14.12 Game Theory
Fall 2005
Answers to Midterm 1, Fall 2005
Answer to Problem 1
a)
Player
1
has
the
same
payoﬀ
function
in
both
games,
so
player
1
trivially
has
the
same
preference
relation
over
lotteries
with
strategy
proﬁles
as
their
outcomes.
What
about
player
2?
In
other
words,
are
the
payoﬀs
for
player
2
in
the
game
on
the
right
a
nonnegative
aﬃne
transformation
of
the
payoﬀs
in
the
game
on
the
left?
In
yet
other
words,
do
there
exist
a
0and
b
with
0
a
+
b
=0
,
1
a
+
b
=1
,
4
a
+
b
=3
,
and 2
a
+
b
=
2?
You
can
see
that
we’d need
a
and
b
=
0
in
order
to
satisfy
the
ﬁrst
two
equations,
but
this
does
not
satisfy
the
third equation.
So
there
is
no
such
transformation,
and player
2
does
not
have
the
same
preference
relation
over
lotteries
with
strategy
proﬁles
as
their
outcomes.
b)
Are
the
payoﬀs
for
player
1
in
the
game
on
the
right
a
nonnegative
aﬃne
transformation
of
the
payoﬀs
in
the
game
on
the
left?
In
other
words,
do
there
exist
a
b
with
0
a
+
b
,
6
a
+
b
=4
,
2
a
+
b
=2
,
4
a
+
b
,
4
a
+
b
,
and 2
a
+
b
=
2?
Yes,
you
can
solve
the
equations
and see
that
a
/
2and
b
=
1
are
such
an
a
and
b.
Are
the
payoﬀs
for
player
2
in
the
game
on
the
right
a
nonnegative
aﬃne
transformation
of
the
payoﬀs
in
the
game
on
the
left?
In
other
words,
do
there
exist
a
b
with
1
a
+
b
,
4
a
+
b
,
4
a
+
b
,
7
a
+
b
,
−
2
a
+
b
=
−
1
,
and 1
a
+
b
=
0?
Yes,
you
can
solve
the
equations
and see
that
a
/
3and
b
=
−
1
/
3are
such
an
a
and
b.
So
yes,
both
players
have
the
same
preference
relation
on
lotteries
with
strategy
proﬁles
as
their
outcomes.