Week 1: Basics of Game Theory

1. Preferences and Rationality
   • Completeness: Every pair of choices can be compared (e.g., either "tea is preferred to coffee" or vice versa).
   • Reflexivity: Any good is as good as itself (e.g., "tea is as good as tea").
   • Transitivity: If a person prefers A over B and B over C, then they must prefer A over C. Violations lead to irrational preferences (e.g., preferring tea to coffee, coffee to lassi, but lassi to tea violates transitivity).
2. Nash Equilibrium
   • A situation where no player can benefit by changing their strategy unilaterally.
   • Example: Rock-Paper-Scissors has no pure strategy Nash equilibrium but has a mixed strategy Nash equilibrium where each option is chosen with equal probability.
3. Optimization in Games
   • Allocating limited resources for maximum payoff:
     • Example: Studying for two quizzes with limited hours (e.g., allocate one hour to maximize combined scores in both subjects).

Week 2: Strategic Form Games

1. Prisoner’s Dilemma
   • Payoff Matrix:
     • Two prisoners can either cooperate or defect.
     • Mutual cooperation yields moderate benefits; mutual defection is suboptimal but safe.
     • Defecting while the other cooperates gives the highest individual payoff, but mutual defection leads to a worse collective outcome.
2. Mixed Strategies
   • Players randomize their strategies to keep opponents indifferent.
   • Example: In the "Police Patrol vs. Drug Dealer" game, randomization probabilities (e.g., patrol streets, patrol parks) determine equilibrium.

Week 3: Extensive Form Games

1. Extensive Form Representation
   • Models sequential decision-making.
   • Example: A multi-player game where choices unfold in a tree-like structure, showing each player's decisions and payoffs.
2. Backward Induction
   • Start from the terminal nodes (final outcomes) to determine optimal strategies for preceding decisions.

Week 4: Auctions and Mechanism Design

1. Auction Types
   • First-Price Auction: Highest bidder pays their bid.
   • Second-Price Auction: Highest bidder wins but pays the second-highest bid.
   • All-Pay Auction: Everyone pays their bid, but only the highest bidder wins.
   • Dutch Auction: Prices decrease until a bidder accepts the price.
   • English Auction: Prices increase until no higher bids are made.
2. Revenue Equivalence Theorem
   • All standard auctions yield the same expected revenue under certain conditions (e.g., bidders are risk-neutral).

Week 5: Cooperative Game Theory

1. Coalitions and Allocations
   • Core: Set of payoff distributions where no subset of players can achieve a better outcome by forming a coalition.
   • Efficiency: Total payoff equals the coalition value.
   • Individual Rationality: Payoff for each player is at least their standalone value.