Suppose that the payoffs are given by the matrix Player 1 U D L Player 2 R θ, γ 1,2 -1,7 0,0 where 0 = {0,2} is known by player 1, y = {1,3} is known by player 2, and all pairs of (0,y) have probability 1/4. Provide a formal definition of the Bayesian game and compute the Bayesian Nash equilibrium.

2. Consider the payoff matrix below: Player 1 b) Strategy A Strategy B Player 2 Strategy A Strategy B (2,2) (5,0) (0,5) (3,3) a) What is Player 1's dominant strategy (if they have one)? What is Player 2's dominant strategy (if they have one)? Does this game have a Nash Equilibrium? If so, what is/are they?

We will use Markov chain to model weather XYZ city. According to the city’s meteorologist, every day in XYZ is either sunny, cloudy or rainy.  The meteorologist have informed us that the city never has two consecutive sunny days.  If it is sunny one day, then it is equally likely to be either cloudy or rainy the next day.  If it is rainy or cloudy one day, then there is one chance in two that it will be the same the next possibilities.  In the long run, what proportion of days are cloudy, sunny and rainy? Show the transition matrix.

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).

How does a Nash Equilibrium affect consumers? It makes some customers travel farther than necessary to purchase product. It eliminates the possibility of a socially optimal solution. (c ) It enables customers the ability to compare the products (price, variety, etc.) with ease. (D All of the above

Consider a first-price sealed bid auction of a single object with two bidders j = 1,2 and no reserved price. Bidder 1's valuation is v1 = 2. and bidder 2's valuation is v2= 5. Both v1 and v2 are known to both bidders. Bids must be in whole dollar amounts. In the event of a tie, the object is awarded by a flip of a fair coin. Is there a Nash equilibrium? What is it? Is it unique? Is it efficient?

The weather can be considered a stochastic system, because it evolves in a probabilistic manner from one day to the next. Suppose for a certain location that this probabilistic evolution satisfies the following description: The probability of rain tomorrow is 0.6 if it is raining today. The probability of it being clear (no rain) tomorrow is 0.8 if it is clear today. (For simplicity, you can assume clear = 0 and rain = 1) a. If today’s weather is clear, simulate the weather condition for the next 7 days. b. What kind of probability is used in the table above (Poisson distribution, Normal Distribution, Uniform distribution, Exponential distribution)? Explain your answer.

5) A Professor conducts an exam in a class with 24 boys and 16 girls. 6 boys and 4 girls make an A in the course. a) Write the joint pohahility matrix of the gender of the student and his/her grade (A or no A). b) Find the marignal probabilities of a student being a boy or a girl and the marginal probabilities of a student making or not making an A in the course. Show that the gender of a student and whether he/she makes an A in the course, are independent.

Every day, Eric takes the same street from his home to the university. There are 4 streetlights along his way, and Eric has noticed the following Markov dependence. If he sees agreen light at an intersection, then 60% of time the next light is also green, and 40% of timethe next light is red. However, if he sees a red light, then 70% of time the next light is alsored, and 30% of time the next light is green.(a) Construct the transition probability matrix for the street lights.(b) If the first light is green, what is the probability that the third light is red?(c) Eric’s classmate Jacob has many street lights between his home and the university. Ifthe first street light is green, what is the probability that the last street light is red?(Use the steady-state distribution.)

1. Suppose we have a 2-player zero-sum game where the strategy set of the row player (resp. the column player) is R = {₁,..., rk} (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.] =