Bondareva-Shapley theorem
Encyclopedia
In game theory
, the Bondareva–Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core
of a cooperative game
. 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.
be a cooperative game
, where
is the set of players and where the value function 
is defined on 
's power set (the set of all subsets of 
).
The core of
is non-empty if and only if for every function 
where
the following condition holds:
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, the Bondareva–Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core
Core (economics)
The core is the set of feasible allocations that cannot be improved upon by a subset of the economy's consumers. 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...
of a cooperative game
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
. Specifically, the game's core is non-empty if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the game is balanced. The Bondareva–Shapley theorem implies that market games and convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley
Lloyd Shapley
Lloyd Stowell Shapley is a distinguished American mathematician and economist. He is a Professor Emeritus at University of California, Los Angeles, affiliated with departments of Mathematics and Economics...
in the 1960s.
Theorem
Let the pair be a cooperative gameCooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
, where
is the set of players and where the value function is defined on 's power set (the set of all subsets of ).The core of
is non-empty if and only if for every function where the following condition holds: