8.2 Beliefs, updating, and Perfect Bayesian equilibrium

In dynamic games with incomplete information, players must navigate uncertainty about others' types. Perfect Bayesian Equilibrium (PBE) requires players to form beliefs and update them using Bayes' rule, ensuring strategies are sequentially rational given these beliefs.

PBE refines Nash equilibrium by incorporating belief formation and updating. It's crucial for analyzing games where information is revealed over time, like signaling games or reputation-building scenarios. Understanding PBE helps predict behavior in complex strategic interactions.

Bayesian Equilibrium Concepts

Perfect Bayesian Equilibrium (PBE)

• Equilibrium concept used in dynamic games with incomplete information
• Requires players to have beliefs about the types of other players at each information set
• Players must choose strategies that are sequentially rational given their beliefs
• Beliefs are required to be consistent with the strategies being played according to Bayes' rule whenever possible

Refinements of Perfect Bayesian Equilibrium

• Weak Perfect Bayesian Equilibrium relaxes the requirement of consistency of beliefs
• Allows for a wider range of equilibria compared to PBE
• Sequential rationality ensures players make optimal decisions at each information set given their beliefs
• Refinements aim to eliminate implausible or counterintuitive equilibria
• Examples of refinements include the Intuitive Criterion and Divinity

Belief Formation and Updating

Belief Systems in Dynamic Games

• Players form beliefs about the types or private information of other players
• Beliefs are probability distributions over the possible types at each information set
• Initial beliefs are specified at the start of the game
• Beliefs are updated as the game progresses and new information is revealed

Consistency and Updating of Beliefs

• Consistency requires beliefs to be derived from the strategies being played using Bayes' rule whenever possible
• Bayes' rule is used to update beliefs based on observed actions and prior beliefs
• Formula for Bayes' rule: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• Off-equilibrium beliefs are beliefs at information sets that are not reached in equilibrium
• No restrictions are placed on off-equilibrium beliefs in PBE, allowing for flexibility

Examples of Belief Updating

• In a signaling game, the receiver updates beliefs about the sender's type based on the observed signal
• In a bargaining game with incomplete information, players update beliefs about the other player's valuation based on observed offers and rejections
• In a reputation game, players update beliefs about the long-run player's type based on observed actions in each period