Core (game theory)

The idea of the core already appeared in the writings of Edgeworth (1881) harvtxt error: no target: CITEREFEdgeworth1881 (help), 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 ${\displaystyle (N,v)}$ 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 ${\displaystyle x_{i}<y_{i}}$ 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]

• 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.

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 ${\displaystyle x}$ 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.

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

• 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.
• 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.