Cooperative Games Basics
Cooperative Games, a core branch of Game Theory, focus on scenarios where players form coalitions to achieve shared goals, maximizing collective outcomes through collaboration. Unlike non-cooperative games, cooperative games emphasize teamwork, negotiation, and fair payoff distribution. This guide from MathMultiverse, inspired by the foundational work of John von Neumann and Oskar Morgenstern, explores coalition formation, solution concepts like the Shapley Value and Core, and their applications in economics, politics, and beyond.
Definition and Key Concepts
Cooperative games are defined by their structure, focusing on group dynamics and payoff allocation.
Basic Definition
A cooperative game is a pair \( (N, v) \), where \( N = \{1, 2, \ldots, n\} \) is the player set, and \( v: 2^N \to \mathbb{R} \) assigns a value \( v(S) \) to each coalition \( S \subseteq N \), with \( v(\emptyset) = 0 \).
Superadditivity
A game is superadditive if merging coalitions increases value:
Monotonicity
Larger coalitions yield at least as much value:
Convex Games
A game is convex if:
The Core
An allocation \( x = (x_1, \ldots, x_n) \) is in the core if:
Examples of Cooperative Games
Real-world scenarios illustrate cooperative game dynamics.
Delivery Truck Sharing
Companies A, B, C share a truck. Costs: A=50, B=60, C=70; AB=90, AC=100, BC=110, ABC=120. Savings as values:
Voting Power
Players A(3 votes), B(2 votes), C(1 vote); need 4 to win:
Airport Cost Sharing
Planes need runways of lengths 1, 2, 3; cost = max length:
Resource Pooling
Firms pool resources: \( v(\{1,2\}) = 5 \), \( v(\{1,2,3\}) = 8 \).
Production Alliance
Values: \( v(\{A,B\}) = 10 \), \( v(\{A,B,C\}) = 15 \).
Coalitions and Payoff Allocation
Solution concepts ensure fair payoff distribution.
Shapley Value
Fair allocation for player \( i \):
Core Stability
For truck sharing, \( x = (20, 20, 20) \):
Nucleolus
Minimizes maximum dissatisfaction:
Banzhaf Index
Voting power for player \( i \):
TU Games Stability
Ensures total payoff allocation: