8.1 Bayesian games and incomplete information

Bayesian games introduce incomplete information, where players are uncertain about aspects like payoffs or types of other players. These games model real-world scenarios where decision-makers lack full information, requiring them to form beliefs and make choices based on probabilities.

Players in Bayesian games use their own type and beliefs about others to calculate expected payoffs and choose strategies. The solution concept, Bayesian Nash Equilibrium, represents a set of strategies where no player wants to change their choice given their beliefs about others.

Bayesian Games and Components

Definition and Key Elements

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• Bayesian games are a type of game in which players have incomplete information about some aspect of the game, such as the payoffs, strategies, or types of other players
• Key components of a Bayesian game include:
  • Set of players
  • Possible types for each player
  • Probability distribution over types (common prior)
  • Strategy sets for each player
  • Payoff functions that depend on players' types and chosen strategies

Types and Beliefs

• In a Bayesian game, each player has a set of possible types, which determine their payoffs and available strategies
• The probability distribution over these types is known as the common prior
• Players make decisions based on their own type and their beliefs about the probability distribution of other players' types
• Example: In an auction, a player's type could be their private valuation of the item being auctioned, which affects their willingness to pay and bidding strategy

Decision-Making in Incomplete Information Games

Forming Beliefs and Expected Payoffs

• In a Bayesian game, players form beliefs about the likelihood of other players having certain types based on the common prior probability distribution
• Each player uses their beliefs about others' types to calculate the expected payoff of each available strategy, considering the probabilities of different type combinations
• Players aim to maximize their expected payoffs by choosing the strategy that performs best given their beliefs about the distribution of other players' types

Bayesian Rationality and Belief Updating

• The concept of Bayesian rationality assumes that players update their beliefs using Bayes' rule when new information about types is revealed during the game
• As the game progresses and players observe actions or signals, they may update their beliefs about the likelihood of different type combinations
• Example: In a signaling game, a job applicant's education level (type) affects the employer's belief about their ability, and the employer updates this belief based on the applicant's performance in an interview (signal)

Solving for Bayesian Nash Equilibria

Definition and Equilibrium Condition

• A Bayesian Nash equilibrium (BNE) is a set of strategies, one for each player, that maximizes each player's expected payoff given their beliefs about others' types and assuming other players also play their equilibrium strategies
• In a BNE, no player has an incentive to unilaterally deviate from their equilibrium strategy given their beliefs about others' types and strategies

Finding Bayesian Nash Equilibria

• To find a BNE, players consider their expected payoffs for each possible combination of types and strategies, weighted by the probability of each type combination occurring based on the common prior
• Solving for BNE involves finding a set of strategies that simultaneously satisfy the equilibrium condition for all players and all possible type combinations
• Example: In a two-player Bayesian game with two possible types for each player, finding a BNE requires checking the equilibrium condition for four possible type combinations

Multiplicity and Equilibrium Selection

• In some cases, multiple BNE may exist for a given Bayesian game
• Equilibrium selection techniques may be used to determine the most plausible or relevant equilibrium
• Criteria for equilibrium selection could include payoff dominance, risk dominance, or focal points based on the game's context or players' expectations

Common Prior in Bayesian Games

Definition and Consistency

• The common prior is the probability distribution over the possible types of all players in a Bayesian game, which is assumed to be known by all players
• The common prior captures the initial beliefs that players have about the likelihood of different type combinations before any additional information is revealed during the game
• The assumption of a common prior implies that players' beliefs are consistent with each other and with the actual probability distribution of types

Importance for Bayesian Nash Equilibrium

• The common prior assumption is crucial for the concept of Bayesian Nash equilibrium, as it allows players to form consistent beliefs and make decisions based on those beliefs
• Without a common prior, players may have inconsistent or incompatible beliefs about the distribution of types, which can affect the analysis and outcomes of the game

Relaxing the Common Prior Assumption

• In some cases, the common prior assumption may not hold, leading to situations where players have inconsistent or incompatible beliefs about the distribution of types
• Relaxing the common prior assumption can lead to alternative solution concepts, such as subjective equilibria or self-confirming equilibria, which allow for players to have different beliefs that are consistent with their observed outcomes
• Example: In a game with private information about players' abilities, if players have different prior beliefs about the distribution of abilities, their choices may not align with a Bayesian Nash equilibrium based on a common prior