8.1 Sequential games with incomplete information

Sequential games with incomplete information 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 beliefs and information sets.

As the game progresses, players update their beliefs using Bayesian updating. 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

• Type space refers to the set of possible types for each player, which capture the private information or characteristics that are relevant to the game (risk preferences, skill levels)
• The type space is equipped with a probability distribution, assigning each type a likelihood of occurring
• Conditional probability 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 strategy profile 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