Kaushik Basu
Spring 2008
Econ 367: Game-Theoretic Methods
Problem Set 14
[This was the final examination in 2006.]
1.
Consider the two-player normal-form game described below.
L
R
T
4, 4
0, 0
B
0, 0
2, 2
(a)
Locate all the pure and mixed-strategy Nash equilibria of this game.
Answer:
There are three NE – (T,L), (B,R), {(T with prob. 1/3, B with prob. 2/3);
(L with prob. 1/3, R with prob. 2/3)}.
(b)
Suppose we use
p
to denote the probability of player 1 playing
T
and
q
to
denote the probability of player 2 playing
L
.
Using a diagram where the
horizontal axis represents
q
and the vertical axis,
p
, draw the ‘reaction
function’ of player 1, that is, a graph showing the optimal choice of
p
for
every value of
q
.
Answer:
Expected pay-off for choosing T = 4q,
expected pay-off for choosing B = 2(1 – q).
Therefore, BR
1
(L,q) = 1 if 4q > 2(1 – q),
= p є [0,1] if 4q = 2(1 – q),
= 0 if 4q < 2(1 – q).
Or, BR
1
(L,q) = 1 if q > 1/3,
= p є [0,1] if q = 1/3,
= 0 if q < 1/3.