The Core

Game Theory · Axiom Academy

Understanding stability through coalition rationality in cooperative games 1. Core Definition and Coalition Rationality For a cooperative game with player set N and characteristic function v , an allocation x = ( x 1 , x 2 , ..., x n ) is in the core if it satisfies two conditions: 1. Efficiency (Individual Rationality for Grand Coalition): The total allocated value equals what the grand coalition can achieve. 2. Coalition Rationality: Every coalition S receives at least what it could guarantee itself by acting independently. 2. The Core as Intersection of Half-Spaces The core has an elegant geometric interpretation: it is the intersection of half-spaces , where each coalition's rationality constraint defines a half-space. For each coalition S ⊆ N , the constraint creates a region in allocation space. The core is where all these constraints are simultaneously satisfied. 3. Visualizing the Core: Two-Player Game In a two-player game with v ( 1 ) = 2, v ( 2 ) = 3, and v ( 1,2 ) = 10, we can visualize the core in 2D space where x 1 and x 2 represent payoffs to players 1 and 2. Individual rationality for player 1: x 1 ≥ 2 Individual rationality for player 2: x 2 ≥ 3 The core is the line segment where all three constraints hold simultaneously. 4. When the Core is Empty vs. Non-Empty Not all cooperative games have a non-empty core. The core can be empty when the coalition rationality constraints are too demanding to satisfy simultaneously.

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