Question

Consider the following game. Ann chooses rows, and Bob chooses columns.

[12 marks]

LMR

3 4	2 0	1 3
3 2	4 4	4 0
0 2	1 1	0 1

U

C

D

(a) Find all pure strategy Nash Equilibria.

(b) Find the set of Rationalizable strategies for each player. Find a dominating strategy for each deletion.

(c) Is there a Nash equilibrium in which Bob puts strictly positive probability on L and M but none on R? If so find it; if not explain why not. (Explanation should be 3 sentances or fewer.)

(d) Is there a mixed strategy Nash equlibrium in which Bob puts strictly positive prob- ability on L and R but none on M? If so find it; If not explain why not. (Explanation should be 3 sentances or fewer.)

(e) Is there a mixed strategy Nash equlibrium in which Bob puts strictly positive prob- ability on M and R but none on L AND Ann puts strictly positive probability on both U and C? If so find it; If not explain why not. (Explanation should be 3 sentances or fewer.)

Answer 1

		Bob
		L	M	R
Ann	U	3,4	2,0	1,3
	C	3,2	4,4	4,0
	D	0,2	1,1	0,1

a) Considering the best responses to opponent's strategy, I have highlighted the payoffs.

Based on them we can see that Pure NE = UL , CM

b)Rationalizable strategies. We will go on finding dominant strategies and delete dominated ones.

If we consider Bob, we'll see that L dominated R. That is, for whatever choice Ann plays, Bob choosing L has greater payoffs than choosing R. Thus L is dominant for Bob. We' ll delete R.

		Bob
		L	M
Ann	U	3,4	2,0
	C	3,2	4,4
	D	0,2	1,1

If we consider Ann , we'll see that C dominated D. That is, for whatever choice Bob plays, Ann choosing C has greater payoffs than choosing D. Thus C is dominant for Ann. We' ll delete D.

		Bob
		L	M
Ann	U	3,4	2,0
	C	3,2	4,4

Now Ann would never play C because C dominates U. So we delete U.

		Bob
		L	M
Ann
	C	3,2	4,4

Now based on that, Bob will play M as it dominates L for best playoffs.

		Bob
			M
Ann
	C		4,4

Thus, Ann plays C and Bob plays M

c)

		Bob
		L	M	R
Ann	U	3,4	2,0	1,3
	C	3,2	4,4	4,0
	D	0,2	1,1	0,1

As it can be seen above, for different choices of Ann, Bob gets best payoffs in L and M but never in R. So, Bob will never choose R for a NE payoff.

There is a Nash equilibrium in which Bob puts strictly positive probability on L and M but none on R.

		Bob
		L	M
Ann	U	3,4	2,0
	C	3,2	4,4
	D	0,2	1,1

NE is : UL and CM.

d)

		Bob
	Probability	p	q	r
		L	M	R
Ann	U	3,4	2,0	1,3
	C	3,2	4,4	4,0
	D	0,2	1,1	0,1

In a mixed strategy NE, bob plays L, M, R with probabilities p , q, r respectively. p + q + r = 1

The probabilities must be such that Ann must be indifferent of her choices. That is, the utility for playing U , C and D by Ann must be same.

Based on the payoffs, equating utilities :

3p + 2q + 1r = 3p + 4q + 4r = q

In question, it is given that probability on M = 0 , i.e q = 0

Therefore, 3p + 1r = 3p + 4r = 0. For this to be true, either p or r needs to be negative. But question mentions they are strictly positive.

Thus, no such NE exists.

e)

			Bob
	Probabilities		p	q	r
			L	M	R
Ann	a	U	3,4	2,0	1,3
	b	C	3,2	4,4	4,0
	c	D	0,2	1,1	0,1

As given : p = 0 and c = 0

			Bob
	Probabilities			q	r
				M	R
Ann	a	U		2,0	1,3
	b	C		4,4	4,0

Thus, we can write r = 1-q

b = 1 - a

Now, Ann must make Bob indifferent in choices and Bob must make Ann indifferent in choices :

Thus writing payoff equations based on probability:

Bob :

0a + 4b = 3a + 0b

i.e , 4 - 4a = 3a

a = 4/7 b = 3/7 - Satisfactory

Ann : 2q + r = 4q + 4r

i,e : 2q + 1 - q = 4q + 4 - 4q

q = 3 . This cannot be. Because this would make r negative.

Thus, no such NE exists.