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 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]

Properties

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

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 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 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.

See also

Related Research Articles

Minimax is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for minimizing the possible loss for a worst case scenario. When dealing with gains, it is referred to as "maximin"—to maximize the minimum gain. Originally formulated for two-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty.

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion 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:

The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951. 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.

In game theory, folk theorems are a class of theorems about possible 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 equilibrium.

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 the traditional concept of economic equilibrium, 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 game theory, decision-makers deduce strategies for how to behave within the constraints of a game. Market design is the flip side of that coin: given a set of agents, market design seeks to identify the game rules a market designer might implement that would produce the desired behaviors in the players. 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.

Shapley–Folkman lemma

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:

Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose sums belong to the same subset of the partition. The theorem had been discovered and proved independently by several mathematicians, before it was named "Folkman's theorem", as a memorial to Jon Folkman, by Graham, Rothschild, and Spencer.

Jean-François Mertens Belgian game theorist

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: 10338.dmlcz/135729 .

Further reading