🎱Game Theory Unit 14 – Advanced Game Theory: Frontiers and Topics

Key Concepts and Foundations

• Game theory studies strategic interactions between rational decision-makers
• Players in a game aim to maximize their payoffs by choosing optimal strategies
• Nash equilibrium is a fundamental concept where each player's strategy is a best response to others' strategies
• Extensive form games represent sequential decision-making using game trees
  • Nodes represent decision points and edges represent actions
  • Payoffs are assigned to terminal nodes
• Normal form games describe simultaneous decision-making using payoff matrices
• Dominant strategies are optimal regardless of other players' choices (Prisoner's Dilemma)
• Pareto efficiency occurs when no player can improve their payoff without harming others
• Zero-sum games have a fixed total payoff that players compete for (Matching Pennies)

Advanced Equilibrium Concepts

• Subgame perfect equilibrium refines Nash equilibrium for sequential games
  • Strategies must be optimal at every decision point (subgame)
  • Eliminates non-credible threats
• Perfect Bayesian equilibrium combines subgame perfection with Bayesian updating in games of incomplete information
• Correlated equilibrium allows players to coordinate strategies through a mediator or public signal
• Equilibrium refinements (trembling hand, proper equilibrium) address issues of implausible equilibria
• Risk dominance selects equilibria based on players' risk attitudes
• Quantal response equilibrium models players' bounded rationality and noisy decision-making
• Epsilon-equilibrium relaxes the strict best-response requirement, allowing for small deviations

Dynamic Games and Repeated Interactions

• Repeated games model long-term interactions between players
• Folk Theorem shows that cooperation can be sustained in infinitely repeated games with patient players
  • Trigger strategies (Grim Trigger, Tit-for-Tat) punish deviations from cooperation
• Finitely repeated games often unravel to non-cooperative outcomes due to backward induction
• Stochastic games generalize repeated games with evolving game states
• Markov perfect equilibrium is a refinement for stochastic games based on state-dependent strategies
• Reputation effects can sustain cooperation in finitely repeated games with incomplete information
• Renegotiation-proofness ensures equilibria are immune to jointly beneficial deviations

Games with Incomplete Information

• Incomplete information arises when players are uncertain about others' payoffs or types
• Bayesian games model incomplete information using probability distributions over types
• Harsanyi transformation converts a game of incomplete information into one of imperfect information
• Bayes-Nash equilibrium extends Nash equilibrium to Bayesian games
  • Players maximize expected payoffs given their beliefs about others' types
• Signaling games involve informed senders communicating with uninformed receivers (Job Market Signaling)
• Cheap talk refers to costless, non-binding communication before or during a game
• Screening games have uninformed players designing mechanisms to reveal informed players' types
• Adverse selection occurs when informed players' actions depend on their private information (Insurance Markets)

Mechanism Design and Auction Theory

• Mechanism design aims to create rules and incentives to achieve desired outcomes
• Revelation Principle states that any mechanism can be replicated by an incentive-compatible direct mechanism
• Vickrey-Clarke-Groves (VCG) mechanisms ensure truthful reporting and efficient outcomes in private value settings
• Auctions are common mechanisms for allocating goods and services
  • First-price sealed-bid auctions award the item to the highest bidder at their bid price
  • Second-price sealed-bid (Vickrey) auctions award the item to the highest bidder at the second-highest bid price
• Revenue equivalence theorem shows that various auction formats yield the same expected revenue under certain conditions
• Optimal auction design maximizes the auctioneer's expected revenue (Reserve Prices)
• Combinatorial auctions allow bidders to express preferences for bundles of items

Evolutionary Game Theory

• Evolutionary game theory studies the dynamics of strategy adoption in populations
• Replicator dynamics describe how strategy frequencies evolve based on their relative payoffs
  • Strategies with above-average payoffs grow in popularity
• Evolutionarily stable strategies (ESS) resist invasion by mutant strategies
• Nash equilibria are not always evolutionarily stable (Hawk-Dove Game)
• Evolutionary stability concepts (Neutrally Stable Strategies, Convergence Stability) refine the ESS notion
• Adaptive dynamics model the evolution of continuous traits in a population
• Evolutionary branching can lead to the divergence of traits and speciation
• Evolutionary game theory has applications in biology, ecology, and social sciences (Animal Contests, Social Norms)

Applications in Economics and Social Sciences

• Oligopoly models analyze strategic interactions among firms in markets (Cournot, Bertrand Competition)
• Bargaining theory studies how players divide a surplus through negotiation (Nash Bargaining Solution)
• Matching markets involve pairing agents based on preferences (Stable Marriage Problem, College Admissions)
• Social choice theory examines collective decision-making and voting systems (Arrow's Impossibility Theorem)
• Public goods games model the provision of goods that are non-excludable and non-rivalrous (Voluntary Contributions Mechanism)
• Network formation games analyze the creation and stability of social and economic networks
• Game theory informs the design of market mechanisms and institutions (Spectrum Auctions, Kidney Exchanges)
• Behavioral game theory incorporates insights from psychology and bounded rationality (Ultimatum Game, Level-k Reasoning)

Current Research and Future Directions

• Algorithmic game theory develops efficient algorithms for computing equilibria and mechanism design
• Dynamic mechanism design extends static mechanisms to settings with evolving private information
• Robust mechanism design aims to create mechanisms that perform well under various assumptions
• Learning in games studies how players adapt and learn in repeated interactions (No-Regret Learning, Reinforcement Learning)
• Cooperative game theory focuses on coalition formation and stability concepts (Shapley Value, Core)
• Mean-field games model strategic interactions in large populations using continuous approximations
• Quantum game theory explores the implications of quantum mechanics for strategic decision-making
• Interdisciplinary applications of game theory continue to emerge in fields such as computer science, biology, and political science