62
CHAPTER
7.
PRELIMINARY
NOTIONS
IN
GAME
THEORY
player
i
, and
u
i
:
S
1
×
. . .
×
S
n
→
R
is
player
i
’s
von
Neumann-Morgenstern
utility
function.
Notice
that
a
player’s
utility
depends
not
only
on
his
own
strategy
but
also
on
the
strategies
played
by
other
players.
Moreover,
each
player
i
tries
to
maximize
the
expected
value
of
u
i
(where
the
expected
values
are
computed
with
respect
to
his
own
beliefs);
in
other
words,
u
i
is
a
von
Neumann-Morgenstern
utility
function.
We
will
say
that
player
i
is
rational
i
ff
he
tries to maximize the
expected
value of
u
i
(given
his
beliefs).
It
is
also
assumed
that
it
is
common
knowledge
that
the
players
are
N
=
{
1
, . . . , n
}
,
that
the
set
of
strategies
available
to
each
player
i
is
S
i
, and
that
each
i
tries
to
maximize
expected
value
of
u
i
given
his
beliefs.
When
there
are
only
two
players,
we
can
represent
the
(normal
form)
game
by
a
bimatrix
(i.e.,
by
two
matrices):
1
\
2
1,1
4,1
3,2
left
right
up
0,2
down
Here,
Player
1
has
strategies
up
and
down,
and
Player
2
has
the
strategies
left
and
right.
In
each
box
the
fi
rst number is
Player 1’s
payo
ff
and
the
second
one
is
Player
2’s
(e.g.,
u
1
(
up,left
) = 0
,
u
2
(
up,left
) = 2
.)
I
will
use
the
following
notational
convention
throughout
the
course.
Given
any
list
X
1
, . . . , X
n
of
sets
with
generic
elements
x
1
, . . . , x
n
,
I
will
•
write
X
=
X
1
× · · · ×
X
n
and
designate
x
= (
x
1
, . . . , x
n
)
as
the
generic
element,
write
X
−
i
=
Q
j
=
i
X
j
and
designate
x
−
i
= (
x
1
, . . . , x
i
−
1
, x
i
+1
, . . . , x
n
)
as
the
generic
•
6
element
for
any
i
, and
•
write
(
x
i
0
, x
−
i
) = (
x
1
, . . . , x
i
−
1
, x
0
i
, x
i
+1
, . . . , x
n
)
.
For
example,
•
S
=
S
1
× · · · ×
S
n
is
the
set
of
strategy
pro
fi
les
s
= (
s
1
, . . . , s
n
)
,
•
S
−
i
is
the
set
of
strategy
pro
fi
les
s
−
i
= (
s
1
, . . . , s
i
−
1
, s
i
+1
, . . . , s
n
)
other
than
player
i
, and