The core is the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's consumers. 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.

Origin

The idea of the core already appeared in the writings of , 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 .

Definition

Consider a transferable utility cooperative game $(N,v)$ where $N$ denotes the set of players and $v$ is the characteristic function. An imputation $xinmathbb\{R\}^N$ is dominated by another imputation $y$ if there exists a coalition $C$, such that each player in $C$ prefers $y$, formally: $x\_ileq\; y\_i$ for all $iin\; N$ and there exists $iin\; N$ such that dominated if there exists an imputation $y$ dominating it.

The core collects imputations that are not dominated.

Properties

• It is equivalent to the definition to say that the core is the collection of payoff allocations $xinmathbb\{R\}^N$ satisfying
• Efficiency: $sum\_\{iin\; N\}x\_i=v(N)$,
• Coalitional rationality: $sum\_\{iin\; C\}x\_igeq\; v(C)$ for all subsets (coalitions) $Csubseteq\; N$.
• The core is a set which satisfies a system of weak linear inequalities, so it is closed and convex.
• The core is well-defined, but can be empty.
• The Bondareva-Shapley theorem:The core of the game is non-empty if and only if the game is balanced (,).
• 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.
• 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

Example 1: Miners

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 two 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.

Example 2: Gloves

Mrs A and Mrs B are knitting gloves. The gloves are one-size-fits-all, and two gloves make a pair that they sell for €5. They have each made 3 gloves. How to share the proceeds from the sale? The problem can be described by a characteristic function form game with the following characteristic function: Each lady has 3 gloves, that is 1 pair with a market value of €5. Together, they have 6 gloves or 3 pair, having a market value of €15. Since the singleton coalitions (consisting of a single lady) are the only non-trivial coalitions of the game all possible distributions of this sum belong to the core, provided both ladies get at least €5, the amount they can achieve on their own. For instance (7.5, 7.5) belongs to the core, but so is (5, 10) or (9, 6).

Example 3: Shoes

For the moment ignore shoe sizes: a pair consists of a left and a right shoe, which can then be sold for €10. Consider a game with 2001 players: 1000 of them have 1 left shoe, 1001 have 1 right shoe. The core of this game is somewhat surprising: it consists of a single imputation that gives 10 to those having a (scarce) left shoe, and 0 to those owning an (oversupplied) right shoe.

We verify that this is indeed the case. Observe that any pair having a left and a right shoe can form a coalition and sell their pair for €10, so any pair getting less than that will block the imputation. So if an imputation is in the core, we can write down left-right pairs and any of these pairs will get at least 10, in fact, exactly 10, since on the end we can only sell 1000 pairs, making the total budget equal to 10000. This leaves a right-shoe owner with 0 payment. Now go through the pairs: if there is a left-shoe owner who has less than 10, say 8, then it can join this poor player, sell their shoes, give him 1, and keep 9 to herself. This way both are better off. For stability such a left-shoe owner cannot exist: all left shoe owners get already 10.

The message remains the same, even if we increase the numbers as long as left shoes are scarcer. The core has been criticized for being so extremely sensitive to oversupply of one type of player.

See also

• Welfare economics
• Pareto efficiency

References

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• Osborne, Martin J. Rubinstein, Ariel. A Course in Game Theory. The MIT Press (1994)
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• Telser, Lester G. "The Usefulness of Core Theory in Economics." The Journal of Economic Perspectives, 1994, 8(2), pp. 151-64