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6.207/14.15:
Networks
Lecture
11:
Introduction
to
Game
Theory—3
Daron
Acemoglu
and
Asu
Ozdaglar
MIT
October
19,
2009
1
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View Full Document Networks:
Lecture
11
Introduction
Outline
Existence
of
Nash
Equilibrium
in
InFnite
Games
Extensive
±orm
and
Dynamic
Games
Subgame
Perfect
Nash
Equilibrium
Applications
Reading:
Osborne,
Chapters
56.
2
1
2
3
Networks:
Lecture
11
Nash
Equilibrium
Existence
of
Equilibria
for
InFnite
Games
A
similar
theorem
to
Nash’s
existence
theorem
applies
for
pure
strategy
existence
in
inFnite
games.
Theorem
(Debreu,
Glicksberg,
Fan)
Consider
an
infnite
strategic
Form
game
I
,
(
S
i
)
i
∈I
,
(
u
i
)
i
∈I
±
such
that
For
each
i
∈ I
S
i
is
compact
and
convex;
u
i
(
s
i
,
s
−
i
)
is
continuous
in
s
−
i
;
u
i
(
s
i
,
s
−
i
)
is
continuous
and
concave
in
s
i
[in
Fact
quasiconcavity
suﬃces].
Then
a
pure
strategy
Nash
equilibrium
exists.
3
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View Full Document concave function
not a concave function
Networks:
Lecture
11
Nash
Equilibrium
Defnitions
Suppose
S
is
a
convex
set.
Then
a
function
f
:
S
R
is
concave
if
→
for
any
x
,
y
∈
S
and
any
λ
∈
[0
,
1],
we
have
f
(
λ
x
+
(1
−
λ
)
y
)
≥
λ
f
(
x
)
+
(1
−
λ
)
f
(
y
)
.
4
±
Networks:
Lecture
11
Nash
Equilibrium
Proof
Now
defne
the
best
response
correspondence
For
player
i
,
B
i
:
S
−
i
S
i
,
B
i
(
s
−
i
) =
s
i
∈
S
i

u
i
(
s
i
,
s
−
i
)
≥
u
i
(
s
i
,
s
−
i
)
For
all
s
i
∈
S
i
.
Thus
restriction
to
pure
strategies.
Defne
the
set
oF
best
response
correspondences
as
B
(
s
) = [
B
i
(
s
−
i
)]
i
∈I
.
and
B
:
S
S
.
5
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2
Networks:
Lecture
11
Nash
Equilibrium
Proof
(continued)
We
will
again
apply
Kakutani’s
theorem
to
the
best
response
correspondence
B
:
S
S
by
showing
that
B
(
s
)
satisfes
the
conditions
oF
Kakutani’s
theorem.
S
is
compact,
convex,
and
nonempty.
By
defnition
S
=
S
i
i
∈I
since
each
S
i
is
compact
[convex,
nonempty]
and
fnite
product
oF
compact
[convex,
nonempty]
sets
is
compact
[convex,
nonempty].
B
(
s
)
is
nonempty.
By
defnition,
B
i
(
s
−
i
)
=
arg
max
u
i
(
s
,
s
−
i
)
s
∈
S
i
where
S
i
is
nonempty
and
compact,
and
u
i
is
continuous
in
s
by
assumption.
Then
by
Weirstrass’s
theorem
B
(
s
)
is
nonempty.
6
Networks:
Lecture
11
Nash
Equilibrium
Proof
(continued)
3.
B
(
s
)
is
a
convexvalued
correspondence.
This
follows
from
the
fact
that
u
i
(
s
i
,
s
−
i
)
is
concave
[or
quasiconcave]
in
s
i
.
Suppose
not,
then
there
exists
some
i
and
some
s
−
i
∈
S
−
i
such
that
B
i
(
s
−
i
)
∈
arg
max
s
∈
S
i
u
i
(
s
,
s
−
i
)
is
not
convex.
This
implies
that
there
exists
s
i
,
s
i
∈
S
i
such
that
s
i
,
s
i
∈
B
i
(
s
−
i
)
and
λ
s
i
+
(1
−
λ
)
s
i
∈
/
B
i
(
s
−
i
).
In
other
words,
λ
u
i
(
s
i
,
s
−
i
)
+
(1
−
λ
)
u
i
(
s
i
,
s
−
i
)
>
u
i
(
λ
s
i
+
(1
−
λ
)
s
i
,
s
−
i
)
.
But
this
violates
the
concavity
of
u
i
(
s
i
,
s
−
i
)
in
s
i
[recall
that
for
a
concave
function
f
(
λ
x
+
(1
−
λ
)
y
)
≥
λ
f
(
x
)
+
(1
−
λ
)
f
(
y
)].
Therefore
B
(
s
)
is
convexvalued.
4.
The
proof
that
B
(
s
)
has
a
closed
graph
is
identical
to
the
previous
proof.
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This note was uploaded on 06/12/2010 for the course EECS 6.207J taught by Professor Acemoglu during the Fall '09 term at MIT.
 Fall '09
 Acemoglu

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