Extensive Games with Possibly Unaware Players (0704.2014v1)

Abstract: Standard game theory assumes that the structure of the game is common knowledge among players. We relax this assumption by considering extensive games where agents may be unaware of the complete structure of the game. In particular, they may not be aware of moves that they and other agents can make. We show how such games can be represented; the key idea is to describe the game from the point of view of every agent at every node of the game tree. We provide a generalization of Nash equilibrium and show that every game with awareness has a generalized Nash equilibrium. Finally, we extend these results to games with awareness of unawareness, where a player i may be aware that a player j can make moves that i is not aware of, and to subjective games, where payers may have no common knowledge regarding the actual game and their beliefs are incompatible with a common prior.

Overview of Extensive Games with Possibly Unaware Players

The paper explores a significant extension to traditional game-theory models by introducing the concept of extensive games with players who may lack awareness of certain actions or other features within the game. Typical assumptions in game theory, such as full awareness and common knowledge among players, are relaxed, allowing for scenarios where players may be unaware of certain moves or strategies available to themselves or others, thereby impacting strategic decision-making. The authors offer a formalized representation for modeling awareness using sets of augmented games based on a shared extensive-form game. This representation seeks to encapsulate the various degrees of awareness each player possesses at different stages of the game, leading to a generalization of the Nash equilibrium that accommodates unawareness.

Key Contributions

1. Augmented Games Framework: The paper introduces a structure for modeling varying levels of awareness among players using augmented extensive games. Each augmented game reflects the awareness state of a player, capturing not only known actions but also the changes in awareness over the course of the game. This is crucial for dealing with players who gain awareness during gameplay, influencing their strategic choices.
2. Generalized Nash Equilibrium: The authors generalize the concept of Nash equilibrium to handle games where players have differential awareness. In this framework, each player's strategy is considered optimal based on their subjective view of the game—taking into account their current awareness level. This equilibrium ensures that even when players are unaware of certain moves or outcomes, they make decisions consistent with their beliefs about the game and other players' strategies.
3. Incorporation of Awareness of Unawareness: Building on previous models, the paper extends the augmented games framework to account for players being aware of their unawareness—recognizing that they might not know all possible game moves. This allows for a more nuanced and realistic depiction of strategic decision-making in complex environments like technological warfare or financial markets where unknown strategies might exist.
4. Consideration of Lack of Common Knowledge: The paper challenges the typical assumption of common knowledge about the game among players. By weakening conditions related to awareness, this model allows players to hold conflicting beliefs or misunderstandings about game structure and payoffs, representing scenarios where societal or cultural differences influence awareness.

Practical and Theoretical Implications

The theoretical construct provided by the authors has profound implications. In practical terms, it is adaptative for real-world situations involving asymmetric information, such as consumer markets with obscure product features or financial systems where some agents use unconventional strategies. The framework also prompts reconsideration of strategic interactions in situations where players might not have perfect information or complete cognitive resources to compute outcomes—such as in protracted chess games, where players might struggle to analyze all potential move consequences.

The generalized Nash equilibrium model might pave the way for more sophisticated AI systems in gaming and economics, allowing them to tailor strategies based on diverse levels of player awareness. Future research could explore advanced computational techniques for determining equilibria in these settings, addressing scenarios with extensive unawareness and large player sets. Moreover, nuanced models of unawareness could enable AI and game theory to better align with real-world human decision-making, where strategic foresight and awareness are often limited.

Conclusion

The generalization of extensive games to include possibly unaware players represents a significant advance in game theory, challenging longstanding assumptions of common knowledge and awareness. By constructing a more flexible framework that incorporates varying levels of awareness and potential unawareness, the paper extends the applicability of game-theoretic analyses to more realistic and complex decision-making environments. This work creates pathways for further exploration into understanding strategic interactions where players operate under incomplete or incorrect information, generating insights into both human behavior and artificial intelligence systems operating in uncertain environments.