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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. In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ...
Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
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 - 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.
General Equilbrium (linear) supply and demand curves. ...
In economics, the Edgeworth Conjecture is the idea, named after Francis Ysidro Edgeworth, that the Core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity. ...
The word linear comes from the Latin word linearis, which means created by lines. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...
Example 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 Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ...
Pareto efficiency, or Pareto optimality, is a central theory in economics with broad applications in game theory, engineering and the social sciences. ...
References - 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)
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