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Intro; Preface; Contents; Symbols; Symbols for Extensive Form Game Theory; Part I Overview; 1 Introduction and Summary; 1.1 The Key Concept; 1.2 Historical Background; 1.3 Chapter Contents; 1.3.1 Chapter 2: Planes, Polyhedra, and Polytopes; 1.3.2 Chapter 3: Computing Fixed Points; 1.3.3 Chapter 4: Topologies on Sets; 1.3.4 Chapter 5: Topologies on Functions and Correspondences; 1.3.5 Chapter 6: Metric Space Theory; 1.3.6 Chapter 7: Essential Sets of Fixed Points; 1.3.7 Chapter 8: Retracts; 1.3.8 Chapter 9: Approximation; 1.3.9 Chapter 10: Manifolds; 1.3.10 Chapter 11: Sard's Theorem
1.3.11 Chapter 12: Degree Theory1.3.12 Chapter 13: The Fixed Point Index; 1.3.13 Chapter 14: Topological Consequences; 1.3.14 Chapter 15: Dynamical Systems; 1.3.15 Chapter 16: Extensive Form Games; 1.3.16 Chapter 17: Monotone Equilibria; Part II Combinatoric Geometry; 2 Planes, Polyhedra, and Polytopes; 2.1 Affine Subspaces; 2.2 Convex Sets and Cones; 2.3 Polyhedra; 2.4 Polytopes and Polyhedral Cones; 2.5 Polyhedral Complexes; 2.6 Simplicial Approximation; 2.7 Graphs; 3 Computing Fixed Points; 3.1 The Axiom of Choice, Subsequences, and Computation; 3.2 Sperner's Lemma; 3.3 The Scarf Algorithm
3.4 Primitive Sets3.5 The Lemke-Howson Algorithm; 3.6 Implementation and Degeneracy Resolution; 3.7 Using Games to Find Fixed Points; 3.8 Homotopy; 3.9 Remarks on Computation; Part III Topological Methods; 4 Topologies on Spaces of Sets; 4.1 Topological Terminology; 4.2 Spaces of Closed and Compact Sets; 4.3 Vietoris' Theorem; 4.4 Hausdorff Distance; 4.5 Basic Operations on Subsets; 4.5.1 Continuity of Union; 4.5.2 Continuity of Intersection; 4.5.3 Singletons; 4.5.4 Continuity of the Cartesian Product; 4.5.5 The Action of a Function; 4.5.6 The Union of the Elements
5 Topologies on Functions and Correspondences5.1 Upper and Lower Hemicontinuity; 5.2 The Strong Upper Topology; 5.3 The Weak Upper Topology; 5.4 The Homotopy Principle; 5.5 Continuous Functions; 6 Metric Space Theory; 6.1 Paracompactness; 6.2 Partitions of Unity; 6.3 Topological Vector Spaces; 6.4 Banach and Hilbert Spaces; 6.5 Embedding Theorems; 6.6 Dugundji's Theorem; 7 Essential Sets of Fixed Points; 7.1 The Fan-Glicksberg Theorem; 7.2 Convex Valued Correspondences; 7.3 Convex Combinations of Correspondences; 7.4 Kinoshita's Theorem; 7.5 Minimal Q-Robust Sets; 8 Retracts
8.1 Kinoshita's Example8.2 Retracts; 8.3 Euclidean Neighborhood Retracts; 8.4 Absolute Neighborhood Retracts; 8.5 Absolute Retracts; 8.6 Domination; 9 Approximation of Correspondences by Functions; 9.1 The Approximation Result; 9.2 Technical Lemmas; 9.3 Proofs of the Propositions; Part IV Smooth Methods; 10 Differentiable Manifolds; 10.1 Review of Multivariate Calculus; 10.2 Smooth Partitions of Unity; 10.3 Manifolds; 10.4 Smooth Maps; 10.5 Tangent Vectors and Derivatives; 10.6 Submanifolds; 10.7 Tubular Neighborhoods; 10.8 Manifolds with Boundary; 10.9 Classification of Compact 1-Manifolds
1.3.11 Chapter 12: Degree Theory1.3.12 Chapter 13: The Fixed Point Index; 1.3.13 Chapter 14: Topological Consequences; 1.3.14 Chapter 15: Dynamical Systems; 1.3.15 Chapter 16: Extensive Form Games; 1.3.16 Chapter 17: Monotone Equilibria; Part II Combinatoric Geometry; 2 Planes, Polyhedra, and Polytopes; 2.1 Affine Subspaces; 2.2 Convex Sets and Cones; 2.3 Polyhedra; 2.4 Polytopes and Polyhedral Cones; 2.5 Polyhedral Complexes; 2.6 Simplicial Approximation; 2.7 Graphs; 3 Computing Fixed Points; 3.1 The Axiom of Choice, Subsequences, and Computation; 3.2 Sperner's Lemma; 3.3 The Scarf Algorithm
3.4 Primitive Sets3.5 The Lemke-Howson Algorithm; 3.6 Implementation and Degeneracy Resolution; 3.7 Using Games to Find Fixed Points; 3.8 Homotopy; 3.9 Remarks on Computation; Part III Topological Methods; 4 Topologies on Spaces of Sets; 4.1 Topological Terminology; 4.2 Spaces of Closed and Compact Sets; 4.3 Vietoris' Theorem; 4.4 Hausdorff Distance; 4.5 Basic Operations on Subsets; 4.5.1 Continuity of Union; 4.5.2 Continuity of Intersection; 4.5.3 Singletons; 4.5.4 Continuity of the Cartesian Product; 4.5.5 The Action of a Function; 4.5.6 The Union of the Elements
5 Topologies on Functions and Correspondences5.1 Upper and Lower Hemicontinuity; 5.2 The Strong Upper Topology; 5.3 The Weak Upper Topology; 5.4 The Homotopy Principle; 5.5 Continuous Functions; 6 Metric Space Theory; 6.1 Paracompactness; 6.2 Partitions of Unity; 6.3 Topological Vector Spaces; 6.4 Banach and Hilbert Spaces; 6.5 Embedding Theorems; 6.6 Dugundji's Theorem; 7 Essential Sets of Fixed Points; 7.1 The Fan-Glicksberg Theorem; 7.2 Convex Valued Correspondences; 7.3 Convex Combinations of Correspondences; 7.4 Kinoshita's Theorem; 7.5 Minimal Q-Robust Sets; 8 Retracts
8.1 Kinoshita's Example8.2 Retracts; 8.3 Euclidean Neighborhood Retracts; 8.4 Absolute Neighborhood Retracts; 8.5 Absolute Retracts; 8.6 Domination; 9 Approximation of Correspondences by Functions; 9.1 The Approximation Result; 9.2 Technical Lemmas; 9.3 Proofs of the Propositions; Part IV Smooth Methods; 10 Differentiable Manifolds; 10.1 Review of Multivariate Calculus; 10.2 Smooth Partitions of Unity; 10.3 Manifolds; 10.4 Smooth Maps; 10.5 Tangent Vectors and Derivatives; 10.6 Submanifolds; 10.7 Tubular Neighborhoods; 10.8 Manifolds with Boundary; 10.9 Classification of Compact 1-Manifolds