🎱Game Theory Unit 8 – Imperfect and Incomplete Information Games

Key Concepts

• Imperfect information games involve situations where players do not have complete knowledge of the actions taken by other players
• Incomplete information games refer to scenarios where players lack information about certain aspects of the game, such as the payoffs or strategies available to other players
• Information sets represent the different possible states of the game that a player cannot distinguish between when making a decision
• Beliefs and probabilities play a crucial role in decision-making under uncertainty, as players must assign probabilities to different possible scenarios
• Bayesian equilibrium is a solution concept used in games with incomplete information, where players update their beliefs based on observed actions and maximize their expected payoffs
• Signaling and screening are mechanisms used by players to convey or obtain information about hidden characteristics or intentions
• Real-world applications of imperfect and incomplete information games include auctions, bargaining, insurance markets, and political campaigns

Types of Imperfect Information Games

• Simultaneous move games, such as the Prisoner's Dilemma or Battle of the Sexes, where players make decisions simultaneously without knowing the choices of other players
• Sequential games with imperfect information, like the Signaling Game or the Screening Game, involve players making moves in a specific order but without complete knowledge of previous actions
• Repeated games with imperfect monitoring, where players engage in multiple rounds of interaction but cannot perfectly observe the actions of other players in each round
• Stochastic games with imperfect information introduce random elements or chance moves that affect the game's outcome, and players have incomplete knowledge of these random events
• Extensive form games with imperfect information represent the game as a decision tree, with information sets indicating the different possible states that a player cannot distinguish between
  • Example: In a poker game, players do not have perfect information about the cards held by their opponents, leading to imperfect information sets at each decision point

Incomplete Information Games Explained

• Incomplete information games involve situations where players lack knowledge about certain aspects of the game, such as the payoffs, strategies, or types of other players
• Types refer to the different possible characteristics or attributes of players that are not directly observable, such as their preferences, abilities, or private information
• Harsanyi transformation is a technique used to convert an incomplete information game into an imperfect information game by introducing a chance move that determines the types of players
• Beliefs and updating involve players forming initial beliefs about the types of other players and updating these beliefs based on observed actions or signals
• Expected utility maximization under incomplete information requires players to make decisions based on their beliefs and the expected payoffs associated with different actions
• Private value auctions, such as sealed-bid auctions, are examples of incomplete information games where players have different valuations for the item being auctioned, and these valuations are private information
• Adverse selection and moral hazard are common problems in incomplete information settings, where one party has private information that can lead to inefficient outcomes or strategic behavior

Strategies and Decision-Making

• Pure strategies specify a single action to be taken in each possible situation or information set of the game
• Mixed strategies involve players randomizing over multiple actions according to a probability distribution, allowing for more flexible and unpredictable behavior
• Behavioral strategies are used in extensive form games and specify a probability distribution over actions at each information set, rather than a complete plan of action
• Beliefs and updating are crucial for decision-making in imperfect and incomplete information games, as players must form and revise their beliefs about the likely actions or types of other players
• Bayes' rule is used to update beliefs based on observed actions or signals, allowing players to make more informed decisions
• Expected utility calculation involves weighing the payoffs associated with different actions by the probabilities of those actions occurring, based on the player's beliefs
• Rationalizability is a solution concept that eliminates strategies that are never best responses to any belief about the strategies of other players, narrowing down the set of plausible strategies

Bayesian Games

• Bayesian games are a class of games with incomplete information, where players have private information about their own types but uncertainty about the types of other players
• Type spaces define the possible types of each player and the probability distribution over these types, capturing the incomplete information aspect of the game
• Common prior assumption states that all players start with the same initial beliefs about the probability distribution over types, before receiving their private information
• Bayesian Nash equilibrium is an extension of Nash equilibrium to Bayesian games, where each player's strategy maximizes their expected payoff given their beliefs about the types and strategies of other players
  • In a Bayesian Nash equilibrium, players' beliefs are consistent with the strategies being played, and no player has an incentive to deviate from their strategy given their beliefs
• Revelation principle suggests that any Bayesian Nash equilibrium outcome can be achieved through a direct revelation mechanism, where players truthfully report their types to a central planner
• Auction theory heavily relies on Bayesian games to analyze bidding behavior and optimal auction design, considering the incomplete information about bidders' valuations

Signaling and Screening

• Signaling games model situations where an informed player (sender) takes an action to convey information about their type to an uninformed player (receiver), who then takes an action based on the signal
  • Example: In a job market signaling game, a job candidate (sender) may obtain a higher education degree to signal their ability to potential employers (receivers)
• Screening games involve an uninformed player (principal) designing a contract or mechanism to induce informed players (agents) to reveal their types through self-selection
  • Example: In an insurance market screening game, an insurance company (principal) may offer different coverage plans to separate high-risk and low-risk individuals (agents)
• Separating equilibrium occurs when different types of senders choose different actions, allowing the receiver to perfectly infer the sender's type based on the observed action
• Pooling equilibrium arises when all types of senders choose the same action, making it impossible for the receiver to distinguish between types based on the observed action
• Costly signaling refers to situations where the signaling action is more costly for some types than others, allowing for credible information transmission
• Incentive compatibility ensures that agents have an incentive to truthfully reveal their types in a screening mechanism, as lying would not lead to a better outcome for them

Real-World Applications

• Auctions: Imperfect and incomplete information games are prevalent in auction settings, where bidders have private valuations and uncertainty about the valuations of other bidders
  • Example: In a sealed-bid first-price auction, bidders submit their bids without knowing the bids of others, leading to incomplete information and strategic bidding behavior
• Bargaining and negotiation: Bargaining situations often involve imperfect information about the preferences and reservation prices of the parties involved, leading to strategic behavior and potential inefficiencies
  • Example: In a labor union negotiation, the union and the employer may have incomplete information about each other's willingness to compromise, affecting the bargaining process
• Insurance markets: Adverse selection and moral hazard are common problems in insurance markets, arising from the incomplete information about the risk types and actions of insured individuals
  • Example: In a health insurance market, individuals have private information about their health status, leading to potential adverse selection if high-risk individuals are more likely to purchase insurance
• Political campaigns and lobbying: Political actors often have incomplete information about the preferences and strategies of their opponents, leading to strategic behavior and signaling through campaign promises or lobbying efforts
  • Example: In a political campaign, candidates may make promises or take positions to signal their type (e.g., being pro-environment) to voters who have incomplete information about their true intentions
• Financial markets: Imperfect and incomplete information can lead to information asymmetries and strategic behavior in financial markets, such as insider trading or market signaling
  • Example: In a corporate takeover scenario, the acquiring firm may have incomplete information about the target firm's true value, leading to potential overbidding or underbidding

Advanced Topics and Extensions

• Repeated games with imperfect monitoring: Analyzing the effects of imperfect information in repeated interactions, such as collusion in oligopolies or cooperation in social dilemmas
• Evolutionary game theory and learning: Investigating how players adapt and learn in games with imperfect and incomplete information, using concepts like replicator dynamics and reinforcement learning
• Mechanism design with incomplete information: Designing optimal mechanisms (e.g., auctions, voting systems) that incentivize truthful revelation of private information and achieve desired outcomes
• Robust mechanism design: Developing mechanisms that perform well even in the presence of imperfect or incomplete information about players' beliefs or behavior
• Higher-order beliefs and interactive epistemology: Exploring the role of players' beliefs about the beliefs of others, and the implications for strategic reasoning and equilibrium concepts
• Global games and coordination: Analyzing coordination problems in the presence of imperfect information, using tools like global game theory to study equilibrium selection and efficiency
• Experiments and empirical studies: Conducting laboratory experiments or field studies to test the predictions of imperfect and incomplete information game models and investigate actual human behavior in these settings