Eco 354K
Game theory
Solutions to Problem Set 9
1.(a) The set of complete contingency plans for each player.
S
1
= {T,B},S
2
= {L,B}.
1.(b) Normal form of this game.
L
R(q
R
)
T(p
T
)
18
, 12
12,
15
B
16,
13
13
, 12
1.(c) Solve for the unique Nash equilibrium
No pure strategy Nash equilibrium. Thus,
12 p
T
+13 (1- p
T
) = 15 p
T
+ 12 (1- p
T
) => p
T
=1/4
18 (1- q
R
) + 12 q
R
= 16 (1- q
R
) +13 q
R
=> q
R
= 2/3
1.(d) The sequentially rational beliefs of player 2.
Prob(d|2.1) (3/4)/[3/4 + (1/4)*(3/4)] = 4/5. Thus,
EU
2
(L,p
T
) = (1/5)*28 + (4/5)*8 = 12 = (1/5)*12 + (4/5)*12
= EU
2
(R,p
T
).
2.(a) The set of complete contingency plans for each player.
S
1
= {AC,AD,BC,BD},S
2
= {L,R}.
2.(b) Normal form of this game.
L
R
AC
1,2
5,
4
AD
3,0
3,0
BC
0,
5
7
,4
BD
2,
3
5,0
2.(c) Solve for the Nash equilibrium:
by checking off best responses,
you find that {AD,L} is a pure-strategy Nash equilibrium.
There is also a continuum of mixed-strategy NE.