Honors Economics Unit 18 ReviewGame Theory and Strategic Interactions

Key Concepts and Definitions

• Game theory studies strategic decision-making and interactions between rational agents (players) in various settings
• Players are individuals, firms, or any decision-making entities involved in a game or strategic interaction
• Strategies are complete plans of action that specify what a player will do in every possible situation throughout the game
• Payoffs are the outcomes or rewards that players receive based on the strategies chosen by all players involved
• Common knowledge assumes that all players know the rules of the game, available strategies, and payoffs associated with each outcome
• Rationality assumes that players always make decisions to maximize their own payoffs given their beliefs about other players' strategies
• Zero-sum games have outcomes where one player's gain is always equal to another player's loss, resulting in a constant sum of payoffs
  • In contrast, non-zero-sum games allow for outcomes where all players can gain or lose simultaneously

Game Theory Fundamentals

• Game theory analyzes strategic interactions by constructing mathematical models of conflict and cooperation between rational decision-makers
• The basic elements of a game include players, strategies, payoffs, and information structure
• Players are assumed to be rational, meaning they make decisions to maximize their own payoffs based on available information
• Strategies can be pure (choosing a single action) or mixed (randomly selecting actions based on a probability distribution)
• Payoff matrices represent the outcomes for each combination of strategies chosen by players in a game
• Simultaneous games involve players making decisions without knowing the choices of other players (e.g., Prisoner's Dilemma)
• Sequential games involve players making decisions in a specific order, with later players observing the actions of earlier players (e.g., Stackelberg competition)
• Information structure refers to what each player knows about the game, including other players' strategies and payoffs
  • Games can have perfect information (all players know everything) or imperfect information (some aspects are unknown)

Types of Games and Strategies

• Static games are one-shot interactions where players make simultaneous decisions without observing others' choices (e.g., Rock-Paper-Scissors)
• Dynamic games involve players making decisions sequentially, with later players observing earlier actions (e.g., chess)
• Cooperative games allow players to communicate and make binding agreements to coordinate their strategies for mutual benefit
• Non-cooperative games do not permit enforceable agreements, so players must independently choose their strategies
• Dominant strategies are optimal choices for a player regardless of what other players do (e.g., confessing in Prisoner's Dilemma)
• Dominated strategies are always inferior to other available strategies and should never be chosen by rational players
• Coordination games have multiple Nash equilibria, and players must coordinate their choices to achieve the best outcome (e.g., choosing which side of the road to drive on)
• Signaling games involve one player (the sender) having private information that can be conveyed to another player (the receiver) through observable actions or signals

Nash Equilibrium and Dominant Strategies

• Nash equilibrium is a set of strategies where no player can unilaterally improve their payoff by changing their strategy, given the strategies of other players
• In a Nash equilibrium, each player's strategy is a best response to the strategies of other players
• Dominant strategy equilibrium occurs when all players have a dominant strategy, resulting in a unique Nash equilibrium
• Iterated elimination of dominated strategies is a process of removing dominated strategies from a game to simplify the analysis and identify Nash equilibria
• Mixed strategy Nash equilibria involve players randomizing their strategies according to specific probabilities to make others indifferent between their available strategies
• Pareto optimality is a state where no player can be made better off without making another player worse off
• Nash equilibria are not always Pareto optimal, as players may get stuck in suboptimal outcomes due to strategic considerations (e.g., Prisoner's Dilemma)
• Focal points are salient or prominent outcomes that players may coordinate on in the absence of communication, helping to resolve coordination problems

Applications in Economics and Business

• Oligopoly markets can be modeled using game theory to analyze pricing strategies, quantity competition, and market entry decisions
• Bertrand competition involves firms competing on price, often leading to a Nash equilibrium where prices equal marginal costs (perfect competition outcome)
• Cournot competition involves firms simultaneously choosing quantities, resulting in a Nash equilibrium with prices above marginal costs but below monopoly levels
• Stackelberg competition has a leader firm moving first and follower firms responding, allowing the leader to gain a first-mover advantage
• Auction theory uses game theory to design optimal auction mechanisms and analyze bidding strategies for various auction formats (e.g., first-price sealed-bid, second-price sealed-bid, English auction)
• Bargaining theory models negotiations between parties over the division of a surplus, such as in labor-management negotiations or international trade agreements
• Principal-agent problems arise when there is information asymmetry and misaligned incentives between a principal (e.g., employer) and an agent (e.g., employee)
  • Game theory can help design optimal contracts and incentive schemes to mitigate moral hazard and adverse selection issues

Decision-Making Models

• Expected utility theory assumes that rational decision-makers choose actions to maximize their expected utility, which is the sum of utilities of all possible outcomes weighted by their probabilities
• Prospect theory is a behavioral model that accounts for observed deviations from expected utility theory, such as loss aversion and reference dependence
• Bounded rationality recognizes that decision-makers have limited cognitive abilities and often use heuristics or rules of thumb to simplify complex decisions
• Satisficing is a decision-making strategy where individuals choose the first satisfactory option rather than seeking the optimal solution
• Evolutionary game theory studies how strategies evolve over time in populations of players, using concepts from evolutionary biology such as replicator dynamics and evolutionary stable strategies
• Behavioral game theory incorporates insights from psychology and behavioral economics to model how real people make decisions in strategic situations
• Hyperbolic discounting is a time-inconsistent preference where individuals exhibit higher discount rates for near-term trade-offs compared to future trade-offs
• Quantal response equilibrium is a generalization of Nash equilibrium that allows for players to make small mistakes or exhibit bounded rationality

Real-World Case Studies

• The Cuban Missile Crisis (1962) can be analyzed using game theory to understand the strategic interactions between the United States and the Soviet Union during the Cold War
• Spectrum auctions, such as those conducted by the Federal Communications Commission (FCC), use game theory to design efficient allocation mechanisms for scarce radio frequencies
• Matching markets, such as those for medical residencies (NRMP) and school choice programs, use game-theoretic algorithms to find stable matches between participants
• Tragedy of the commons occurs when individuals acting in their own self-interest deplete a shared resource, leading to suboptimal outcomes for all (e.g., overfishing, deforestation)
• Volunteer's dilemma arises when a group would benefit from a costly action, but each individual prefers that someone else volunteers to bear the cost (e.g., bystander effect)
• Hotelling's law predicts that competing firms will tend to locate near each other in order to capture the maximum market share (e.g., gas stations, fast food restaurants)
• Braess's paradox shows that adding a new road to a congested traffic network can sometimes increase overall travel times due to individual drivers' selfish routing decisions
• Mechanism design theory aims to create incentive-compatible rules and institutions that align individual incentives with social objectives (e.g., auction design, voting systems)

Advanced Topics and Extensions

• Repeated games involve players interacting over multiple rounds, allowing for the possibility of cooperation through strategies like tit-for-tat or grim trigger
• Stochastic games are repeated games where the state of the game evolves according to a probability distribution based on players' actions
• Bayesian games incorporate incomplete information, where players have private information about their own types or payoffs
• Correlated equilibrium is a generalization of Nash equilibrium that allows players to coordinate their strategies based on a common public signal or mediator
• Cooperative game theory focuses on coalition formation and the distribution of payoffs among players in a cooperative setting (e.g., Shapley value, core)
• Algorithmic game theory studies the computational complexity of finding equilibria and designing efficient algorithms for solving game-theoretic problems
• Evolutionary stable strategies (ESS) are strategies that, if adopted by a population, cannot be invaded by any alternative strategy
• Mean field game theory analyzes strategic interactions in large populations where individual players have negligible impact on others, using tools from statistical physics and optimal control theory