Core Examples

Game Theory · Axiom Academy

Unit 6 - Cooperative Game Theory: Finding and Analyzing the Core of a Coalition Game Excellent work! You've completed this core analysis example. Here's what we learned: Characteristic Function: The foundation of cooperative game theory - v(S) represents the maximum value coalition S can achieve independently. Core Constraints: An allocation must be efficient (sum to v(N)) and satisfy both individual rationality ( x_i v( i ) ) and coalition rationality ( x_S v(S) for all coalitions). Testing Membership: To verify if an allocation is in the core, check all 2 n - 2 proper coalition constraints systematically. Geometric Interpretation: The core is a convex polytope in the payoff simplex, bounded by coalition constraint hyperplanes. Stability Analysis: A non-empty core guarantees at least one stable allocation that no coalition wants to deviate from. Empty cores indicate fundamental instability. Practical Importance: The core concept is essential for fair division problems, cost allocation, voting games, and market equilibria. This systematic approach to analyzing the core will help you solve complex coalition games. Practice with different characteristic functions to build confidence in identifying stability conditions!

This is the written version of the interactive lesson above. See the full Game Theory course.