Core (game theory)

In game theory, the core is the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's agents. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition. In game theory, the core is the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's agents. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition. An allocation is said to have the core property if there is no coalition that can improve upon it. The core is the set of all feasible allocations with the core property. The idea of the core already appeared in the writings of Edgeworth (1881), at the time referred to as the contract curve. Even if 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 ( N , v ) {displaystyle (N,v)} where N {displaystyle N} denotes the set of players and v {displaystyle v} is the characteristic function. An imputation x ∈ R N {displaystyle xin mathbb {R} ^{N}} is dominated by another imputation y {displaystyle y} if there exists a coalition C {displaystyle C} , such that each player in C {displaystyle C} prefers y {displaystyle y} , formally: x i ≤ y i {displaystyle x_{i}leq y_{i}} for all i ∈ C {displaystyle iin C} and there exists i ∈ C {displaystyle iin C} such that x i < y i {displaystyle x_{i}<y_{i}} and C {displaystyle C} can enforce y {displaystyle y} (by threatening to leave the grand coalition to form C {displaystyle C} ), formally: ∑ i ∈ C y i ≤ v ( C ) {displaystyle sum _{iin C}y_{i}leq v(C)} . An imputation x {displaystyle x} is dominated if there exists an imputation y {displaystyle y} dominating it. When the core exists and is not empty, it is the set of imputations that are not dominated. Consider a group of n miners, who have discovered large bars of gold. If two miners can carry one piece of gold, then the payoff of a coalition S is