Soln:
In period
T
, player 1 will accept any offer.
So player 2 offers 0 to player 1.
Therefore, if the game reaches period
T
, player 2 gets the whole pie. Knowing this, in
period
T
°
1
;
player 2 will accept if and only if he is offered at least
°
2
; so player 1 will
offer him exactly
°
2
. In period
T
°
2
;
player 2 will accept if and only if he is offered at
least
°
2
2
;
so player 1 will offer him
°
2
2
:
Proceeding backwards, we see that in any period
T
°
k
, player 2 will accept if and only if he is offered at least
°
k
2
;
so player 1 will offer
him
°
k
2
:
In the first period, player 2 will accept if and only if he is offered at least
°
T
°
1
2
;
so player 1 will offer him
°
T
°
1
2
:
Therefore, the subgame perfect equilibrium is:
´
In any period
T
°
k
,
k
D
1
; :::;
T
°
1
;
the offer strategy of player 1 is to offer
°
k
2
regardless of previous play, and the acceptance strategy of player 2 is to accept if
and only if he is offered at least
°
k
2
I
´
In period
T
, the offer strategy of player 2 is to offer 0 to player 1, regardless of pre-
vious play, and the acceptance strategy of player 1 is to accept any offer, regardless
of previous play.
The outcome of this equilibrium is agreement in the first period, with player 1 receiving
1
°
°
T
°
1
2
and player 2 getting
°
T
°
1
2
:
5. (20 pts) Andy and Betty can together choose any action
a
2
[0
;
10]
:
Given a choice of
a
;
their payoffs are
u
A
D
a
and
u
B
D
100
°
a
2
:
(a) (10 pts) Find all Pareto efficient values of
a
:
Soln:
Any value of
a
2
[0
;
10] is Pareto efficient: making Andy better off would
require increasing
a
, but this would make Betty worse off, and vice versa.
(b) (10 pts) Now assume transferable utility: in addition to choosing
a
;
a transfer of
any amount
t
2
R
can be made from Andy to Betty, resulting in payoffs
u
A
D
a
°
t
and
u
B
D
100
°
a
2
C
t
:
Find all Pareto efficient pairs
.
a
;
t
/:
Soln:
Given the transferable utility assumption, we know that
.
a
;
t
/
is Pareto effi-
cient if and only if
a
maximizes the joint payoff,
a
C
100
°
a
2
:
This is maximized by
a
D
1
2
:
Hence,
.
a
;
t
/
is Pareto efficient if and only if
a
D
1
2
:
3

6. (20 pts) Find all pure strategy subgame perfect equilibria, and one other Nash equilibrium,
of this game:
1
2
B
S
b
s
b
s
3,1
0,0
0,0
1,3
1
C
D
C
D
c
d
3,3
4,0
0,4
2,2
1
2
X
Y
Soln:
The subgame following
X
is
b
s
B
3,1
0,0
S
0,0
1,3
Its Nash equilibria are (B,b) and (S,s). Note that (B,b) yields payoff 3 to player 1, and
(S,s) yields payoff 1 to player 1.
The subgame following
Y
is
c
d
C
3,3
0,4
D
4,0
2,2
It has a unique Nash equilibrium: (D,d). It yields payoff 2 to player 1.
Player 1 prefers to play
X
if the equilibrium (B,b) will be played in the resulting sub-
game, and otherwise prefers to play
Y
. Therefore, the subgame perfect equilibria are:
.
X BD
;
bd
/
and
.
Y SD
;
sd
/
.
.
Y BD
;
sd
/
is a Nash equilibrium that is not subgame perfect. Here, a Nash equilibrium
gets played in the subgame that is in fact reached after player 1 plays
Y
. Furthermore,
deviating to
X
would give player 1 a payoff of at most 1
<
2, so he is playing a best
response. This equilibrium is not subgame perfect because it prescribes the play of (B,s)