🆚Game Theory and Economic Behavior Unit 8 – Dynamic Games: Perfect Bayesian Equilibrium

Key Concepts

• Dynamic games involve sequential decision-making where players move in a specific order and have information about previous moves
• Players in dynamic games have beliefs about the other players' types or strategies based on their observed actions
• Perfect Bayesian Equilibrium (PBE) is a solution concept for dynamic games with incomplete information
• PBE requires players' strategies to be sequentially rational given their beliefs at each information set
• Beliefs in PBE are determined by Bayes' rule whenever possible and are consistent with the players' equilibrium strategies
• Subgame perfection is a refinement of Nash equilibrium used in dynamic games of complete information
• Backward induction is a technique for solving dynamic games by starting at the end and working backwards to determine optimal strategies
• Information sets represent the knowledge a player has when making a decision at a particular point in the game

Foundations of Dynamic Games

• Dynamic games are represented using game trees that specify the order of moves, players' choices, and payoffs
• Players have perfect recall, meaning they remember their own past actions and observations
• Information sets partition the nodes in the game tree where a player has the same information and available actions
• Strategies in dynamic games specify a player's action at each information set
• Mixed strategies assign probabilities to actions at each information set
• Extensive form represents the complete structure of a dynamic game, including the game tree, information sets, and payoffs
• Dynamic games can have perfect or imperfect information depending on whether players observe all previous moves
• Subgames are parts of the game tree that can be treated as separate games starting from a specific node

Perfect Bayesian Equilibrium (PBE) Explained

• PBE is an equilibrium concept for dynamic games with incomplete information where players have beliefs about others' types
• In PBE, players' strategies are optimal given their beliefs, and beliefs are consistent with the strategies being played
• PBE requires specifying both the players' strategies and their beliefs at each information set
• Sequential rationality means that players' strategies are optimal at each information set, given their beliefs
• Consistency requires that beliefs are updated using Bayes' rule whenever possible, based on the observed actions
• PBE is a refinement of Bayesian Nash Equilibrium (BNE) that incorporates sequential rationality and consistency
• In PBE, players' beliefs about others' types may change as the game progresses and new information is revealed
• PBE can involve pooling equilibria where different types take the same action, or separating equilibria where types take different actions

Components of PBE

• Players' types represent their private information or characteristics that affect their payoffs or available actions
• Prior beliefs specify the initial probabilities assigned to each player's possible types before the game begins
• Posterior beliefs are updated probabilities of types after observing actions, calculated using Bayes' rule when possible
• Players' strategies specify their actions at each information set for each possible type
• Payoffs represent the utility or rewards players receive at the end of the game based on the actions taken and types realized
• Information sets partition the nodes where a player has the same information and available actions
  • They can be singleton (containing a single node) or non-singleton (containing multiple nodes)
• Bayes' rule is used to update beliefs about types based on observed actions and the prior probabilities of types
  • It is applied when the observed action is consistent with the equilibrium strategies for some types

Strategies and Beliefs in PBE

• Strategies in PBE specify a player's action at each information set for each possible type they might have
• Beliefs in PBE assign probabilities to the nodes within each information set, representing the likelihood of being at each node
• Off-path beliefs are assigned to information sets reached after observing actions that should not occur in equilibrium
  • These beliefs must still be consistent with the players' strategies and Bayes' rule when possible
• Beliefs must be consistent with the strategies being played, meaning they are derived from the prior probabilities and equilibrium strategies
• Players' strategies must be sequentially rational, maximizing their expected payoff at each information set given their beliefs
• In pooling equilibria, different types choose the same action, making it impossible to infer the type from the observed action
• In separating equilibria, different types choose different actions, allowing beliefs to be updated accurately using Bayes' rule
• Semi-separating equilibria involve some types playing mixed strategies, partially revealing information about their types

Solving PBE Problems

• Begin by specifying the game tree, including players, types, actions, and payoffs
• Assign prior probabilities to each player's types based on the initial information
• Identify the information sets for each player and the available actions at each information set
• Start at the end of the game tree and work backwards, determining optimal actions for each type at each information set
  • This process is similar to backward induction but incorporates beliefs about types
• At each information set, calculate the expected payoff for each action based on the beliefs and the optimal actions in future subgames
• Choose the action that maximizes the expected payoff at each information set for each type
• Update beliefs using Bayes' rule whenever possible, based on the observed actions and the prior probabilities of types
• Check for consistency between the calculated strategies and beliefs
  • Ensure that the beliefs are derived from the strategies and that the strategies are optimal given the beliefs
• Verify that no player has an incentive to deviate from their strategy at any information set, given their beliefs

Applications and Examples

• PBE is widely used in economics, political science, and other fields to analyze strategic interactions with incomplete information
• Signaling games, such as job market signaling and advertising, involve players sending costly signals to reveal their types
  • In a job market signaling game, workers can signal their ability through education, and employers update their beliefs based on the education level
• Reputation games, such as the chain store paradox, involve players trying to establish or maintain a reputation over repeated interactions
  • In the chain store paradox, an incumbent firm can deter entry by establishing a reputation for aggressive behavior
• Bargaining games with incomplete information, such as the ultimatum game, involve players negotiating over the division of a resource
  • In the ultimatum game, the proposer's offer can signal their beliefs about the responder's minimum acceptable offer
• Principal-agent problems, such as moral hazard and adverse selection, involve designing contracts to incentivize agents with private information
  • In a moral hazard problem, the principal can design a contract that induces the agent to choose the desired action based on their unobserved type

Common Pitfalls and Misconceptions

• Failing to specify beliefs at all information sets, including those off the equilibrium path
• Assigning arbitrary beliefs that are not consistent with the strategies being played
• Updating beliefs using Bayes' rule when the observed action is not consistent with any type's equilibrium strategy
• Ignoring the sequential rationality requirement and choosing strategies that are not optimal given the beliefs at each information set
• Confusing PBE with other solution concepts, such as subgame perfect equilibrium or Bayesian Nash equilibrium
• Assuming that players always have an incentive to reveal their true types through their actions
  • In some cases, players may benefit from concealing or misrepresenting their types
• Neglecting to consider off-path beliefs and their impact on the equilibrium strategies
• Focusing solely on pure strategy equilibria and overlooking the possibility of mixed strategy or semi-separating equilibria