Game Theory

1. Mathematical Tools in Game Theory
   1. Linear Programming
      1. Formulation of Game Theory Problems as Linear Programs
         1. Objective Functions
         2. Constraints
      2. Duality in Linear Programming
         1. Primal-Dual Relationships
         2. Applications in Game Theory
      3. Applications in Game Strategy Optimization
         1. Simplex Method
         2. Interior Point Methods
   2. Fixed Point Theorems
      1. Brouwer's Fixed Point Theorem
         1. Background and Statement of the Theorem
         2. Applications in Proving Equilibrium Existence
      2. Kakutani’s Fixed Point Theorem
         1. Generalization for Set-Valued Functions
         2. Use in Proving Nash Equilibria in Games
   3. Utility Theory
      1. Representation of Preferences
         1. Utility Functions
         2. Ordinal vs. Cardinal Utility
      2. Expected Utility Theory
         1. Risk and Uncertainty
         2. Von Neumann-Morgenstern Utility Theorem
      3. Utility in Cooperative and Non-Cooperative Games
         1. Transferable Utility
         2. Non-transferable Utility
   4. Game Trees and Graphs
      1. Representation of Sequential Games
         1. Nodes and Edges as Decision Points and Actions
         2. Terminal Nodes and Payoffs
      2. Solving Games with Backward Induction
         1. Subgame Perfect Equilibrium Analysis
      3. Graph Theoretical Approaches
         1. Network Flows and Connectivity
         2. Algorithmic Solutions for Game Graphs
   5. Probability and Statistics in Game Theory
      1. Role of Randomness in Mixed Strategies
         1. Probability Distributions Over Strategies
      2. Statistical Analysis of Game Outcomes
         1. Experimental Games and Analysis
         2. Data-Driven Strategy Refinement
   6. Convex Analysis
      1. Convex Sets and Functions
         1. Properties and Representation
      2. Applications in Strategy Space Analysis
         1. Convex Hulls of Strategy Sets
      3. Use in Optimization Problems within Games
         1. Convex Optimization Methods
   7. Differential Equations and Dynamics
      1. Modeling Evolutionary Game Dynamics
         1. Differential Equations for Population Games
      2. Stability Analysis in Dynamic Games
         1. Stability Theorems
         2. Phase Space Analysis
   8. Computational Methods
      1. Algorithms for Equilibrium Computation
         1. Polynomial Time Algorithms
         2. Approximation Algorithms for Complex Games
      2. Computational Complexity of Game-Theoretic Problems
         1. Classifications (P, NP, etc.)
         2. Hardness Results for Computing Solutions