Lecture 6 - Bayesian Games, Repeated Games, and Cooperative Games

03.25

Instructor: Yaodong Yang

Topics Covered

1. Bayesian Game

   • 1.1 Example: Chicken Game
     • 1.1.1 Incomplete Information Game
     • 1.1.2 Harsanyi Transformation
     • 1.1.3 Players’ Decision-Making
     • 1.1.4 “Natural” Selection
     • 1.1.5 Definition
     • 1.1.6 Strategy
   • 1.2 Bayesian Nash Equilibrium
     • 1.2.1 Harsanyi Transformation: Incomplete → Imperfect Information
     • 1.2.2 Conditional Payoff
     • 1.2.3 Pure Strategy Bayesian Nash Equilibrium: Existence & Solution
     • 1.2.4 Simplified Solution for Pure Strategy Bayesian Nash Equilibrium

2. Repeated Game

   • 2.1 Finite Repeated Games
     • 2.1.1 Example: The Prisoner’s Dilemma
     • 2.1.2 Twice Repeated Game
     • 2.1.3 Multi-Stage Repeated Game
   • 2.2 Definition
   • 2.3 Characteristics of Finite Repeated Games
     • 2.3.1 Identical Stage Game Structure
     • 2.3.2 Consensus & Subgames
     • 2.3.3 Discount Factor
   • 2.4 Unique Nash Equilibrium
   • 2.5 Multiple Nash Equilibria
     • 2.5.1 Trigger Strategy: Not a Nash Equilibrium but Possibly Pareto Optimal
     • 2.5.2 Factors Affecting Equilibrium: Number of Repetitions & Information Completeness
     • 2.5.3 Two-Stage Repeated Game with Multiple Nash Equilibria
   • 2.6 Infinite Repeated Games
     • 2.6.1 Same as Initial Stage Game
     • 2.6.2 Discounted Payoff & Average Payoff
     • 2.6.3 Short-Term vs. Long-Term Considerations
     • 2.6.4 Construction of Trigger Strategy
     • 2.6.5 Solution for Discount Factors
   • 2.7 Feasible Payoff and Average Payoff
   • 2.8 Folk Theorem and Its Proof
   • 2.9 Feasible Payoff Set: Anything Can Be an Equilibrium!
   • 2.10 Further Discussions
     • 2.10.1 Impact of Discount Factor
     • 2.10.2 Abreu’s Harshest Credible Punishment
   • 2.11 Experimental Study: Repeated Prisoner’s Dilemma
     • 2.11.1 Grim Trigger Strategy
     • 2.11.2 Tit-for-Tat Strategy
     • 2.11.3 Analysis and Experiments

3. Cooperative Game

   • 3.1 Basic Concepts and Characteristics
     • 3.1.1 Goal: Achieving Optimal Solutions
     • 3.1.2 Collective Rationality: Efficiency and Fairness
     • 3.1.3 Cooperative Games vs. Non-Cooperative Games
     • 3.1.4 Example: Factory Cooperation Game
   • 3.2 Coalition Games
     • 3.2.1 Coalition and Grand Coalition
     • 3.2.2 Characteristic Function: Superadditivity and Monotonicity
     • 3.2.3 Constant-Sum Games and Essential Games
     • 3.2.4 Example: Garbage Disposal Game
   • 3.3 The Kernel Solution
     • 3.3.1 Solution Concept
     • 3.3.2 Two Basic Principles
     • 3.3.3 Classification of Cooperative Games
     • 3.3.4 Superior Relation (confirm if you intended “Superrelation”)
     • 3.3.5 Kernel Solution: Social Stability
     • 3.3.6 Simple Games
   • 3.4 Shapley Value
     • 3.4.1 Concept: Fair Allocation
     • 3.4.2 Carrier Concept
     • 3.4.3 Shapley Value and Shapley Index
     • 3.4.4 Existence and Uniqueness
     • 3.4.5 Example: Factory Cooperation Game
     • 3.4.6 Kernel Solution vs. Shapley Value