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Concept
Core (game theory)
Summary
In cooperative game theory, the core is the set of feasible allocations or imputations where no coalition of agents can benefit by breaking away from the grand coalition. One can think of the core corresponding to situations where it is possible to sustain cooperation among all agents. A coalition is said to improve upon or block a feasible allocation if the members of that coalition can generate more value among themselves than they are allocated in the original allocation. As such, that coalition is not incentivized to stay with the grand coalition.
An allocation is said to be in the core of a game if there is no coalition that can improve upon it. The core is then the set of all feasible allocations .
The idea of the core already appeared in the writings of , at the time referred to as the contract curve. Even though von Neumann and Morgenstern considered it an interesting concept, they only worked with zero-sum games where the core is always empty. The modern definition of the core is due to Gillies.
Consider a transferable utility cooperative game where denotes the set of players and is the characteristic function. An imputation is dominated by another imputation if there exists a coalition , such that each player in prefers , formally: for all and there exists such that and can enforce (by threatening to leave the grand coalition to form ), formally: . An imputation is dominated if there exists an imputation dominating it.
The core is the set of imputations that are not dominated.
Another definition, equivalent to the one above, states that the core is a set of payoff allocations satisfying
Efficiency: ,
Coalitional rationality: for all subsets (coalitions) .
The core is always well-defined, but can be empty.
The core is a set which satisfies a system of weak linear inequalities. Hence the core is closed and convex.
The Bondareva–Shapley theorem: the core of a game is nonempty if and only if the game is "balanced".
Every Walrasian equilibrium has the core property, but not vice versa.
Official source
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