In
cooperative game theory
In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to ...
, the core is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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.
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 though
von Neumann Von Neumann may refer to:
* John von Neumann (1903–1957), a Hungarian American mathematician
* Von Neumann family
* Von Neumann (surname), a German surname
* Von Neumann (crater), a lunar impact crater
See also
* Von Neumann algebra
* Von Neu ...
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.
Definition
Consider a
transferable utility cooperative game where
denotes the set of players and
is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
. 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