Game Theory
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due March 8th, in the discussion section.
Homework 8
Exercise 1.
Prove the following Lemma from the notes: The set of functions
{
W
S
}
S
⊆{
1
,...,n
}
is an orthonormal basis for the space of functions from
{-
1
,
1
}
n
→
R
, with respect to the
inner product defined in the notes. (When we write
S
⊆ {
1
, . . . , n
}
, we include the empty
set
∅
as a subset of
{
1
, . . . , n
}
.) (Also, for any
x
∈ {-
1
,
1
}
n
,
W
S
(
x
) =
Q
i
∈
S
x
i
.)
Exercise 2.
Let
f
:
{-
1
,
1
}
2
→ {-
1
,
1
}
such that
f
(
x
) = 1 for all
x
∈ {-
1
,
1
}
2
. Compute
b
f
(
S
) for all
S
⊆ {
1
,
2
}
.
Let
f
:
{-
1
,
1
}
3
→ {-
1
,
1
}
such that
f
(
x
1
, x
2
, x
3
) = sign(
x
1
+
x
2
+
x
3
) for all (
x
1
, x
2
, x
3
)
∈
{-
1
,
1
}
3
. Compute
b
f
(
S
) for all
S
⊆ {
1
,
2
,
3
}
. The function
f
is called a
majority function
.
Exercise 3.
Let
f
:
{-
1
,
1
}
3
→ {-
1
,
1
}
such that
f
(
x
1
, x
2
, x
3
) = sign(
x
1
+
x
2
+
x
3
) for all
(
x
1
, x
2
, x
3
)
∈ {-
1
,
1
}
3
. In the previous homework, we computed
b
f
(
S
) for all
S
⊆ {
1
,
2
,
3
}
.