8.1 Sequential games with incomplete information
3 min read•august 7, 2024
with add complexity to decision-making. Players take turns acting without full knowledge of the game's structure or other players' types. This uncertainty forces them to rely on and .
As the game progresses, players update their beliefs using . This process allows them to incorporate new information from observed actions, refining their strategies and adapting to the evolving game state.
Sequential Games with Incomplete Information
Game Structure and Information
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- Sequential games involve players taking turns making decisions, with each player's move influenced by the previous player's actions
- Incomplete information refers to situations where players do not have full knowledge about the game's structure, payoffs, or other players' types (preferences, abilities, or beliefs)
- Information sets group together decision nodes that are indistinguishable to a player, representing their lack of perfect information at that point in the game (poker)
- Beliefs represent a player's subjective probabilities about which decision node within an information set they are currently at, based on the available information and the other players' strategies
Key Concepts and Terminology
- The term "sequential" emphasizes the step-by-step nature of the game, with players acting one after another rather than simultaneously
- "Incomplete information" captures the uncertainty players face about various aspects of the game, which can significantly impact their decision-making process
- Information sets are a crucial concept for modeling the limited information available to players at different stages of the game (card games, auctions)
- Beliefs play a central role in guiding players' choices, as they must rely on their assessment of the likelihood of different scenarios given the incomplete information
Player Types and Strategies
Type Space and Probability
- refers to the set of possible types for each player, which capture the or characteristics that are relevant to the game (risk preferences, skill levels)
- The type space is equipped with a , assigning each type a likelihood of occurring
- allows players to update their beliefs about other players' types based on observed actions, using Bayes' rule to incorporate new information
Strategies and Profiles
- In games with incomplete information, a player's strategy must specify their action at each information set, contingent on their type
- A is a collection of strategies, one for each player, that fully specifies how the game will be played out given the realization of player types
- The concept of strategy profiles is essential for analyzing the equilibria of the game, as it allows for the evaluation of each player's best response to the strategies of others
Belief Updating
Bayesian Updating Process
- Bayesian updating is a rational process for revising beliefs in light of new evidence, following the principles of conditional probability
- Players start with prior beliefs about the likelihood of different types or states of nature, based on their initial information
- As the game progresses and players observe actions or signals, they use Bayes' rule to compute posterior beliefs, which combine the prior beliefs with the new information (medical diagnosis, stock market trading)
- The updated beliefs then inform the players' subsequent decisions, allowing them to adapt their strategies to the evolving information structure of the game
Importance of Belief Updating
- Belief updating is a key aspect of dynamic games with incomplete information, enabling players to learn and adjust their behavior over time
- Properly incorporating new information through Bayesian updating allows players to make more informed decisions and improve their expected payoffs
- The process of belief updating can lead to interesting strategic considerations, such as signaling and reputation-building, as players try to influence the beliefs of others through their actions (entry deterrence, bargaining)
- Understanding how beliefs evolve is crucial for predicting the outcome of the game and analyzing the incentives of players at different stages
Key Terms to Review (20)
Asymmetric information: Asymmetric information refers to a situation where one party in a transaction has more or better information than the other party, leading to an imbalance in decision-making. This imbalance can create problems such as adverse selection and moral hazard, affecting how parties interact in strategic scenarios. Understanding asymmetric information is crucial for analyzing behavior in situations with incomplete information, negotiation dynamics, and designing mechanisms that ensure efficient outcomes.
Auction Theory: Auction theory studies how people bid in auctions and how these bidding processes affect the allocation of goods and services. This theory explores various auction formats, bidder strategies, and outcomes, providing insights into decision-making under competition and uncertainty. It connects to key concepts such as incomplete information and strategic interactions, making it essential for understanding economic behaviors in competitive markets.
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from the end to the beginning. It involves looking at the last possible moves of players and determining their best responses, then moving sequentially backward through the game tree to deduce the optimal actions of earlier moves. This technique is particularly relevant in analyzing strategic interactions in sequential games and helps in identifying subgame perfect equilibria.
Bayesian updating: Bayesian updating is a statistical technique used to revise existing beliefs or probabilities based on new evidence or information. This approach allows individuals to adjust their initial beliefs, or prior probabilities, in light of new data to form updated beliefs, or posterior probabilities. It plays a critical role in decision-making under uncertainty and is foundational for understanding concepts like beliefs, sequential games, and equilibrium in strategic interactions.
Beliefs: In game theory, beliefs refer to the subjective probabilities that players assign to the types or actions of other players in a strategic setting. These beliefs are crucial as they influence players' decisions and strategies, especially in scenarios where information is incomplete or uncertain. Understanding beliefs allows players to formulate expectations about others' behavior, impacting the overall outcome of the game.
Conditional Probability: Conditional probability is the likelihood of an event occurring given that another event has already occurred. It plays a crucial role in understanding how information updates our beliefs about the world, especially in situations where information is incomplete, such as in sequential games where players make decisions based on what they know at different stages.
Incentive Compatibility: Incentive compatibility is a condition in which individuals have the motivation to act in accordance with the rules of a mechanism or system, ensuring that truthfully revealing their private information is in their best interest. This concept is crucial for designing systems where participants are expected to make choices based on their private information, as it guarantees that those choices align with the intended outcomes. The effectiveness of various strategic interactions and mechanisms hinges on this principle, making it a foundational concept in many economic models.
Incomplete Information: Incomplete information refers to a situation in a game where players do not have perfect knowledge about the other players' characteristics, strategies, or payoffs. This lack of information influences how players form strategies and make decisions, leading to uncertainties in predictions about opponents' behavior and outcomes in various interactions.
Information Sets: Information sets are collections of decision nodes in game theory that represent a player's knowledge at a given point in the game. They are crucial for understanding situations of imperfect information, as they help distinguish between different possible scenarios that a player might face, influencing their strategy and decisions. When players cannot see each other's choices or payoffs, these sets enable them to formulate expectations based on available information.
Job market signaling: Job market signaling refers to the actions taken by individuals to convey their qualities or abilities to potential employers, often through educational credentials or work experiences. This signaling helps to reduce information asymmetry in the job market, where employers may not have complete knowledge about a candidate's true capabilities. By providing signals, candidates can influence hiring decisions and earn a competitive advantage.
Payoff matrices: Payoff matrices are tools used in game theory to represent the outcomes of different strategies chosen by players in a game, allowing for visual comparison of payoffs associated with each strategy combination. These matrices display the gains or losses for each player based on the strategies they choose, making it easier to analyze strategic interactions and decisions. In sequential games with incomplete information, payoff matrices help clarify how players make decisions when they lack knowledge about the other players' types or strategies.
Private information: Private information refers to knowledge or data that is not shared openly among all participants in a game or economic scenario, leading to situations of incomplete information. This concept is vital for understanding how players make decisions based on their beliefs about others' actions or types, which can influence the strategies they choose. In contexts with private information, individuals must often rely on Bayesian reasoning and signals to infer the unknown aspects of others’ preferences or actions.
Probability Distribution: A probability distribution is a mathematical function that describes the likelihood of different outcomes in a random variable. It provides a complete description of the probabilities associated with each possible outcome, whether in discrete or continuous form. This concept is crucial for understanding decision-making under uncertainty, especially in games where players must consider mixed strategies, make decisions based on beliefs about other players' types, and handle situations involving sequential moves with incomplete information.
Revelation Principle: The revelation principle is a key concept in mechanism design that asserts if a desired outcome can be achieved through a game with incomplete information, then there exists a direct mechanism where players truthfully reveal their private information. This principle helps simplify the analysis of strategic interactions by transforming complex games into more manageable forms where honesty is incentivized.
Sequential games: Sequential games are a type of game in game theory where players make decisions one after another, rather than simultaneously. In these games, the order of moves matters and players can take into account the previous actions of others when deciding their next move. This structure allows for strategies that can exploit the timing of decisions, leading to various outcomes based on the knowledge and expectations players have about each other's actions.
Signaling Theory: Signaling theory is a concept in economics and game theory that explains how individuals or organizations convey information about themselves to influence the actions of others in situations where information is asymmetric. It plays a crucial role in decision-making processes, especially in contexts where one party has more or better information than the other, leading to potential misinterpretations or misalignments in expectations.
Strategy Profile: A strategy profile is a complete description of the strategies chosen by each player in a game, detailing what action each player will take in every possible situation they may encounter. This concept is crucial as it helps analyze the overall outcome of strategic interactions and assists in determining the equilibrium points within various game formats. Understanding strategy profiles is essential for evaluating decision-making processes, especially when converting between different game representations or analyzing equilibria in games with incomplete information.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.
Type space: A type space is a concept used in game theory to represent the different types of players in a game, particularly when there is incomplete information about their characteristics or preferences. It helps to model scenarios where players have private information, allowing them to possess different beliefs and strategies based on their type. This framework is essential for understanding how players form expectations and make decisions in settings like Bayesian games and sequential games with incomplete information.
Utility Functions: Utility functions are mathematical representations that describe how individuals or agents prioritize and rank their preferences over different outcomes or bundles of goods. These functions help to quantify satisfaction or happiness derived from consuming goods and services, which is essential in analyzing decision-making processes in various contexts, including strategic interactions and bargaining scenarios.