8.2 Beliefs, updating, and Perfect Bayesian equilibrium
2 min read•august 7, 2024
In with , players must navigate uncertainty about others' types. (PBE) requires players to form beliefs and update them using , 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
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
relaxes the requirement of consistency of beliefs
Allows for a wider range of equilibria compared to PBE
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 and
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
Formula for Bayes' rule: P(A∣B)=P(B)P(B∣A)P(A)
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 , the receiver updates beliefs about the sender's type based on the observed signal
In a with incomplete information, players update beliefs about the other player's valuation based on observed offers and rejections
In a , players update beliefs about the long-run player's type based on observed actions in each period
Key Terms to Review (20)
Bargaining game: A bargaining game is a strategic interaction where two or more players negotiate the division of a resource or the terms of an agreement, each aiming to maximize their own payoff. The game's outcome often depends on players' beliefs about each other's intentions and strategies, highlighting the importance of information updating and the concept of Perfect Bayesian equilibrium in determining the optimal strategies during negotiation.
Bayes' Rule: Bayes' Rule is a fundamental theorem in probability theory that describes how to update the probability of a hypothesis based on new evidence. It emphasizes the relationship between prior beliefs and new information, allowing individuals to adjust their expectations accordingly. This concept plays a crucial role in understanding incomplete information situations, evaluating strategic behavior in games, and establishing coherent beliefs when faced with uncertainty.
Bayesian updating: Bayesian updating is a statistical technique used to revise existing beliefs or probabilities based on new evidence or information. This approach allows individuals to adjust their initial beliefs, or prior probabilities, in light of new data to form updated beliefs, or posterior probabilities. It plays a critical role in decision-making under uncertainty and is foundational for understanding concepts like beliefs, sequential games, and equilibrium in strategic interactions.
Common prior assumption: The common prior assumption is a foundational concept in Bayesian game theory that posits all players share a common initial belief about the state of the world before any private information is revealed. This assumption plays a critical role in how players update their beliefs as they receive new information, ensuring that any differences in beliefs among players arise solely from their unique observations rather than differing starting points.
Divinity: Divinity refers to the quality or state of being divine, often associated with a higher power or god-like attributes. In game theory, particularly in relation to beliefs and updating, divinity can signify the inherent nature of the players' beliefs about the strategies and actions of others in a strategic interaction. This understanding can impact how players update their beliefs over time, particularly in contexts where they may not have complete information about others' intentions.
Dynamic Games: Dynamic games are a class of games where players make decisions at different points in time, taking into account the actions and reactions of other players over the course of the game. These games are characterized by the sequential nature of decision-making and often utilize extensive form representation, allowing for strategies that evolve as the game progresses. The concepts of equilibrium, beliefs, and information updates become crucial in understanding how players interact within these dynamic frameworks.
Expected Utility: Expected utility is a concept in economics and decision theory that represents the anticipated satisfaction or value derived from different choices, taking into account the probabilities of various outcomes. This idea helps individuals and organizations make rational decisions under uncertainty by calculating the expected value of potential consequences associated with each option. It forms a foundational element for various theories that involve strategic interactions, beliefs updating, and revenue determination.
Incomplete Information: Incomplete information refers to a situation in a game where players do not have perfect knowledge about the other players' characteristics, strategies, or payoffs. This lack of information influences how players form strategies and make decisions, leading to uncertainties in predictions about opponents' behavior and outcomes in various interactions.
Information Set: An information set is a collection of decision nodes in a game where a player cannot distinguish between them, meaning the player does not know which node they are at when making a decision. This concept is crucial in understanding how players make strategic choices under uncertainty and helps to illustrate the differences between normal and extensive form representations of games, sequential decision-making processes, and the formation of beliefs in dynamic environments.
Intuitive Criterion: The intuitive criterion is a method used to evaluate the consistency of beliefs and strategies in game theory, particularly in the context of Perfect Bayesian Equilibrium. It suggests that players should form beliefs about the types of other players based on their observed actions, and these beliefs should align with the strategies that those players would choose under those beliefs. This alignment is crucial for maintaining coherence in decision-making processes.
Off-equilibrium beliefs: Off-equilibrium beliefs refer to the assumptions or expectations that players hold about others' strategies in a game when those strategies are not currently being played. These beliefs are crucial in dynamic games where players may deviate from equilibrium strategies, and understanding them helps in predicting how players will respond to unexpected moves. In the context of Perfect Bayesian equilibrium, off-equilibrium beliefs play a significant role in how players update their beliefs based on observed actions.
Perfect Bayesian Equilibrium: Perfect Bayesian Equilibrium is a solution concept in game theory that applies to games with incomplete information, where players have beliefs about the types of other players and update these beliefs based on observed actions. This concept integrates both strategies and beliefs, ensuring that players' strategies are optimal given their beliefs, and their beliefs are consistent with the actual strategies played. It highlights how players make decisions when they are uncertain about others' types, providing a framework to analyze dynamic games where information is revealed over time.
Posterior beliefs: Posterior beliefs refer to the updated beliefs or probabilities that an individual holds after considering new evidence or information. These beliefs are central in decision-making processes, especially in situations where individuals face uncertainty, allowing them to adjust their initial assumptions based on what they learn. In game theory, understanding how players form these beliefs is crucial when dealing with imperfect and incomplete information, as it impacts their strategies and outcomes.
Prior Beliefs: Prior beliefs are the initial assumptions or convictions that a player holds about the types and characteristics of other players in a game, particularly when dealing with incomplete information. These beliefs shape how players strategize and make decisions based on their expectations regarding others' actions. In settings with imperfect information, prior beliefs act as a foundation for players to update their knowledge as new information becomes available, influencing their future strategies in games.
Repeated games: Repeated games are strategic situations where the same game is played multiple times by the same players, allowing for the possibility of strategies to evolve based on past interactions. This framework enables players to build reputations, establish trust, and potentially achieve cooperative outcomes that would not be attainable in a one-shot game. The dynamics of repeated interactions can lead to various equilibria, including the possibility of sustaining cooperation over time.
Reputation game: A reputation game refers to a strategic interaction where individuals or entities must consider the perceptions and beliefs that others have about them when making decisions. This type of game emphasizes the importance of maintaining a positive reputation, as it can influence future interactions and outcomes, especially in situations involving trust and cooperation.
Sequential Rationality: Sequential rationality is a concept in game theory that describes a player's strategy as being optimal at every stage of a decision-making process, considering the future actions of other players. It implies that players make decisions not only based on the current situation but also take into account how their choices will affect subsequent moves in the game. This concept connects closely with the ideas of backward induction and how beliefs are updated in response to others' actions within strategic interactions.
Signaling game: A signaling game is a strategic interaction where one player, the sender, conveys information to another player, the receiver, through a signal. This type of game involves asymmetric information, where the sender has more knowledge about a particular aspect than the receiver. The signals can be interpreted by the receiver to update their beliefs about the sender's type or intentions, playing a crucial role in determining the outcome of the game.
Type space: A type space is a concept used in game theory to represent the different types of players in a game, particularly when there is incomplete information about their characteristics or preferences. It helps to model scenarios where players have private information, allowing them to possess different beliefs and strategies based on their type. This framework is essential for understanding how players form expectations and make decisions in settings like Bayesian games and sequential games with incomplete information.
Weak Perfect Bayesian Equilibrium: Weak Perfect Bayesian Equilibrium is a refinement of Bayesian Nash Equilibrium that combines the concepts of beliefs, strategies, and updating in a dynamic game context. It requires players to have consistent beliefs about the state of the game and to update these beliefs according to Bayes' rule, even when faced with off-path actions. This equilibrium concept helps in analyzing situations where players may have incomplete information, ensuring that their strategies are not only optimal given their beliefs but also robust against deviations by other players.