Competitive Markov Decision Processes

1 Introduction.- 1.0 Background.- 1.1 Raison d'Etre and Limitations.- 1.2 A Menu of Courses and Prerequisites.- 1.3 For the Cognoscenti.- 1.4 Style and Nomenclature.- I Mathematical Programming Perspective.- 2 Markov Decision Processes: The Noncompetitive Case.- 2.0 Introduction.- 2.1 The Summable Markov Decision Processes.- 2.2 The Finite Horizon Markov Decision Process.- 2.3 Linear Programming and the Summable Markov Decision Models.- 2.4 The Irreducible Limiting Average Process.- 2.5 Application: The Hamiltonian Cycle Problem.- 2.6 Behavior and Markov Strategies.- 2.7 Policy Improvement and Newton's Method in Summable MDPs.- 2.8 Connection Between the Discounted and the Limiting Average Models.- 2.9 Linear Programming and the Multichain Limiting Average Process.- 2.10 Bibliographic Notes.- 2.11 Problems.- 3 Stochastic Games via Mathematical Programming.- 3.0 Introduction.- 3.1 The Discounted Stochastic Games.- 3.2 Linear Programming and the Discounted Stochastic Games.- 3.3 Modified Newton's Method and the Discounted Stochastic Games.- 3.4 Limiting Average Stochastic Games: The Issues.- 3.5 Zero-Sum Single-Controller Limiting Average Game.- 3.6 Application: The Travelling Inspector Model.- 3.7 Nonlinear Programming and Zero-Sum Stochastic Games.- 3.8 Nonlinear Programming and General-Sum Stochastic Games.- 3.9 Shapley's Theorem via Mathematical Programming.- 3.10 Bibliographic Notes.- 3.11 Problems.- II Existence, Structure and Applications.- 4 Summable Stochastic Games.- 4.0 Introduction.- 4.1 The Stochastic Game Model.- 4.2 Transient Stochastic Games.- 4.2.1 Stationary Strategies.- 4.2.2 Extension to Nonstationary Strategies.- 4.3 Discounted Stochastic Games.- 4.3.1 Introduction.- 4.3.2 Solutions of Discounted Stochastic Games.- 4.3.3 Structural Properties.- 4.3.4 The Limit Discount Equation.- 4.4 Positive Stochastic Games.- 4.5 Total Reward Stochastic Games.- 4.6 Nonzero-Sum Discounted Stochastic Games.- 4.6.1 Existence of Equilibrium Points.- 4.6.2 A Nonlinear Compementarity Problem.- 4.6.3 Perfect Equilibrium Points.- 4.7 Bibliographic Notes.- 4.8 Problems.- 5 Average Reward Stochastic Games.- 5.0 Introduction.- 5.1 Irreducible Stochastic Games.- 5.2 Existence of the Value.- 5.3 Stationary Strategies.- 5.4 Equilibrium Points.- 5.5 Bibliographic Notes.- 5.6 Problems.- 6 Applications and Special Classes of Stochastic Games.- 6.0 Introduction.- 6.1 Economic Competition and Stochastic Games.- 6.2 Inspection Problems and Single-Control Games.- 6.3 The Presidency Game and Switching-Control Games.- 6.4 Fishery Games and AR-AT Games.- 6.5 Applications of SER-SIT Games.- 6.6 Advertisement Models and Myopic Strategies.- 6.7 Spend and Save Games and the Weighted Reward Criterion.- 6.8 Bibliographic Notes.- 6.9 Problems.- Appendix G Matrix and Bimatrix Games and Mathematical Programming.- G.1 Introduction.- G.2 Matrix Game.- G.3 Linear Programming.- G.4 Bimatrix Games.- G.5 Mangasarian-Stone Algorithm for Bimatrix Games.- G.6 Bibliographic Notes.- Appendix H A Theorem of Hardy and Littlewood.- H.1 Introduction.- H.2 Preliminaries, Results and Examples.- H.3 Proof of the Hardy-Littlewood Theorem.- Appendix M Markov Chains.- M.1 Introduction.- M.2 Stochastic Matrix.- M.3 Invariant Distribution.- M.4 Limit Discounting.- M.5 The Fundamental Matrix.- M.6 Bibliographic Notes.- Appendix P Complex Varieties and the Limit Discount Equation.- P.1 Background.- P.2 Limit Discount Equation as a Set of Simultaneous Polynomials.- P.3 Algebraic and Analytic Varieties.- P.4 Solution of the Limit Discount Equation via Analytic Varieties.- References.