8.3 Perfect Bayesian equilibrium

Perfect Bayesian equilibrium (PBE) is a game theory concept that combines strategic thinking with belief updating. It's used in games where players have incomplete information about each other, like in job market signaling or auctions.

PBE requires players' strategies to be optimal given their beliefs, and beliefs to be consistent with strategies. This concept helps analyze complex scenarios where information is asymmetric and actions can reveal hidden information about players' types.

Perfect Bayesian Equilibrium

Definition and Requirements

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• Perfect Bayesian equilibrium (PBE) extends subgame perfect equilibrium to games with incomplete information
• PBE requires players' strategies and beliefs about other players' types to be consistent with each other and the equilibrium being played
• In a PBE, players' strategies must be sequentially rational
  • Strategies are optimal given their beliefs at each information set
• Players' beliefs are determined by Bayes' rule and the equilibrium strategies whenever possible
• At information sets off the equilibrium path, beliefs must be consistent with the equilibrium strategies but are otherwise unrestricted

Applications and Examples

• PBE is commonly used to analyze dynamic games with asymmetric information (auctions, bargaining, signaling games)
• Example: In a job market signaling game, a job seeker's education level (type) is unknown to employers
  • The job seeker's strategy is the level of education to obtain, while the employer's strategy is the wage to offer based on the observed education
  • In a PBE, the job seeker's education choice must be optimal given the employer's wage strategy, and the employer's beliefs about the job seeker's type must be consistent with the education choice and Bayes' rule

Beliefs and Bayesian Updating

Forming and Updating Beliefs

• In dynamic games with incomplete information, players form beliefs about the types of other players based on their observed actions
• Beliefs are probability distributions over the possible types of other players at each information set
• Players update their beliefs using Bayes' rule whenever they receive new information, such as observing the actions of other players
• Bayesian updating involves calculating the posterior probability of a player's type given the prior probability and the observed action
  • Formula: $P(type|action) = P(action|type) * P(type) / P(action)$

Consistencies and Restrictions

• If an observed action has zero probability under the equilibrium strategies, Bayes' rule cannot be applied
  • In this case, beliefs are unrestricted but must be consistent with the equilibrium
• Consistency requires that beliefs assign positive probability only to types that could have taken the observed action under the equilibrium strategies
• Example: In a signaling game, if the equilibrium specifies that only high types take a certain action, then observing that action must lead to the belief that the player is a high type with probability 1

Perfect Bayesian Equilibria in Dynamic Games

Solving for PBE

• To find a PBE, first determine the possible types of players and their probabilities, as well as the available strategies and payoffs for each type
• Represent the game in extensive form, showing the moves of nature that determine players' types, the actions available to each player at each information set, and the payoffs for each outcome
• Use backward induction to solve for the optimal strategies at each information set, starting from the end of the game and working backwards
  • At each information set, calculate the expected payoff of each action based on the player's beliefs about the other players' types and their expected future actions
  • Choose the action that maximizes the player's expected payoff at each information set, given their beliefs

Verifying PBE

• Check that the resulting strategies and beliefs form a PBE by verifying that they are sequentially rational and consistent with each other and Bayes' rule
• Sequential rationality: Players' strategies must be optimal at every information set, given their beliefs
• Consistency: Beliefs must be updated according to Bayes' rule whenever possible, and must assign positive probability only to types that could have taken the observed actions under the equilibrium strategies
• Example: In a two-stage signaling game, verify that the sender's strategy is optimal given the receiver's response strategy, and that the receiver's beliefs and response strategy are optimal given the sender's strategy and Bayesian updating

Perfect Bayesian Equilibrium vs Other Concepts

Comparison to Subgame Perfect Equilibrium

• PBE is an extension of subgame perfect equilibrium (SPE) to games with incomplete information, where players have beliefs about each other's types
• Like SPE, PBE requires that strategies are optimal at every information set, not just at the beginning of the game
• However, PBE allows for imperfect information and Bayesian updating of beliefs, while SPE assumes perfect information

Relationship to Bayesian Nash Equilibrium

• PBE is a refinement of Bayesian Nash equilibrium (BNE), which is the Nash equilibrium of a game with incomplete information
• BNE requires that strategies are optimal given beliefs, but does not require sequential rationality or consistency of beliefs off the equilibrium path
• PBE is a stronger solution concept than BNE, as it imposes additional requirements on beliefs and strategies
• PBE can be seen as a combination of SPE and BNE, incorporating elements of both sequential rationality and Bayesian updating in games with imperfect information