🆚Game Theory and Economic Behavior Unit 2 – Games: Normal and Extensive Forms

What's This Unit About?

• Focuses on the fundamental concepts and techniques used to analyze strategic interactions between rational decision-makers
• Introduces two main types of game representations: normal form and extensive form games
• Explores various strategies players can employ to maximize their payoffs in different game scenarios
• Teaches methods for solving games and finding equilibrium solutions (Nash equilibrium)
• Discusses real-world applications of game theory in economics, business, and other fields
• Highlights common pitfalls and misconceptions when analyzing games and provides guidance on avoiding them

Key Concepts and Definitions

• Players: Individuals or entities involved in a game who make decisions based on their preferences and available information
• Strategies: Complete plans of action that specify what a player will do in every possible situation throughout the game
  • Pure strategy: A deterministic plan of action that specifies a single action for each decision point
  • Mixed strategy: A randomized plan of action that assigns probabilities to different pure strategies
• Payoffs: Numerical values representing the outcomes or utilities that players receive based on their chosen strategies
• Rationality: The assumption that players make decisions to maximize their expected payoffs given their beliefs about other players' strategies
• Dominance: A situation where one strategy yields a higher payoff than another strategy, regardless of the strategies chosen by other players
  • Strictly dominated strategy: A strategy that always results in lower payoffs compared to another strategy, irrespective of other players' choices
  • Weakly dominated strategy: A strategy that never yields a higher payoff and sometimes results in a lower payoff compared to another strategy
• Nash equilibrium: A set of strategies, one for each player, where no player has an incentive to unilaterally deviate from their chosen strategy given the strategies of other players

Normal Form Games Explained

• Normal form games represent strategic interactions where players make simultaneous decisions without knowing the choices of other players
• Key components of a normal form game include players, strategies, and payoffs
• Payoffs are typically presented in a matrix format, with each cell representing the payoffs for each player based on their chosen strategies
• Example: The classic Prisoner's Dilemma is a normal form game where two suspects must choose between confessing (defecting) or remaining silent (cooperating)
  • If both confess, they each receive a moderate sentence (e.g., 5 years)
  • If one confesses and the other remains silent, the confessor goes free while the silent suspect receives a harsh sentence (e.g., 10 years)
  • If both remain silent, they each receive a light sentence (e.g., 1 year)
• Dominant strategies and Nash equilibria can be identified by analyzing the payoff matrix and considering each player's best response to the other player's strategies

Extensive Form Games Breakdown

• Extensive form games represent strategic interactions where players make sequential decisions, and the order of moves is explicitly modeled
• Key components of an extensive form game include players, decision nodes, branches, payoffs, and information sets
• The game is represented using a tree-like structure called a game tree, which captures the sequence of moves and possible outcomes
  • Decision nodes represent points where a player must make a choice
  • Branches emanating from a decision node represent the available actions or moves
  • Terminal nodes at the end of each path represent the final outcomes and payoffs for each player
• Information sets group together decision nodes where a player has the same information and available actions
  • Perfect information games: Players have complete knowledge of all previous moves when making their decisions (e.g., chess, tic-tac-toe)
  • Imperfect information games: Players may not have complete knowledge of all previous moves (e.g., poker, auctions)
• Backward induction is a common technique used to solve extensive form games with perfect information by starting at the terminal nodes and working backwards to determine optimal strategies

Strategies and Payoffs

• Strategies in game theory are complete plans of action that specify what a player will do in every possible situation throughout the game
• Pure strategies define a specific action to be taken at each decision point, while mixed strategies assign probabilities to different pure strategies
• Payoffs represent the outcomes or utilities that players receive based on the strategies chosen by all players involved in the game
  • Ordinal payoffs: Rank outcomes in order of preference without specifying exact numerical values
  • Cardinal payoffs: Assign specific numerical values to outcomes, allowing for more precise comparisons and calculations
• Expected payoffs can be calculated for mixed strategies by multiplying the probability of each pure strategy by its corresponding payoff and summing the results
• Dominant strategies and Nash equilibria are determined by comparing payoffs across different strategy profiles and identifying best responses for each player
  • Iterated elimination of strictly dominated strategies: Repeatedly remove strategies that are strictly dominated until only undominated strategies remain
  • Best response analysis: For each player, identify the strategy that yields the highest payoff given the strategies of other players

Solving Different Game Types

• Solution concepts in game theory aim to predict the outcomes of games and identify stable strategy profiles
• Nash equilibrium is the most widely used solution concept, representing a set of strategies where no player has an incentive to unilaterally deviate
  • Pure strategy Nash equilibrium: Each player plays a specific pure strategy
  • Mixed strategy Nash equilibrium: Players assign probabilities to their pure strategies
• Pareto optimality is another solution concept that focuses on outcomes where no player can be made better off without making another player worse off
• Cooperative games allow players to communicate and make binding agreements, while non-cooperative games assume players make independent decisions
• Repeated games involve players interacting over multiple rounds, allowing for the emergence of cooperation and punishment strategies (e.g., tit-for-tat)
• Bayesian games incorporate incomplete information, where players have different types or private information that affects their payoffs and strategies

Real-World Applications

• Game theory has numerous applications in economics, business, political science, and other fields where strategic interactions occur
• Oligopolistic competition: Firms in an oligopoly can be modeled as players in a game, with their strategies representing pricing or output decisions (Cournot, Bertrand models)
• Auction design: Game theory helps analyze bidding strategies and design optimal auction mechanisms (first-price, second-price, all-pay auctions)
• Bargaining and negotiations: Models such as the Ultimatum Game and Nash Bargaining Solution provide insights into distributive and integrative negotiation strategies
• Public goods and externalities: Game theory helps explain the free-rider problem and design mechanisms to encourage cooperation (Prisoner's Dilemma, tragedy of the commons)
• Voting and political competition: Game-theoretic models analyze strategic voting behavior, campaign strategies, and the formation of coalitions in political systems

Common Pitfalls and How to Avoid Them

• Assuming players have complete information when they may actually have imperfect or incomplete information
  • Carefully consider the information available to players and use appropriate game structures (e.g., Bayesian games) to model incomplete information
• Neglecting the possibility of multiple Nash equilibria or focusing solely on pure strategy equilibria
  • Analyze both pure and mixed strategy equilibria and consider the plausibility and stability of each equilibrium in the context of the game
• Ignoring the potential for irrational or boundedly rational behavior by players
  • Incorporate concepts from behavioral game theory, such as cognitive biases and learning, to account for deviations from perfect rationality
• Overlooking the impact of repeated interactions and the potential for cooperation or punishment strategies
  • Consider the possibility of repeated games and analyze the sustainability of cooperation using concepts like the folk theorem and trigger strategies
• Failing to consider the limitations and assumptions of game-theoretic models when applying them to real-world situations
  • Recognize that models are simplifications of reality and may not capture all relevant factors; use game theory as a tool for insight rather than a definitive prediction