Historical Context
Cooperative games are a subset of game theory, a field that delves into strategic interactions among rational decision-makers. The concept became prominent with the development of game theory in the 20th century, largely attributed to the works of John von Neumann and Oskar Morgenstern. Their seminal book, “Theory of Games and Economic Behavior” (1944), laid the groundwork for the mathematical study of cooperative behavior in games.
Types and Categories
Cooperative games can be broadly categorized into:
- Coalition Games: Players can form coalitions, and the value of the coalition depends on the collective strategy and effort.
- Bargaining Games: Players negotiate the division of a fixed set of resources or payoffs.
- Market Games: Players represent different market participants, collaborating to maximize collective benefits.
Key Events in Cooperative Game Theory
- 1944: Publication of “Theory of Games and Economic Behavior” by von Neumann and Morgenstern.
- 1953: Introduction of the concept of the Shapley value by Lloyd Shapley.
- 1960s-1980s: Development of the Nash bargaining solution by John Nash and subsequent extensions.
Detailed Explanations
Mathematical Models and Formulas
-
Characteristic Function: Defines the worth of a coalition.
$$ v(S) $$where \( S \) is any subset of players (coalition) and \( v \) denotes the value function. -
The Shapley Value: A solution concept for fairly distributing the total gains among players.
$$ \phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|! (n - |S| - 1)!}{n!} [v(S \cup \{i\}) - v(S)] $$
Charts and Diagrams
graph TD
A[Players] -->|Form| B[Coalitions]
B -->|Negotiate| C[Collective Strategies]
C -->|Optimize| D[Payoffs]
Importance and Applicability
Cooperative games are vital in various fields, including:
- Economics: Market negotiations, cost-sharing problems.
- Politics: Coalition formation in parliaments.
- Business: Strategic alliances and joint ventures.
Examples and Considerations
Real-World Examples
- Political Coalitions: Formed to achieve a majority.
- Corporate Alliances: Companies forming coalitions to enter new markets.
Related Terms and Comparisons
Related Terms
- Non-cooperative Games: Where players make decisions independently.
- Nash Equilibrium: A solution concept in non-cooperative games.
Comparisons
- Cooperative vs. Non-cooperative Games: Cooperative games focus on coalition formation and collective strategies, while non-cooperative games center on individual strategies.
Interesting Facts
- The Shapley value is named after Lloyd Shapley, who won the Nobel Prize in Economics in 2012.
- John Nash’s contributions to game theory were dramatized in the film “A Beautiful Mind.”
Inspirational Stories
Lloyd Shapley, despite being relatively unknown outside academic circles, significantly impacted economics and operations research, showing that the most profound contributions often come from collaborative efforts.
Famous Quotes
- “The only thing that will redeem mankind is cooperation.” – Bertrand Russell
- “In cooperative games, every player’s contribution is crucial for success.” – Anonymous
Proverbs and Clichés
- “Two heads are better than one.”
- “Unity is strength.”
Expressions, Jargon, and Slang
- Coalition: A group of players working together.
- Synergy: The combined effect of a coalition that is greater than the sum of individual efforts.
FAQs
What is a cooperative game?
What is the Shapley value?
References
- von Neumann, John, and Oskar Morgenstern. “Theory of Games and Economic Behavior.” 1944.
- Shapley, Lloyd. “A Value for N-Person Games.” 1953.
Summary
Cooperative games emphasize the importance of collaboration and negotiation among players to achieve common goals. Understanding these concepts aids in various practical scenarios, from economic markets to political coalitions, showcasing the power of unity and collective effort. By exploring mathematical models like the Shapley value and real-world applications, cooperative games offer insightful strategies for resolving complex, multi-agent interactions.
