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.

- Origin
- Definition
- Properties
- Example
- Example 1: Miners
- Example 2: Gloves
- Example 3: Shoes
- The core in general equilibrium theory
- The core in voting theory
- See also
- References
- Further reading

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*.^{ [1] } 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.^{ [2] }

Consider a transferable utility cooperative game where denotes the set of players and is the characteristic function. An imputation 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 to form ), formally: . An imputation is *dominated* if there exists an imputation dominating it.

The **core** is the set of imputations that are not dominated.^{ [3] }

- Another definition, equivalent to the one above, states that the core is a set of payoff allocations satisfying

- Efficiency: ,
- Coalitional rationality: for all subsets (coalitions) .

- The core is always well-defined, but can be empty.
- The core is a set which satisfies a system of weak linear inequalities. Hence the core is closed and convex.
- The Bondareva–Shapley theorem: the core of a game is nonempty if and only if the game is "balanced".
^{ [4] }^{ [5] } - Every Walrasian equilibrium has the core property, but not
*vice versa*. The Edgeworth conjecture 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.

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.

Mr A and Mr 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 three 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 man has three gloves, that is one 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 man) are the only non-trivial coalitions of the game all possible distributions of this sum belong to the core, provided both men 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).

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. No coalition can block this outcome, because no left shoe owner will accept less than 10, and any imputation that pays a positive amount to any right shoe owner must pay less than 10000 in total to the other players, who can get 10000 on their own. So, there is just one imputation in the core.

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 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.

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 for the simple game.

- Welfare economics
- Pareto efficiency
- Knaster–Kuratowski–Mazurkiewicz–Shapley theorem - instrumental in proving the non-emptiness of the core.

In mathematical logic, the **Peano axioms**, also known as the **Dedekind–Peano axioms** or the **Peano postulates**, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

**Pareto efficiency** or **Pareto optimality** is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

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In combinatorial mathematics, the **Bell numbers** count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.

In game theory, a **cooperative game** is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior. Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing.

**Mechanism design** is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called **reverse game theory**. It has broad applications, from economics and politics to networked-systems.

In mathematics, a **Markov decision process** (**MDP**) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via dynamic programming and reinforcement learning. MDPs were known at least as early as the 1950s; a core body of research on Markov decision processes resulted from Ronald Howard's 1960 book, *Dynamic Programming and Markov Processes*. They are used in many disciplines, including robotics, automatic control, economics and manufacturing. The name of MDPs comes from the Russian mathematician Andrey Markov as they are an extension of Markov chains.

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- There are no externalities and no transaction costs, and each actor has perfect information.
- Firms take prices – ie. behavior is competitive rather than monopolistic.

In game theory, **folk theorems** are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

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 **Knaster–Kuratowski–Mazurkiewicz lemma** is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz.

**Competitive equilibrium** is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated.

In mathematics, a **vector measure** is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

The **Bondareva–Shapley theorem**, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. 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 games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.

A **topological game** is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and convergence.

**Linear production game** is a N-person game in which the value of a coalition can be obtained by solving a linear programming problem. It is widely used in the context of resource allocation and payoff distribution. Mathematically, there are *m* types of resources and *n* products can be produced out of them. Product *j* requires amount of the *kth* resource. The products can be sold at a given market price while the resources themselves can not. Each of the *N* players is given a vector of resources. The value of a coalition *S* is the maximum profit it can achieve with all the resources possessed by its members. It can be obtained by solving a corresponding linear programming problem as follows.

In cooperative game theory and social choice theory, the **Nakamura number** measures the degree of rationality of preference aggregation rules, such as voting rules. It is an indicator of the extent to which an aggregation rule can yield well-defined choices.

The **Shapley–Folkman lemma** is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. *Minkowski addition* is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:

In theoretical economics, an **abstract economy** is a model that generalizes both the standard model of an exchange economy in microeconomics, and the standard model of a game in game theory. An *equilibrium* in an abstract economy generalizes both a Walrasian equilibrium in microeconomics, and a Nash equilibrium in game-theory.

- ↑ Kannai, Y. (1992). "The core and balancedness". In Aumann, Robert J.; Hart, Sergiu (eds.).
*Handbook of Game Theory with Economic Applications*.**I**. Amsterdam: Elsevier. pp. 355–395. ISBN 978-0-444-88098-7.CS1 maint: ref=harv (link) - ↑ Gillies, D. B. (1959). "Solutions to general non-zero-sum games". In Tucker, A. W.; Luce, R. D. (eds.).
*Contributions to the Theory of Games IV*. Annals of Mathematics Studies.**40**. Princeton: Princeton University Press. pp. 47–85.CS1 maint: ref=harv (link) - ↑ As noted by Shapley, L. S.; Shubik, M. (1969). "On Market Games".
*Journal of Economic Theory*.**1**(1): 9–25. doi:10.1016/0022-0531(69)90008-8. due to the contribution of Mr. E. Kohlberg - ↑ Bondareva, Olga N. (1963). "Some applications of linear programming methods to the theory of cooperative games (In Russian)".
*Problemy Kybernetiki*.**10**: 119–139.CS1 maint: ref=harv (link) - ↑ Shapley, Lloyd S. (1967). "On balanced sets and cores".
*Naval Research Logistics Quarterly*.**14**(4): 453–460. doi:10.1002/nav.3800140404. hdl: 10338.dmlcz/135729 .CS1 maint: ref=harv (link)

- Ichiishi, Tatsuro (1983). "Cooperative Behavior and Stability".
*Game Theory for Economic Analysis*. New York: Academic Press. pp. 77–117. ISBN 0-12-370180-5. - Osborne, Martin J.; Rubinstein, Ariel (1994).
*A Course in Game Theory*. The MIT Press. - Peleg, B (1992). "Axiomatizations of the Core". In Aumann, Robert J.; Hart, Sergiu (eds.).
*Handbook of Game Theory with Economic Applications*.**I**. Amsterdam: Elsevier. pp. 397–412. ISBN 978-0-444-88098-7.CS1 maint: ref=harv (link) - Shoham, Yoav; Leyton-Brown, Kevin (2009).
*Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations*. New York: Cambridge University Press. ISBN 978-0-521-89943-7. - Telser, Lester G. (1994). "The Usefulness of Core Theory in Economics".
*Journal of Economic Perspectives*.**8**(2): 151–164. doi: 10.1257/jep.8.2.151 .CS1 maint: ref=harv (link)

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