Game theory: Finding Nash equilibrium

Hello!

I'm not looking for the answer per-say, but if someone could possibly prod me in the right direction(and explain if i'm doing the right thing) i'd really appreciate it.

It’s the second two parts(b and c) i’m unsure about.

Also, apologies but for some reason the latex/tex tags won't work for me :S

(6; 5) (4; 8) (6; 4) (9; 2)

(4; 6) (7; 4) (7; 5) (4; 4)

(4; 7) (4; 4) (9; 5) (2; 6)

(5; 9) (4; 10) (4; 9) (8; 9)

Question 1:

a.) Find all Nash equilibrium for the above game.

b.) For any profile x that you claim is N.E, prove that it is an NE

c.) Prove there are no other profiles(either mixed or pure) that are N.E, other than one's you claim.

So, in terms of finding all the Nash Equilibrium, I’m using dominance. None of the above have any strict dominance, but the last two column are weakly dominated by column 1.

Which gives:

(6; 5) (4; 8)

(4; 6) (7; 4)

(4; 7) (4; 4)

(5; 9) (4; 10)

Now row 4 and 3 are dominated by row 1 which leaves:

C D

A (6; 5) (4; 8)

B (4; 6) (7; 4)

Now. Let player 1 play A with probability p and B with (1-p)

Let player 2 play C with probability q and D with (1 - q)

Getting the equations and answers:

6q + 4(1 - q) = 4q + 7(1 - q)

13q = 8q + 3

q = 3/5

5p + 6(1 - p) = 8p + 4(1 - p)

5p = 2

p = 2 / 5

So(if I have this the right way round). Player 2 must play profile = {⅗, ⅖, 0, 0 } to make it indifferent to player 1’s choice. Player 1 must play profile {⅖, ⅗, 0, 0} to make it indifferent to player 2’s strategies.

Now in regards to the later parts of the question:

b.) For any profile x that you claim is N.E, prove that it is an NE.

I’m not sure how to prove it’s an N.E.

c.) Prove there are no other profiles(either mixed or pure) that are N.E, other than one's you claim.

And i’m not sure how to prove there are no other mixed strategy profiles because i’m not even sure i’ve found them all myself...the matrix was reduced by weak dominance which would mean there are possible other strategies?

Plus, aren't there an infinite number of possibilities for mixed strategy nash?