Intro to Mathematical Economics Unit 6 ReviewGame Theory: Strategic Decision-Making

What's Game Theory?

• Mathematical framework for analyzing strategic interactions between rational decision-makers
• Studies how individuals make decisions in situations where the outcome depends on the choices of others
• Assumes players are rational, meaning they aim to maximize their own payoffs or utilities
• Provides a systematic approach to understanding and predicting behavior in various contexts (economics, political science, psychology, computer science)
• Helps identify optimal strategies for players to achieve their goals
• Considers factors such as available information, possible actions, and potential payoffs
• Enables the analysis of complex decision-making scenarios and the identification of equilibrium outcomes

Key Concepts and Terms

• Players: Individuals or entities involved in the game who make decisions
• Strategies: Plans of action or choices available to each player
• Payoffs: Outcomes or rewards associated with each combination of strategies chosen by the players
• Rationality: Assumption that players make decisions to maximize their own payoffs
• Common knowledge: Information that all players know, and all players know that all players know, and so on
• Dominant strategy: A strategy that yields the highest payoff for a player, regardless of the strategies chosen by other players
• Nash equilibrium: A situation where no player can improve their payoff by unilaterally changing their strategy, given the strategies of the other players

Players and Strategies

• Players are the decision-makers in a game, which can be individuals, firms, or other entities
• Each player has a set of available strategies or actions they can choose from
  • Pure strategies: A single, specific action a player can take
  • Mixed strategies: A probability distribution over a player's pure strategies
• Players' strategies can be simultaneous (chosen at the same time) or sequential (chosen in a specific order)
• The combination of strategies chosen by all players determines the outcome and payoffs of the game
• Players are assumed to be rational and aim to maximize their own payoffs
• Players may have complete or incomplete information about the game and other players' strategies

Types of Games

• Normal-form games: Games represented by a matrix showing the payoffs for each combination of strategies
  • Also known as strategic-form games
  • Example: Prisoner's Dilemma, where two suspects must choose between confessing or remaining silent
• Extensive-form games: Games represented by a decision tree, showing the sequence of moves and possible outcomes
  • Allows for the analysis of games with sequential moves and imperfect information
  • Example: Stackelberg competition, where a leader firm moves first, followed by a follower firm
• Cooperative games: Games where players can form binding agreements and collaborate to achieve a common goal
  • Focus on how players can work together to create value and distribute the resulting payoffs
• Non-cooperative games: Games where players make decisions independently and cannot form binding agreements
  • Focus on individual decision-making and the resulting equilibrium outcomes

Nash Equilibrium

• A key concept in game theory, named after mathematician John Nash
• Represents a stable state in a game where no player has an incentive to unilaterally change their strategy
• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
• Can be pure-strategy Nash equilibrium (players choose specific actions) or mixed-strategy Nash equilibrium (players choose probability distributions over actions)
• Nash equilibrium provides a way to predict the outcome of a game and analyze the strategic behavior of players
• In some games, there may be multiple Nash equilibria or no Nash equilibrium at all
• Finding Nash equilibria helps identify the most likely outcomes of a game and the strategies players should adopt

Applications in Economics

• Game theory is widely applied in various areas of economics to analyze strategic interactions and decision-making
• Oligopoly markets: Studying the behavior of firms in markets with a small number of competitors
  • Example: Cournot competition (firms simultaneously choose quantities) and Bertrand competition (firms simultaneously choose prices)
• Auction theory: Analyzing the design and outcomes of different auction formats
  • Example: First-price sealed-bid auctions and second-price sealed-bid auctions
• Bargaining and negotiation: Examining how parties can reach agreements when they have conflicting interests
  • Example: Nash bargaining solution, which maximizes the product of the players' gains from cooperation
• Public goods and externalities: Investigating the provision of goods that benefit society as a whole and the effects of individual actions on others
  • Example: Free-rider problem in the provision of public goods
• Mechanism design: Designing rules and incentives to achieve desired outcomes in strategic settings
  • Example: Designing auction rules to maximize revenue or achieve efficiency

Solving Game Theory Problems

• Identify the players, their available strategies, and the payoffs associated with each combination of strategies
• Determine the type of game (normal-form, extensive-form, cooperative, or non-cooperative)
• If the game is in normal form, represent it using a payoff matrix
• If the game is in extensive form, represent it using a decision tree
• Look for dominant strategies, if any, which can simplify the analysis
• Find the Nash equilibrium or equilibria of the game
  • In a normal-form game, find the best response for each player given the other players' strategies
  • In an extensive-form game, use backward induction to determine the subgame perfect Nash equilibrium
• Interpret the results and draw conclusions about the strategic behavior of the players and the likely outcomes of the game
• Consider any potential extensions or variations of the game that may provide additional insights

Real-World Examples

• Pricing strategies in oligopolistic markets (Coca-Cola and Pepsi)
• Advertising campaigns among competing firms (Apple and Samsung)
• International trade negotiations between countries (US and China)
• Arms races and military strategy (US and Soviet Union during the Cold War)
• Voting and political competition in elections (Democrats and Republicans)
• Bargaining between employers and unions in labor markets (United Auto Workers and General Motors)
• Resource allocation in environmental conservation efforts (countries agreeing to reduce carbon emissions)
• Spectrum auctions for the allocation of radio frequencies (telecom companies bidding for 5G spectrum)