A core is the set of feasible allocations in an economy that cannot be improved upon by subset of the set of the economy's consumers (a coalition). Thus it is analogous to a Nash equilibrium of a noncooperative game in game theory: an outcome is stable if no deviation is profitable for any player. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under a different 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.

Properties

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• Every Walrasian equilibrium has the core property, but not vice versa. However, under some assumptions, as the number of consumers in the economy tends to infinity, the core tends to a set of Walrasian equilibria, a result known as the Edgeworth conjecture.
• The core is a set which satisfies a system of weak linear inequalities, so it is closed and convex.
• For a group of n players, with n odd, seeking to divide one unit among some coalition which consists of a majority, the core is empty, that is, no stable coalition will arise.

Example

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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 .

If there are more than 2 miners and there are an even number of miners, then the core consists of the single payoff where each miner gets $1/2$. If there are an odd number of miners, then the core is empty.

See also

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• Welfare economics
• Pareto efficiency

References

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• Osborne, Martin J. Rubinstein, Ariel. A Course in Game Theory. The MIT Press (1994)
• Peleg, B. "Axiomatizations of the Core." Handbook of Game Theory. Volumee 1 (R. J. Aumann and S. Hart, eds) North-Holland Publisher: Amsterdam (1992)
• Telser, Lester G. "The Usefulness of Core Theory in Economics." The Journal of Economic Perspectives, 1994, 8(2), pp. 151-64

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