How to Identify Information Sets in Extensive Form Games

Understanding information sets in game theory is fundamental for analyzing strategic interactions in extensive form games where players move sequentially. In contrast, normal form games, which are used to describe simultaneity in decision-making, do not involve the concept of information sets. This article delves into the intricacies of identifying information sets, their significance, and how to use them effectively in game analysis.

Introduction to Extensive Form Games

In game theory, the extensive form is a way of representing a game that follows a sequence of moves. It is particularly useful for analyzing strategic interactions in situations where players have perfect information about the previous moves. The extensive form is typically represented as a game tree, where each node represents a decision point or information set, and edges represent the possible actions that can be taken.

The Concept of Information Sets

Information sets in extensive form games are crucial because they capture the player's uncertainty about the current state of the game. An information set is a collection of nodes (decision points) that a player cannot distinguish between. This is because the player does not have perfect information about the sequence of moves that have occurred leading up to these nodes. The concept is particularly important in games with incomplete information, where players are not sure about the strategies or types of other players.

Identifying Information Sets

To identify information sets in an extensive form game, follow these steps:

Label each decision node: Start by labeling each decision node in the game tree. Each node represents a decision point where a player must choose a move. Mark indistinguishable nodes: Identify nodes that cannot be distinguished by the current player. These nodes are indistinguishable because the player has no information about the past moves that have led to these nodes. Mark these nodes with the same label to indicate they form an information set. Verify player's information: Ensure that the player cannot determine the exact history that led to the current state of the game from their viewpoint. If such indistinguishability exists, nodes form an information set.

For example, in a game where a player does not know the exact sequence of moves leading to their current decision point, the nodes at that point would form an information set. This information set would represent the player's uncertainty about the game's history.

Significance of Information Sets

Information sets are significant in game theory because they help in defining player strategies and expected payoffs. By understanding information sets, players can better anticipate their opponents' moves and make more informed decisions. In particular, information sets are crucial in ensuring players adhere to their strategies as specified in the game, even when they are unsure of the exact sequence of moves.

For instance, in a poker game, a player's information set might include all the possible hands they believe their opponent could have. This helps the player to make better decisions about which moves to take, even when they do not have complete information about the game. In fact, many advanced strategic models used in poker rely heavily on the concept of information sets to analyze the game's complex dynamics.

Comparison with Normal Form Games

Normal form games, on the other hand, use matrices to represent the strategic interactions between players. Each player's strategy is a mix of pure and mixed strategies, and the payoffs are described in a table. In normal form games, there is no concept of sequential moves or information sets because all players make their decisions simultaneously and have perfect information about the game's structure.

Because normal form games do not involve the concept of information sets, the analysis of such games focuses more on pure and mixed strategies, as well as the Nash equilibria. In contrast, in extensive form games, the concept of information sets is used to analyze imperfect information and how it influences player behavior and the expected payoffs.

Conclusion

Understanding information sets in extensive form games is essential for game theorists, economists, and strategists who analyze strategic interactions where players make decisions sequentially. While normal form games provide a way to analyze simultaneous moves with perfect information, the concept of information sets is critical in extensive form games where players have uncertainty about the game's history.

Mastering the identification of information sets allows players to better understand and predict their opponents' behaviors and to develop more effective strategies. By applying these concepts, one can gain a deeper insight into the dynamics of strategic games and improve their decision-making in various real-world scenarios.