Kaushik Basu
Spring
2008
Econ 367: Game-Theoretic Methods
Problem Set 4
[This is a take-home examination.—Total 10 points. Your answer must be turned in (hard
copies, please) by 21
st
February, 5 pm. You may hand it over to Vidya or Sarah after class
on 21
st
Feb. Do not discuss the questions with anybody till the answers are posted on the
course website.]
1. (a)
Describe the
mixed strategy
Nash equilibrium (that is, one that actually
involves mixing two strategies) in the game shown below.
L
R
U
4,
4
0,
0
D
0,
0
2,
2
Answer:
Suppose P1 plays U with probability p and P2 plays L with probability q.
P1 will be indifferent between the strategies U and D when
4q = 2(1-q), i.e., when q = 1/3.
Similarly, P2 will be indifferent between the strategies L and R when
4p = 2(1-p), i.e., when p = 1/3.
Therefore, the
mixed strategy
Nash equilibrium (that is, one that actually
involves mixing two strategies) in the game above is (p*, q*) = (1/3, 1/3).
(b)
In the game described below (as always, player 1 chooses between rows and
2 between columns) locate all the pure strategy Nash equilibria. Now allow
players to mix their strategies. Is there a Nash equilibrium in which player 1
mixes the two strategies? Is there one in which player 2 mixes strategies?
L
R
U
4,
2
0,
2
D
6,
0
0,
2
Answer:
There are two pore strategy Nash equilibria: (U,R) and (D,R).
Yes. Suppose P1 plays U with probability p and P2 plays L with
probability q. Take (p,q) = (1/2, 0). When P2 is playing the pure strategy
R (i.e., playing L with probability 0), P1 is indifferent between his
strategies U and D and thus does not deviate from playing p = ½. On the