Economics 201B
Nonconvex Preferences and
Approximate Equilibria
1
The ShapleyFolkman Theorem
The ShapleyFolkman Theorem is an elementary result in linear algebra, but
it is apparently unknown outside the mathematical economics literature. It
is closely related to Caratheodory’s Theorem, a linear algebra result which
is well known to mathematicians. The ShapleyFolkman Theorem was first
published in Starr [3], an important early paper on existence of approximate
equilibria with nonconvex preferences.
Theorem 1.1 (Caratheodory)
Suppose
x
∈
con
A
, where
A
⊂
R
L
. Then
there are points
a
1
, . . . , a
L
+1
∈
A
such that
x
∈
con
{
a
1
, . . . , a
L
+1
}
.
Theorem 1.2 (ShapleyFolkman)
Suppose
x
∈
con (
A
1
+
· · ·
+
A
I
)
, where
A
i
⊂
R
L
. Then we may write
x
=
a
1
+
· · ·
+
a
I
, where
a
i
∈
con
A
i
for all
i
and
a
i
∈
A
i
for all but
L
values of
i
.
We derive both Caratheodory’s Theorem and the ShapleyFolkman Theorem
from the following lemma:
Lemma 1.3
Suppose
x
∈
con (
A
1
+
· · ·
+
A
I
)
where
A
i
⊂
R
L
. Then we may
write
x
=
I
i
=1
m
i
j
=0
λ
ij
a
ij
(1)
with
∑
I
i
=1
m
i
≤
L
;
a
ij
∈
A
i
and
λ
ij
>
0
for each
i, j
; and
∑
m
i
j
=0
λ
ij
= 1
for
each
i
.
Proof:
1
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1. Suppose
x
∈
con (
A
1
+
· · ·
+
A
I
). Then we may write
x
=
m
j
=0
λ
j
I
i
=1
a
ij
=
I
i
=1
m
j
=0
λ
j
a
ij
(2)
with
λ
j
>
0,
∑
m
j
=0
λ
j
= 1. Letting
λ
ij
=
λ
j
and
m
i
=
m
for each
i
, we
have an expression for
x
in the form of equation 1.
2. Suppose we have any expression for
x
in the form of equation 1 with
∑
I
i
=1
m
i
> L
. Then the set
{
a
ij
−
a
i
0
: 1
≤
i
≤
I,
1
≤
j
≤
m
i
}
(3)
contains
∑
I
i
=1
m
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 Spring '06
 ANDERSON
 Economics, λij aij, I=1 mi

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