The extensive form is a comprehensive representation of a game, employing a tree structure to display decision nodes, strategies, information sets, and pay-offs. This representation allows for deeper insights into certain games beyond what is provided by the pay-off matrix.
Historical Context
The extensive form representation of games was developed as part of the foundational work in game theory, notably by John von Neumann and Oskar Morgenstern in their seminal book “Theory of Games and Economic Behavior” published in 1944. This representation has become a crucial tool in the analysis and understanding of strategic interactions in various fields.
Types/Categories of Extensive Form
- Perfect Information Games: Games where all players are fully informed about all previous actions.
- Imperfect Information Games: Games where some or all players do not have complete information about previous actions.
- Simultaneous-Move Games: Games where players make decisions without knowing the choices of other players.
Key Events
- 1944: Publication of “Theory of Games and Economic Behavior” by von Neumann and Morgenstern.
- 1950s-1960s: Further development and refinement of game theory concepts by scholars such as John Nash.
- 1994: John Nash, John Harsanyi, and Reinhard Selten awarded the Nobel Prize in Economic Sciences for their contributions to game theory.
Detailed Explanations
Decision Nodes
A decision node represents a point in the game where a player must choose an action. Each decision node branches out to subsequent nodes or terminal nodes, representing different outcomes.
Strategies
A strategy in the extensive form game refers to a complete plan of action for a player, detailing what they will do at each decision node they encounter.
Information Sets
An information set groups decision nodes that a player cannot distinguish between, indicating that the player has the same information at each node within the set.
Pay-offs
The pay-off is the outcome each player receives at the end of the game. It can be represented in terms of utility, profits, or other relevant metrics depending on the context of the game.
Mathematical Models
In extensive form games, mathematical representation often involves the following components:
- \( (N, H, P, {u_i}) \) where \(N\): Set of players, \(H\): Set of nodes, \(P\): Player function, and \({u_i}\): Utility functions.
- Utility Function: \( u_i : Z \rightarrow \mathbb{R} \) where \( Z \) is the set of terminal nodes, giving a numerical value to each outcome for player \( i \).
Charts and Diagrams
Using Mermaid syntax, an example of an extensive form game tree is illustrated below:
graph TD;
A(Start) --> B[Player 1]
B --> C[Player 2, Action 1]
B --> D[Player 2, Action 2]
C --> E[Pay-off (3, 2)]
C --> F[Pay-off (1, 4)]
D --> G[Pay-off (2, 3)]
D --> H[Pay-off (4, 1)]
Importance and Applicability
The extensive form is essential in analyzing and understanding sequential and strategic interactions in various domains such as economics, political science, and evolutionary biology. It provides a visual and structured method to evaluate different strategies and potential outcomes.
Examples
- Chess: Each move represents a decision node, with strategies and pay-offs based on the game’s progress and final outcome.
- Bargaining: Analyzing negotiations where offers and counteroffers are made sequentially.
- Business Strategy: Firms making sequential investment decisions based on competitors’ actions.
Considerations
When constructing extensive form representations, it is crucial to account for:
- Completeness of the information set.
- Accurate representation of possible actions and outcomes.
- Realistic pay-offs reflecting the preferences of players.
Related Terms
- Nash Equilibrium: A strategy profile where no player can benefit by unilaterally changing their strategy.
- Subgame Perfect Equilibrium: A refinement of Nash Equilibrium applicable to extensive form games ensuring rationality at every decision node.
- Game Tree: A graphical representation of the sequential structure of the game.
Comparisons
- Extensive Form vs. Normal Form: The extensive form represents games as trees with sequential decision-making, while the normal form represents games in a matrix format summarizing strategies and pay-offs.
Interesting Facts
- Game Theory’s Role: Game theory has applications in various fields including economics, biology, and computer science, significantly impacting our understanding of competitive and cooperative behavior.
Inspirational Stories
- John Nash: Overcame schizophrenia to contribute seminal work in game theory, as depicted in the movie “A Beautiful Mind.”
Famous Quotes
- “The only thing that is certain about game theory is that it is complex.” – John von Neumann
Proverbs and Clichés
- “Life is a game, and true love is a trophy.” – Rufus Wainwright
Jargon and Slang
- Node: A point in the game tree where decisions are made.
- Branch: Represents possible actions leading to different nodes or outcomes.
FAQs
Q: What is the extensive form of a game?
Q: How does the extensive form differ from the normal form?
Q: Why is the extensive form useful?
References
- Von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
- Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences.
- Myerson, R. B. (1991). Game Theory: Analysis of Conflict. Harvard University Press.
Summary
The extensive form is a pivotal representation in game theory, capturing the essence of sequential decision-making and strategic interactions. By mapping out decisions, strategies, information sets, and pay-offs in a tree structure, it provides profound insights into the dynamics of various games. Understanding the extensive form not only enhances one’s grasp of game theory but also equips individuals to analyze and navigate complex strategic scenarios in multiple fields.
