# Core (game theory)

Last updated

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.

## Contents

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 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]

## Definition

Consider a transferable utility cooperative game ${\displaystyle (N,v)}$ where ${\displaystyle N}$ denotes the set of players and ${\displaystyle v}$ is the characteristic function. An imputation ${\displaystyle x\in \mathbb {R} ^{N}}$ is dominated by another imputation ${\displaystyle y}$ if there exists a coalition ${\displaystyle C}$, such that each player in ${\displaystyle C}$ prefers ${\displaystyle y}$, formally: ${\displaystyle x_{i}\leq y_{i}}$ for all ${\displaystyle i\in C}$ and there exists ${\displaystyle i\in C}$ such that ${\displaystyle x_{i} and ${\displaystyle C}$ can enforce ${\displaystyle y}$ (by threatening to leave the grand coalition to form ${\displaystyle C}$), formally: ${\displaystyle \sum _{i\in C}y_{i}\leq v(C)}$. An imputation ${\displaystyle x}$ is dominated if there exists an imputation ${\displaystyle y}$ dominating it.

The core is the set of imputations that are not dominated. [3]

## Properties

• Another definition, equivalent to the one above, states that the core is a set of payoff allocations ${\displaystyle x\in \mathbb {R} ^{N}}$ satisfying
1. Efficiency: ${\displaystyle \sum _{i\in N}x_{i}=v(N)}$,
2. Coalitional rationality: ${\displaystyle \sum _{i\in C}x_{i}\geq v(C)}$ for all subsets (coalitions) ${\displaystyle C\subseteq N}$.
• 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.

## 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

${\displaystyle v(S)={\begin{cases}|S|/2,&{\text{if }}|S|{\text{ is even}};\\(|S|-1)/2,&{\text{if }}|S|{\text{ is odd}}.\end{cases}}}$

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

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).

### 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. 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 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 ${\displaystyle x}$ in favor of another alternative ${\displaystyle y}$. 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.

## Related Research Articles

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. The utilization of Nash Equilibriums, and its principles date data back to the time of Cournot, a prominent Philosopher and mathematician who pioneered the understanding of economic equilibria.

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 or without any loss thereof. 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:

The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Prize in Economics for it in 2012. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject.

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.

Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.

There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal. The requirements for perfect competition are these:

1. There are no externalities and each actor has perfect information.
2. Firms and consumers take prices as given.

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 game theory, a stochastic game, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with probabilistic transitions played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some state. The players select actions and each player receives a payoff that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

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.

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 with anyone else, are unacceptable - a condition known as individual rationality. Imputations are distributions that are efficient and are individually rational.

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.

Market design is a practical methodology for creation of markets of certain properties, which is partially based on mechanism design. In some markets, prices may be used to induce the desired outcomes — these markets are the study of auction theory. In other markets, prices may not be used — these markets are the study of matching theory.

Congestion games are a class of games in game theory first proposed by American economist Robert W. Rosenthal in 1973. In a congestion game the payoff of each player depends on the resources it chooses and the number of players choosing the same resource. Congestion games are a special case of potential games. Rosenthal proved that any congestion game is a potential game and Monderer and Shapley (1996) proved the converse: for any potential game, there is a congestion game with the same potential function.

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:

Jean-François Mertens was a Belgian game theorist and mathematical economist.

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.

## References

1. 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.
2. 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.
3. 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
4. Bondareva, Olga N. (1963). "Some applications of linear programming methods to the theory of cooperative games (In Russian)". Problemy Kybernetiki. 10: 119–139.
5. Shapley, Lloyd S. (1967). "On balanced sets and cores". Naval Research Logistics Quarterly. 14 (4): 453–460. doi:10.1002/nav.3800140404. hdl:.