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

John von Neumann

John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 and Morgenstern

Oskar Morgenstern

Oskar Morgenstern was a German-born Austrian-School economist. He, along with John von Neumann, helped found the mathematical field of game theory ....

 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

Transferable utility

Transferable utility is a term used in cooperative game theory and in economics. Utility is transferable if one player can losslessly transfer part of its utility to another player. Such transfers are possible if the players have a common currency that is valued equally by all...

 cooperative game

Cooperative game

In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...

  where denotes the set of players and is the characteristic function. An imputation

Imputation (game theory)

In fully cooperative games players act efficiently when they form a single coalition, the grand coalition. The focus of the game is to find acceptable distributions of the payoff of the grand coalition. Distributions where a player receives less than it could obtain on its own, without cooperating...

  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

Grand coalition

A grand coalition is an arrangement in a multi-party parliamentary system in which the two largest political parties of opposing political ideologies unite in a coalition government...

 to form ), formally: . An imputation is dominated if there exists an imputation dominating it.

The core collects imputations that are not dominated.

Properties

• Another definition, equivalent
  Logical equivalence
  In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
  
   to the one above, states that the core is a set of payoff allocations satisfying

1. Efficiency: ,
2. Coalitional rationality: for all subsets (coalitions) .
   • The core is always well-defined, but can be empty
     Empty set
     In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
     
     .
   • The core is a set which satisfies a system of weak linear
     Linear
     In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
     
      inequalities. Hence the core is closed
     Closed set
     In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
     
      and convex
     Convex set
     In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
     
     .
   • The Bondareva-Shapley theorem
     Bondareva-Shapley theorem
     In game theory, the Bondareva–Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva–Shapley theorem implies that market games and convex...
     
     : The core of a game is nonempty iff
     IFF
     IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
     
      the game is "balanced" .
   • Every Walrasian equilibrium has the core property, but not vice versa. The Edgeworth conjecture
     Edgeworth conjecture
     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....
     
      states that, given additional assumptions, the limit of the core as the number of consumers goes to infinity is a set of Walrasian equilibria.
   • Let there be n players, where n is odd. A game that proposes to divide one unit of a good among a coalition having at least (n+1)/2 members has an empty core. That is, no stable coalition exists.

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 is an even number of miners, then the core consists of the single payoff where each miner gets 1/2. If there is 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 does (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.

The core in general equilibrium theory

The Walrasian equilibria of an exchange economy in a general equilibrium model, will lie in the core of the cooperation game between the agents. Graphically, and in a two-agent economy (see Edgeworth Box), the core is the set of points on the contract curve (the set of Pareto optimal allocations) lying between each of the agents' indifference curves defined at the initial endowments.

The core in voting theory

When alternatives are allocations (list of consumption bundles), it is natural to assume that any nonempty subsets of individuals can block a given allocation.
When alternatives are public (such as the amount of a certain public good), however, it is more appropriate to assume that only the coalitions that are large enough can block a given alternative. The collection of such large ("winning") coalitions is called a simple game.
The core of a simple game with respect to a profile of preferences is based on the idea that only winning coalitions can reject an alternative in favor of another alternative . A necessary and sufficient condition for the core to be nonempty for all profile of preferences, is provided in terms of the Nakamura number

Nakamura number

In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....

for the simple game.