i. Consider the following zero-sum game given in tabular form:
A B C
I 2 0 -1
II -2 1 3
III 1 1 -2
a) Find an optimal strategy pair ($p, q$) under the assumption that both strategies have all pure strategies in their support. (5 marks)
b) Verify that the strategy pair ($p, q$) is indeed optimal, and find the value of the game. (5 marks)
ii. Consider the following two-player game given in strategic form in the following table:
A B
I (0,0) (-2,5)
II (4,-1) (-1,-1)
And consider the indefinitely repeated game $G(p)$ consisting of playing $G$ repeatedly, and after each such subgame, stopping with probability $1 - p$.
Consider the row player's strategy $s_1$ consisting of playing I as long as the other player has always played A, and playing II otherwise. Consider also the column player's strategy $s_2$ consisting of playing A as long as the other player has always played I, and playing B otherwise.
For which values of $p$ is ($s_1, s_2$) a Nash equilibrium? (5 marks)
