🆚Game Theory and Economic Behavior Unit 7 – Bayesian Nash Equilibrium in Games

Key Concepts

• Game theory mathematical framework for analyzing strategic interactions between rational decision-makers
• Bayesian games model situations where players have incomplete information about other players' characteristics or payoffs
• Players' beliefs about others' private information represented by probability distributions over possible types
• Bayesian Nash equilibrium extends Nash equilibrium concept to games with incomplete information
• Players' strategies must be optimal given their beliefs and other players' strategies
• Beliefs updated using Bayes' rule as game progresses and new information revealed
• Applications span economics, political science, biology, and computer science

Foundations of Game Theory

• Strategic interactions modeled as games with players, strategies, and payoffs
• Players assumed to be rational utility-maximizers with well-defined preferences
• Nash equilibrium fundamental solution concept where each player's strategy is a best response to others' strategies
  • Example: Prisoner's Dilemma game has a unique Nash equilibrium where both players confess
• Extensive form games represent sequential move games with a game tree
  • Allows for modeling imperfect information and chance moves
• Repeated games involve players interacting over multiple rounds, enabling cooperation and punishment strategies

Bayesian Games Explained

• Players have private information about their own characteristics, payoffs, or strategies
• This private information is referred to as the player's type
• Types are drawn from a commonly known probability distribution
• Each player knows their own type but not the types of other players
• Strategies are type-contingent, specifying an action for each possible type
• Payoffs depend on players' realized types and chosen actions
• Example: In an auction, bidders' valuations for the item are their private types

Formulating Bayesian Nash Equilibrium

• Players choose strategies to maximize their expected payoff given their beliefs about others' types and strategies
• Beliefs are probability distributions over the possible types of other players
• In equilibrium, players' strategies are best responses to each other, and beliefs are consistent with strategies
• Mathematically, a strategy profile $s^*$ and belief system $\mu^*$ form a Bayesian Nash equilibrium if:
  • For each player $i$ and type $t_i$, $s_i^*(t_i)$ maximizes $i$'s expected payoff given $\mu_i^*$ and $s_{-i}^*$
  • $\mu^*$ is derived from $s^*$ using Bayes' rule whenever possible
• Solving for equilibrium involves finding a fixed point in the space of strategies and beliefs

Strategies and Beliefs

• Pure strategies specify a single action for each possible type
• Mixed strategies assign probabilities to actions for each type
• Beliefs are updated as the game progresses and new information is revealed
  • Bayes' rule used to calculate posterior probabilities based on prior beliefs and observed actions
• Equilibrium strategies must be sequentially rational, maximizing expected payoff at every information set
• Perfect Bayesian equilibrium refines Bayesian Nash equilibrium by imposing consistency on beliefs off the equilibrium path
• Example: In a signaling game, the sender's strategy is a mapping from types to messages, and the receiver's strategy is a mapping from messages to actions

Applications and Examples

• Auctions (first-price, second-price, common value) where bidders have private valuations or signals
• Bargaining games with incomplete information about the other party's reservation price
• Principal-agent problems with hidden information (adverse selection) or hidden actions (moral hazard)
  • Example: Job market signaling where education level signals worker productivity
• Reputation formation in repeated games, such as the chain store paradox
• Voting and political competition models with uncertainty about candidates' positions or competence
• Biological applications, such as animal conflicts with unknown fighting ability

Limitations and Criticisms

• Assumes players are perfectly rational and have infinite computational capacity
• Common prior assumption may not hold in reality, as players may have different initial beliefs
• Multiplicity of equilibria can make predictions ambiguous or sensitive to refinements
• Equilibrium selection problem arises when there are multiple equilibria
• Neglects bounded rationality, learning, and psychological factors that affect decision-making
• Focuses on static equilibrium rather than dynamic adjustment processes
• May not capture all relevant aspects of real-world strategic interactions

Further Reading and Resources

• "A Primer on Bayesian Games" by Robert Gibbons (1992) provides an accessible introduction
• "Game Theory: Analysis of Conflict" by Roger Myerson (1991) covers Bayesian games in depth
• "Bayesian Games" chapter in "Game Theory" by Drew Fudenberg and Jean Tirole (1991)
• "Bayesian Games and Mechanism Design" course materials by Rakesh Vohra (available online)
• "Bayesian Games and Incomplete Information" lecture notes by Asu Ozdaglar (MIT OpenCourseWare)
• "Bayesian Games" entry in the Stanford Encyclopedia of Philosophy for a philosophical perspective
• "Bayesian Games and the Smoothness Framework" by Tim Roughgarden (2016) connects Bayesian games to algorithmic game theory