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.

Let the pair
⟨
N
,
v
⟩
be a cooperative game in characteristic function form, where
N
is the set of players and where the *value function*
v
:
2
N
→
R
is defined on
N
's power set (the set of all subsets of
N
).

The core of
⟨
N
,
v
⟩
is non-empty if and only if for every function
α
:
2
N
∖
{
∅
}
→
[
0
,
1
]
where

∀
i
∈
N
:
∑
S
∈
2
N
:
i
∈
S
α
(
S
)
=
1

the following condition holds:

∑
S
∈
2
N
∖
{
∅
}
α
(
S
)
v
(
S
)
≤
v
(
N
)
.