Bayesian Nash equilibrium is a key concept in games with incomplete information. Players make decisions based on their beliefs about others' types, aiming to maximize expected utility. This equilibrium occurs when no one wants to change their strategy.
The concept involves three stages: ex-ante (before learning types), interim (after learning own type), and ex-post (after choices are made). Players use conditional probabilities and Bayes' rule to update beliefs as they gain new information.
Bayesian Nash Equilibrium Fundamentals
Definition and Key Components
- Bayesian Nash equilibrium represents a stable outcome in a game with incomplete information where players have beliefs about the types of other players
- Consists of a strategy profile where each player's strategy is a best response to the strategies of the other players, given their beliefs about the types of the other players
- Players aim to maximize their expected utility based on their beliefs and the strategies of other players
- Beliefs are represented by probability distributions over the possible types of the other players (common prior assumption)
Equilibrium Conditions and Best Response
- In a Bayesian Nash equilibrium, no player has an incentive to unilaterally deviate from their strategy, given their beliefs about the types and strategies of the other players
- Each player's strategy must be a best response to the strategies of the other players, taking into account the player's beliefs about the types of the other players
- Best response maximizes a player's expected utility, which is calculated by weighing the payoffs for each possible combination of types and strategies according to the player's beliefs (probability distribution)
Game Stages and Probabilities
Ex-Ante, Interim, and Ex-Post Stages
- Ex-ante stage occurs before players learn their types and make decisions based on the expected payoffs across all possible type realizations
- Interim stage takes place after players learn their types but before they choose their strategies, allowing them to update their beliefs about the types of other players (conditional probabilities)
- Ex-post stage happens after players have chosen their strategies and the payoffs are realized, revealing the actual types of all players
Conditional Probability and Bayes' Rule
- Conditional probability measures the likelihood of an event occurring given that another event has already occurred (e.g., the probability of player 2 being type A given that player 1 is type B)
- Bayes' rule allows players to update their beliefs about the types of other players based on new information (e.g., player 1 updates their belief about player 2's type after observing player 2's action)
- Bayes' rule states that the updated probability (posterior) is proportional to the product of the prior probability and the likelihood of the observed event given the hypothesis (P(A|B) = P(B|A) * P(A) / P(B))
Advanced Concepts
Mixed Strategies in Bayesian Games
- Mixed strategies involve players randomizing over their available pure strategies according to a probability distribution
- In Bayesian games, players can use mixed strategies to optimize their expected utility based on their beliefs about the types and strategies of other players
- Mixing probabilities are chosen to make the other players indifferent between their available strategies, ensuring that no player has an incentive to deviate (indifference condition)
- Mixed strategy Bayesian Nash equilibria can exist when there is no pure strategy Bayesian Nash equilibrium or when players have incentives to randomize to avoid being predictable (e.g., in a Bayesian game of matching pennies)