Information sets and imperfect information are key concepts in game theory. They help us understand how players make decisions when they don't have all the facts. This topic builds on earlier ideas about game representation, showing how uncertainty shapes strategies.

In real life, we often act without knowing everything. Game theory models this using information sets and imperfect information. These tools let us analyze complex situations like poker games or business competition, where hidden info is crucial.

Information Sets and Game Structures

Representing Player Knowledge in Games

• Information set represents the knowledge a player has about the game at a particular decision point
  • Includes all decision nodes that are indistinguishable to the player given their current information
  • Nodes within an information set are connected by a dashed line (bridge)
• Imperfect information occurs when a player does not have complete knowledge of the game's history when making a decision
  • Results in information sets containing multiple nodes
  • Common in card games (poker) where players are uncertain about opponents' cards
• Simultaneous moves in sequential games can be modeled using information sets
  • Players act without knowing the actions chosen by others
  • Represented by connecting the decision nodes of each player with a dashed line
  • Occurs in situations like sealed-bid auctions or simultaneous investment decisions

Game Structures and Information

• Perfect information games have information sets containing only one node each
  • Players have complete knowledge of the game's history at every decision point
  • Examples include chess and tic-tac-toe
• Imperfect information games have at least one information set with multiple nodes
  • Players face uncertainty about the game's history when making decisions
  • Occurs in many real-world situations (business competition, international relations)
• The structure of information sets impacts the strategic considerations and equilibrium outcomes of a game
  • Players must consider the possible actions and beliefs of opponents under imperfect information
  • Strategies must account for the uncertainty and potential for strategic misrepresentation

Incomplete and Hidden Information

Hidden Information in Games

• Hidden information refers to situations where players have private knowledge that is not shared with others
  • Can include information about a player's type, preferences, or available actions
  • Creates informational asymmetries between players
• Examples of hidden information in games:
  • In a job market signaling game, a worker's true ability is hidden information to employers
  • In a bargaining game, a player's true valuation of the item being divided may be hidden
• Hidden information can lead to strategic behavior such as signaling, screening, or bluffing
  • Players may take actions to reveal or conceal their private information
  • Equilibrium outcomes can be influenced by the presence and nature of hidden information

Incomplete Information and Uncertainty

• Incomplete information occurs when players are uncertain about some aspect of the game itself
  • Can include uncertainty about the payoff structure, available strategies, or the types of other players
  • Differs from imperfect information, which relates to uncertainty about the history of moves
• Uncertainty can arise from various sources:
  • Nature's moves or chance events (weather conditions, economic shocks)
  • Lack of knowledge about other players' characteristics or preferences
• Incorporating incomplete information often requires extending the game model
  • Bayesian games introduce the concept of player types and beliefs over types
  • Players update their beliefs based on observed actions and use expected payoffs to make decisions
• Incomplete information can lead to complex strategic considerations and equilibrium concepts
  • Players must reason about the likely types and actions of others
  • Solution concepts such as Bayesian Nash equilibrium or perfect Bayesian equilibrium are used