The Mathematics of Nash Equilibrium is the tentative title of a project that is currently envisaged as consisting of two books. The first will deal with advanced issues in the theory of fixed points as they relate to the theory of refinements of Nash equilibrium and evolutionary game theory. The second considers Nash equilibrium from the point of view of semi-algebraic geometry, and presents recent results related to the computational complexity of the problem of computing all Nash equilibrium.

The project has not advanced to the point where the manuscripts are ready for wide circulation. The purpose of this page is to allow the posting of certain chapters for teaching purposes. Please bear in mind that the material presented here is in a tentative and unpolished condition.

The Mathematics of Nash Equilibrium I: Fixed Point Theory

• Taking a historical approach, the development of the theory of refinements of Nash equilibrium is intertwined with an exposition of advanced material from the theory of fixed points.
Chapter 7: Consistent Conditional Systems (postscript,pdf)

Chapter 8: Response Sets (postscript,pdf)

Chapter 9: Retracts (postscript,pdf)

Chapter 10: Approximation (postscript,pdf)

Chapter 12: Manifolds and Sard's Theorem (postscript,pdf)

The Mathematics of Nash Equilibrium II: Semi-algebraic Geometry and Computation

• Basic results from semi-algebraic geometry are developed and applied to the set of Nash equilibria. Emphasis is placed on algorithms, and results describing the complexity of the set of Nash equilibria.

A Treatise on the Theory of Fixed Points

Table of Contents and Chapter 1 (postscript,pdf)

Last updated on Oct 2, 2003.