity hierarchy; having both simultaneously places it just above P, at
what can be called the frontier of intractability. Indeed, the tools we develop
in this book to advance on this frontier are useful for proving hardness of
approximation of several other important problems whose complexity lies between
P and NP: Brouwer’s fixed point, market equilibrium, CourseMatch (A-CEEI),
densest k-subgraph, community detection, VC dimension and Littlestone
dimension, and signaling in zero-sum games.

Contents:

Preface

Acknowledgments

__PART I: OVERVIEW__

**Chapter 1. The
Frontier of Intractability** • PPAD: Finding a Needle You *Know*
Is in the Haystack • Quasi-Polynomial Time and the Birthday Paradox •
Approximate Nash Equilibrium

**Chapter 2.
Preliminaries** • Notation • Nash Equilibrium and
Relaxations • PPAD and END-OF-A-LINE • Exponential Time Hypotheses • PCP
Theorems • Learning Theory • Information Theory • Useful Lemmata

__PART
II: COMMUNICATION COMPLEXITY__

**Chapter 3.
Communication Complexity of Approximate Nash Equilibrium** •
Our Results • Uncoupled Dynamics • Techniques • Additional Related Literature •
Proof Overview • Proofs • An Open Problem: Correlated Equilibria in 2-Player
Games

**Chapter 4. Brouwer’s
Fixed Point** • BROUWER with *l*8• Euclidean BROUWER

__PART
III: PPAD__

**Chapter 5.
PPAD-Hardness of Approximation**

**Chapter 6. The
Generalized Circuit Problem** • Our Results • Proof Overview • From
Brouwer to ?-GCIRCUIT • GCIRCUIT with Fan-out 2

**Chapter 7. Many-Player
Games**
• Related Works: Tractable Special Cases • Graphical, Polymatrix Games •
Succinct Games

**Chapter 8. Bayesian
Nash Equilibrium**

**Chapter 9. Market
Equilibrium** • Why Are Non-Monotone Markets Hard? • High-Level
Structure of the Proof • Adaptations for Constant Factor Inapproximability •
Non-Monotone Markets: Proof of Inapproximability

**Chapter 10.
CourseMatch** • The Course Allocation Problem • A-CEEI Is PPAD-Hard •
A-CEEI ? PPAD

__PART
IV: QUASI-POLYNOMIAL TIME__

**Chapter 11. Birthday
Repetition** • Warm-Up: Best *?*-Nash

**Chapter 12. Densest ***k*-Subgraph •
Construction (and Completeness) • Soundness

**Chapter 13. Community
Detection **• Related Works • Overview of Proofs • Hardness of
Counting Communities • Hardness of Detecting Communities

**Chapter 14. VC and
Littlestone’s Dimensions** • Discussion • Techniques • Related
Work • Inapproximability of the VC Dimension • Inapproximability of
Littlestone’s Dimension • Quasi-polynomial Algorithm for Littlestone’s Dimension

**Chapter 15. Signaling** •
Techniques • Near-Optimal Signaling Is Hard

__PART
V: APPROXIMATE NASH EQUILIBRIUM__

**Chapter 16. 2-Player
Approximate Nash Equilibrium** • Additional Related Work • Technical
Overview • END-OF-A-LINE with Local Computation • Holographic Proof •
Polymatrix WeakNash • From Polymatrix to Bimatrix

References

Index

Author Biography

About the Author:

**Aviad Rubinstein** is an Assistant Professor
of Computer Science at Stanford University. Before coming to Stanford, he
received his Ph.D. from the University of California, Berkeley, and spent one
year as a Rabin Postdoctoral Fellow at Harvard University.

Target Audience:

This
book is helpful for people interested in nash equilibrium, information science
& technology, computer science and game theory.