Core (of a Cooperative Game)
The set of coalition-stable payoff allocations.
Definition
No coalition can strictly improve by breaking off — every coalition earns at least what it can guarantee.
Intuition
Allocations in the core are immune to any coalitional deviation.
Think of it as a set-valued solution generalizing individual rationality to every subgroup.
Worked example
In the majority game with three players, the core is empty.
In a market with one seller and many buyers, the core converges to the competitive equilibrium as the market grows (Debreu-Scarf).
The math
Bondareva-Shapley Theorem: the core is nonempty iff the game is balanced — any balanced collection of weights satisfies .
Convex games have nonempty cores with the Shapley value as the barycenter.
Where it is used
Market equilibria, assignment games, and fair resource allocation.
Provides a "no-blocking" stability concept across all coalitions.
More in Game theory
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