Game theory is all about strategic decision-making. It's like playing chess, but with real-life situations. You'll learn how people and businesses make choices when their actions affect each other's outcomes.
This topic covers the building blocks of game theory. You'll explore different types of games, how to represent them, and key concepts like Nash equilibrium. It's crucial for understanding how people interact strategically in various scenarios.
Game Theory Elements
Players, Strategies, and Payoffs
- Players act as decision-makers in games with unique action sets and preferences
- Strategies outline complete action plans for players in all possible game situations
- Payoffs represent outcomes or utilities players receive based on chosen strategy combinations
- Strategic form (normal form) displays strategy combinations and payoffs in a matrix
- Extensive form illustrates game sequences and player information as a tree-like structure
- Information sets determine player knowledge about others' actions during decision-making
- Dominant strategy consistently outperforms other strategies regardless of opponents' choices
- Example: In the Prisoner's Dilemma, confessing dominates staying silent for both players
- Example: In a Cournot duopoly model, firms may have dominant strategies for production levels
Game Representations and Analysis
- Strategic form suits simultaneous decision games (rock-paper-scissors)
- Extensive form accommodates sequential decision games (chess)
- Perfect information games provide complete history to all players (chess)
- Imperfect information games involve uncertainty about game aspects (poker)
- Mixed strategies involve randomized choices based on probability distributions
- Example: In matching pennies, players might choose heads or tails with equal probability
- Subgame perfect equilibrium refines Nash equilibrium for sequential games
- Backward induction solves sequential games by working from end to beginning
- Example: In a two-stage bargaining game, players use backward induction to determine optimal offers
Simultaneous vs Sequential Games
Simultaneous Games
- Players make decisions without knowledge of others' choices
- Often represented in strategic form as a payoff matrix
- Utilize mixed strategies for randomized decision-making
- Example: In rock-paper-scissors, players might randomize their choices to avoid being predictable
- Nash equilibrium serves as a key solution concept
- Example: In a Cournot duopoly, firms simultaneously choose production levels, reaching a Nash equilibrium
Sequential Games
- Players make decisions in a specific order with some information about earlier choices
- Typically represented in extensive form as a game tree
- Introduce subgame perfect equilibrium concept
- Employ backward induction for solution finding
- Example: In the centipede game, players use backward induction to determine optimal stopping points
- Perfect information games reveal complete history at each decision point (chess)
- Imperfect information games involve uncertainty about some game aspects (poker)
Cooperative vs Non-Cooperative Games
Cooperative Games
- Allow binding agreements and coalition formation among players
- Focus on payoff distribution among coalition members
- Utilize the core as a solution concept for stable allocations
- Example: In a three-player voting game, the core represents allocations that no two-player coalition can improve upon
- Employ Shapley value for fair surplus allocation
- Example: In a joint venture, Shapley value determines each partner's contribution to the overall profit
- Classify as transferable utility (TU) or non-transferable utility (NTU) games
- Address bargaining problems modeling negotiations over surplus division
- Example: Nash bargaining solution for two players dividing a fixed amount of money
Non-Cooperative Games
- Assume players cannot form enforceable agreements outside game model
- Emphasize strategic decision-making among players
- Rely on Nash equilibrium and its refinements as solution concepts
- Example: In the Battle of the Sexes game, multiple Nash equilibria may exist
- Model competitive situations without explicit cooperation
- Example: Cournot and Bertrand models of oligopolistic competition
- Analyze strategic interactions in various economic and social contexts
- Example: Analyzing firms' decisions to enter or exit a market using game-theoretic models
Rationality in Game Theory
Rational Decision-Making
- Assumes players maximize payoffs or utilities based on beliefs about others' actions
- Incorporates common knowledge of rationality (CKR) concept
- Utilizes best response strategy optimizing outcomes given beliefs about others
- Example: In a Cournot duopoly, each firm's output decision is a best response to the other's expected output
- Forms consistent beliefs about other players' strategies
- Applies Bayesian equilibrium concept in incomplete information games
- Example: In a first-price sealed-bid auction, bidders form beliefs about others' valuations
Bounded Rationality and Alternative Approaches
- Recognizes limitations in cognitive abilities and information availability
- Considers suboptimal decision-making due to constraints
- Example: In complex games, players might use heuristics or rules of thumb instead of full optimization
- Employs trembling hand perfect equilibrium to test equilibria stability
- Example: In the centipede game, trembling hand perfection refines Nash equilibrium predictions
- Explores evolutionary game theory challenging traditional rationality assumptions
- Example: Analyzing the evolution of cooperation in repeated Prisoner's Dilemma games
- Models strategy adoption through natural selection or social learning
- Example: Studying the emergence of social norms using evolutionary game theory models