🧃Intermediate Microeconomic Theory Unit 11 – Game Theory: Strategic Decision-Making

Key Concepts and Definitions

• Game theory studies strategic interactions between rational decision-makers
• Players are the individuals or entities making decisions in a game
• Strategies are the possible actions or plans of action available to each player
• Payoffs represent the outcomes or rewards resulting from the combination of strategies chosen by all players
• Rationality assumes that players make decisions to maximize their own payoffs
  • Players consider the strategies of other players when making decisions
  • Players have consistent preferences and beliefs about the game
• Common knowledge refers to the information that all players know, and know that all other players know
• Dominant strategy provides a player with the highest payoff regardless of the strategies chosen by other players
• Nash equilibrium is a stable state where no player has an incentive to unilaterally change their strategy

Game Theory Foundations

• Game theory originated from the work of mathematician John von Neumann and economist Oskar Morgenstern in the 1940s
• The field has since expanded to various domains, including economics, political science, psychology, and computer science
• Games can be represented in different forms, such as normal form (matrix) or extensive form (game tree)
• Normal form games are represented by a matrix that lists the players, their strategies, and the corresponding payoffs
  • Each cell in the matrix represents a possible outcome based on the strategies chosen by the players
• Extensive form games are represented by a game tree that illustrates the sequence of moves and decision points for each player
  • The tree structure captures the dynamic nature of the game and the information available to players at each stage
• Game theory assumes that players have complete information about the game structure, available strategies, and payoffs
• Games can be classified based on various criteria, such as the number of players, cooperation or competition, and information availability

Types of Games

• Cooperative games involve players working together to achieve a common goal or maximize collective payoffs
  • Players can communicate and form binding agreements (coalitions)
  • Examples include team sports, business partnerships, and international treaties
• Non-cooperative games involve players making independent decisions without the ability to form binding agreements
  • Players pursue their own interests and cannot rely on the cooperation of others
  • Examples include price competition, political campaigns, and arms races
• Zero-sum games are a special case where the total payoffs of all players always sum to zero
  • One player's gain is always equal to another player's loss
  • Examples include poker, chess, and sports matches
• Non-zero-sum games have outcomes where the total payoffs can be positive, negative, or zero
  • Players' interests are not always directly opposed, allowing for potential win-win or lose-lose situations
  • Examples include the Prisoner's Dilemma, the Stag Hunt, and the Battle of the Sexes
• Simultaneous games involve players making decisions at the same time without knowledge of the other players' choices
• Sequential games involve players making decisions in a specific order, with later players having knowledge of earlier players' choices
• Repeated games involve players engaging in the same game multiple times, allowing for the possibility of cooperation or retaliation based on past interactions

Nash Equilibrium

• Nash equilibrium is a fundamental concept in game theory, named after mathematician John Nash
• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
  • No player has an incentive to unilaterally deviate from their chosen strategy
  • Each player is maximizing their payoff given the strategies of the other players
• Nash equilibrium can be pure (players choose a single strategy) or mixed (players randomize over multiple strategies)
• To find Nash equilibria, players consider their own strategies and the strategies of others, seeking a stable point where no one wants to change
• Nash equilibrium provides a solution concept for predicting the outcomes of strategic interactions
  • It helps identify the likely strategies players will choose and the resulting payoffs
• In some games, there may be multiple Nash equilibria, while in others, there may be none
• The existence of a Nash equilibrium does not necessarily imply that it is the best outcome for all players or society as a whole (Pareto efficiency)
• The concept of Nash equilibrium has been applied to various fields, including economics, political science, and evolutionary biology

Strategies and Payoffs

• Strategies are the possible actions or plans of action available to each player in a game
  • Pure strategies involve choosing a single action with certainty
  • Mixed strategies involve randomizing over multiple actions according to a probability distribution
• Payoffs represent the outcomes or rewards resulting from the combination of strategies chosen by all players
  • Payoffs can be monetary (profits, costs) or non-monetary (utility, satisfaction)
  • Payoff matrices or tables are used to represent the outcomes for each combination of strategies
• Dominant strategy is a strategy that provides a player with the highest payoff regardless of the strategies chosen by other players
  • A strictly dominant strategy always yields a higher payoff than any other strategy
  • A weakly dominant strategy yields payoffs at least as high as any other strategy
• Dominated strategy is a strategy that provides a player with a lower payoff compared to another strategy, regardless of the strategies chosen by other players
  • Rational players will avoid choosing dominated strategies
• Best response is a strategy that maximizes a player's payoff given the strategies chosen by the other players
  • In a Nash equilibrium, each player's strategy is a best response to the strategies of others
• Minimax strategy involves choosing the strategy that minimizes the maximum possible loss (maximin) or maximizes the minimum possible gain (minimax)
  • It is a conservative approach that focuses on avoiding the worst-case scenario
• Pareto optimal outcome is a situation where no player can be made better off without making another player worse off
  • Pareto improvements are changes that make at least one player better off without making any player worse off

Applications in Economics

• Game theory has numerous applications in economics, helping to analyze and predict strategic behavior in various markets and interactions
• Oligopoly markets involve a small number of firms competing against each other
  • Game theory models (Cournot, Bertrand) analyze pricing and output decisions of firms
  • The Prisoner's Dilemma can represent the incentives for firms to engage in price wars or collusion
• Auction theory uses game theory to design and analyze different auction formats (first-price, second-price, English, Dutch)
  • It helps determine optimal bidding strategies and revenue-maximizing auction designs
• Bargaining theory applies game theory to analyze negotiations and the division of surplus between parties
  • The Ultimatum Game and the Nash Bargaining Solution are examples of bargaining models
• Public goods provision can be modeled as a game, where individuals decide whether to contribute to a shared resource
  • The free-rider problem arises when individuals enjoy the benefits without contributing
• Matching markets involve the allocation of resources or partnerships based on preferences (college admissions, medical residencies)
  • Game theory helps design stable and efficient matching mechanisms
• Principal-agent problems arise when there is information asymmetry and conflicting incentives between a principal and an agent (employer-employee, insurer-insured)
  • Game theory helps design contracts and incentive structures to align interests
• Evolutionary game theory applies game theory to study the evolution of strategies in populations over time
  • It analyzes the stability and dynamics of strategies in biological and social contexts

Advanced Game Theory Topics

• Subgame perfect equilibrium is a refinement of Nash equilibrium for sequential games
  • It requires that the strategies chosen by players are optimal at every decision point (subgame) of the game
  • Backward induction is used to solve for subgame perfect equilibria
• Bayesian games incorporate incomplete information, where players have different types or private information
  • Players form beliefs (probabilities) about the types of other players
  • Bayesian Nash equilibrium is a solution concept for games with incomplete information
• Correlated equilibrium is a generalization of Nash equilibrium that allows for players' strategies to be correlated
  • A mediator or external device can provide recommendations to players based on a joint probability distribution
• Mechanism design is the reverse of game theory, focusing on designing rules and incentives to achieve desired outcomes
  • It involves creating games or mechanisms that align players' incentives with the desired goals
  • Examples include auction design, voting systems, and incentive contracts
• Cooperative game theory studies the formation and stability of coalitions among players
  • The core and the Shapley value are solution concepts for allocating payoffs within coalitions
• Repeated games analyze the strategic interactions that occur when players engage in the same game multiple times
  • They allow for the possibility of cooperation, punishment, and reputation-building
  • The Folk Theorem shows that repeated interactions can sustain cooperation in some cases
• Behavioral game theory incorporates insights from psychology and behavioral economics into game-theoretic models
  • It relaxes assumptions of perfect rationality and explores the impact of biases, emotions, and social norms on strategic behavior

Real-World Examples

• Prisoner's Dilemma: Two suspects are interrogated separately, each facing the choice to confess or remain silent. The Nash equilibrium is for both to confess, even though remaining silent would yield a better outcome for both.
• Tragedy of the Commons: A shared resource (pasture, fishery) is overexploited by individuals acting in their own self-interest, leading to the depletion of the resource. This is an example of a social dilemma.
• Arms Race: Two countries engage in a competitive buildup of military capabilities, driven by the fear of being outpaced by the other. The Nash equilibrium is for both countries to invest heavily in arms, even though mutual disarmament would be preferable.
• Price Competition: Firms in an oligopoly market face the temptation to undercut each other's prices to gain market share. The Nash equilibrium is often a price war, leading to lower profits for all firms.
• Advertising Campaigns: Companies must decide how much to invest in advertising to attract customers. The Nash equilibrium may involve excessive spending on advertising, as each company tries to outdo its competitors.
• Climate Change Negotiations: Countries must decide whether to cooperate in reducing greenhouse gas emissions. The free-rider problem arises as each country has an incentive to let others bear the costs of mitigation.
• Voting Paradox: In a group decision-making setting, individual preferences may lead to intransitive collective preferences (A > B, B > C, C > A), highlighting the challenges of aggregating individual choices.
• Poker: Players must make strategic decisions based on incomplete information about their opponents' cards. Bluffing and mixed strategies are key aspects of the game.