Kaushik Basu
Spring
2008
Econ 367: Game-Theoretic Methods
Problem Set 4
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 other hand, when P1 plays U with p = ½, P2’s
expected payoff from playing L is 1 which is less than 2 - his expected payoff