🎱Game Theory Unit 1 – Game Theory: Strategic Decision-Making

Foundations of Game Theory

• Game theory studies strategic interactions between rational decision-makers
• Originated in the work of mathematician John von Neumann and economist Oskar Morgenstern in the 1940s
• Assumes players are rational, intelligent, and aim to maximize their own payoffs
• Payoffs represent the outcomes or utilities that players receive based on their decisions and the decisions of others
• Games can be represented using matrices, trees, or other mathematical structures
• Key components of a game include players, strategies, payoffs, and information sets
• Provides a framework for analyzing conflicts and cooperation in various domains (economics, political science, psychology)

Key Concepts and Terminology

• Players are the decision-makers in a game, which can be individuals, firms, or other entities
• Strategies are the possible actions or plans that players can choose from
  • Pure strategies specify a single action for each decision point
  • Mixed strategies involve probabilistic combinations of pure strategies
• Payoffs are the outcomes or utilities that players receive based on the chosen strategies
• Information sets describe the knowledge available to players at each decision point
  • Perfect information games (chess) have complete knowledge of all previous moves
  • Imperfect information games (poker) involve uncertainty about other players' actions or private information
• Rationality assumes that players make decisions to maximize their expected payoffs
• Common knowledge refers to information that all players know, and all players know that all players know, and so on

Types of Games and Their Structures

• Static games involve players making decisions simultaneously without knowledge of others' choices
  • Example: Prisoner's Dilemma, where two suspects must choose to confess or remain silent
• Dynamic games involve players making decisions sequentially, with knowledge of previous moves
  • Example: Stackelberg competition, where a leader firm moves first and a follower firm responds
• Cooperative games allow players to form binding agreements and coordinate their strategies
  • Example: Formation of coalitions in political negotiations
• Non-cooperative games do not allow for enforceable agreements between players
• Zero-sum games have payoffs that sum to zero, meaning one player's gain is another's loss (matching pennies)
• Non-zero-sum games have payoffs that do not necessarily sum to zero, allowing for mutual gains or losses (Battle of the Sexes)
• Repeated games involve players interacting over multiple rounds, enabling strategies like tit-for-tat

Nash Equilibrium and Strategic Thinking

• Nash equilibrium is a key concept in game theory, representing a stable outcome where no player can improve their payoff by unilaterally changing their strategy
• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
• Nash equilibrium can be pure (players choose a single strategy) or mixed (players randomize over multiple strategies)
• Finding Nash equilibria involves analyzing best responses and iterative reasoning
  • Dominant strategies are optimal regardless of other players' choices
  • Dominated strategies are always inferior and can be eliminated
• Nash equilibrium provides a framework for predicting outcomes and analyzing strategic stability
• Multiple Nash equilibria can exist in a game, leading to coordination challenges
• Refinements of Nash equilibrium (subgame perfect, perfect Bayesian) address dynamic and informational considerations

Decision-Making Under Uncertainty

• Many real-world situations involve uncertainty about payoffs, probabilities, or other players' types
• Expected utility theory provides a framework for decision-making under uncertainty
  • Players assign utilities to outcomes and choose strategies to maximize expected utility
  • Expected utility is calculated as the sum of utilities weighted by their probabilities
• Risk attitudes describe players' preferences for certain vs. uncertain outcomes
  • Risk-averse players prefer certain outcomes to gambles with the same expected value
  • Risk-neutral players are indifferent between certain outcomes and gambles with the same expected value
  • Risk-seeking players prefer gambles to certain outcomes with the same expected value
• Bayesian games incorporate incomplete information about players' types or payoffs
  • Players have prior beliefs about the distribution of types and update beliefs based on observed actions
• Information revelation and signaling can occur in games with uncertainty
  • Players may take actions to reveal or conceal private information strategically

Applications in Economics and Business

• Game theory has wide-ranging applications in economics and business, providing insights into market competition, bargaining, auctions, and more
• Oligopoly models analyze strategic interactions among firms in markets with few competitors
  • Cournot competition involves firms choosing quantities simultaneously (output decisions)
  • Bertrand competition involves firms choosing prices simultaneously (price decisions)
• Bargaining models examine how players divide a surplus or resolve conflicts through negotiation
  • Rubinstein bargaining model analyzes alternating offers and the role of patience
  • Nash bargaining solution predicts outcomes based on axioms of fairness and efficiency
• Auction theory studies the design and outcomes of different auction formats (first-price, second-price, English, Dutch)
• Principal-agent models analyze incentive problems and contract design in situations with asymmetric information (moral hazard, adverse selection)
• Matching markets involve the allocation of resources or partnerships based on preferences and stability criteria (stable marriage problem, college admissions)

Advanced Game Theory Techniques

• Evolutionary game theory studies the dynamics of strategy adoption and adaptation in populations
  • Replicator dynamics describe how strategies' frequencies change based on their relative payoffs
  • Evolutionarily stable strategies (ESS) are robust to invasion by mutant strategies
• Cooperative game theory focuses on coalition formation and the distribution of payoffs among players
  • Shapley value assigns fair payoffs to players based on their marginal contributions to coalitions
  • Core identifies stable allocations that no coalition can improve upon
• Mechanism design aims to create rules and incentives to achieve desired outcomes in strategic settings
  • Revelation principle states that any equilibrium outcome can be achieved through a direct revelation mechanism
  • Vickrey-Clarke-Groves (VCG) mechanism ensures truthful reporting and efficient outcomes in certain settings
• Behavioral game theory incorporates insights from psychology and experimental evidence
  • Bounded rationality models relax assumptions of perfect rationality and optimization
  • Prospect theory captures risk attitudes and reference-dependent preferences
• Learning in games examines how players adapt and converge to equilibria over time
  • Fictitious play assumes players best-respond to the empirical distribution of past actions
  • Reinforcement learning models update strategies based on their past performance

Real-World Case Studies and Examples

• Game theory has been applied to a wide range of real-world situations, providing valuable insights and policy implications
• Auction design: Game-theoretic principles have informed the design of spectrum auctions, online advertising auctions, and government procurement auctions
  • Example: The FCC's spectrum auctions use a simultaneous multiple-round format to allocate licenses efficiently
• Bargaining and negotiations: Game theory has been used to analyze international trade negotiations, labor-management disputes, and peace negotiations
  • Example: The Camp David Accords between Israel and Egypt in 1978 involved strategic concessions and issue linkage
• Market competition: Game-theoretic models have been applied to analyze pricing strategies, entry decisions, and mergers in various industries
  • Example: The airline industry exhibits strategic interactions in pricing, route selection, and capacity decisions
• Voting and political competition: Game theory has been used to study voting systems, campaign strategies, and coalition formation in political settings
  • Example: The U.S. presidential election can be modeled as a game between candidates, with strategies focused on key swing states
• Environmental and resource management: Game theory has been applied to analyze international environmental agreements, fisheries management, and water resource allocation
  • Example: The Paris Agreement on climate change involves strategic considerations and incentives for participation and compliance