Suppose we have a 2-player zero-sum game where the strategy set of the row player (resp. the column player) is R = {₁,..., Tk} (resp. C = {₁,..., ce}) and where the payoff matrix is A = (aij). If (r₁, C₁) and (r2, C₂) are both Nash equilibria, show that they have the same payoff (i.e. a₁1 = a22). [Do this directly using the definitions and without using any theorems from the lectures.]

9. A system contains two components, A and B. The system will function only if both components function. The probability that A functions is 0.98, the probability that B functions is 0.95, and the probability that either A or B functions is 0.99. What is the probability that the system functions? 10. True or False: If A and B are mutually exclusive, a) P(AU B) = 0 b) P(An B) = 0 c) P(AU B) = P(A n B) d) P(AU B) = P(A) + P(B)

4.43. At a local grocery store there are queues for service at the fish counter (1), meat counter (2), and café (3). For i = 1,2,3 customers arrive from outside the system to station i at rate i, and receive service at rate 4 + i. A customer leaving station i goes to j with probabilities p(i, j) given the following matrix 1 2 3 0 1/4 1/2 2 1/5 0 1/5 3 1/3 1/3 0 1 In equilibrium what is the probability no one is in the system, i.e., 7(0,0,0).

3. suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system by (x, y), where x and y, respectively, take on the values 1 or 0 depending upon whether the primary machine (x) and the spare machine (y) are operational (value of 1) or not operational (value of 0) at the end of the day. [Hint: Note that (0, 0) is not a possible state.] (a) Construct the (one-step) transition matrix for this Markov chain. (b) Find the expected recurrence time for the state (1, 0).