Title Hardness of Approximation Between P and NP
Author Aviad Rubinstein
ISBN 9781947487208
List price USD 89.95
Price outside India Available on Request
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Binding Paperback
No of pages 322
Book size 191 X 235 mm
Publishing year 2019
Original publisher Morgan & Claypool Publishers (Eurospan Group)
Published in India by .
Exclusive distributors Viva Books Private Limited
Sales territory India, Sri Lanka, Bangladesh, Pakistan, Nepal, .
Status New Arrival
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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.






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



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 l8• Euclidean BROUWER



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



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



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



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.


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