Exam 9: Markov Chains and the Theory of Games

At a certain university, three bookstores - the University Bookstore, the Campus Bookstore, and the Book Mart - currently serve the university community. From a survey conducted at the beginning of the fall quarter, it was found that the University Bookstore and the Campus Bookstore each had 30% of the market, whereas the Book Mart had 40% of the market. Each quarter the University Bookstore retains 80% of its customers, but loses 10% to the Campus Bookstore and 10% to the Book Mart. The Campus Bookstore retains 90% of its customers, but loses 5% to the University Bookstore and 5% to the Book Mart. The Book Mart retains 70% of its customers, but loses 10% to the University Bookstore and 20% to the Campus Bookstore. ​ What percent of the market will each store have at the beginning of the third quarter? ​

Compute the steady-state matrix of the given stochastic matrix. ​ ​

Determine which of the following is an absorbing stochastic matrix. ​ ​

Find the steady-state vector for the transition matrix. ​

The transition matrix for a Markov process and the initial-state distribution vector are given by ​ State 1 State 2 and ​ Find .

Determine whether the given two-person, zero-sum matrix game is strictly determined. ​

Determine whether the matrix is regular. ​ ​

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game. ​ ​

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game.

The registrar of a law school has compiled the following statistics on the progress of the school's students working toward the LLB degree: Of the first-year students in a particular year, 88% successfully complete their course of studies and move on to the second year, whereas 12% drop out of the program; of the second-year students in a particular year, 92% go on to the third year, whereas 8% drop out of the program; of the third-year students in a particular year, 95% go on to graduate at the end of the year, whereas 5% drop out of the program. Determine the probability that a beginning law student enrolled in the program will go on to graduate.

Answer whether the matrix ​ ​ is the steady-state matrix of the given stochastic matrix ​

As more and more old cars are taken off the road and replaced by late models that use unleaded fuel, the consumption of leaded gasoline will continue to drop. Suppose the transition matrix L UL describes this Markov process, where L denotes leaded gasoline and UL denotes unleaded gasoline. Rewrite the matrix A so that the absorbing states appear first, and identify the submatrices R and S.

Determine whether the matrix is regular. ​ ​

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​ ​

Find the optimal strategies, P and Q, for the row and column players, respectively. Also compute the expected payoff E of the matrix game and determine which player it favors, if any, if the row and column players use their optimal strategies. ​ ​

As more and more people switch to broadband Internet service, the demand for dial-up Internet service will continue to drop. Suppose the transition matrix B D describes this Markov process, where B denotes broadband Internet services and D denotes dial-up Internet service. Compute the steady-state matrix of A.

Determine which of the following is an absorbing stochastic matrix. ​

The payoff matrix for a game is ​ ​ Find the expected payoff to the row player if the row player R uses the maximin pure strategy and the column player C uses the minimax pure strategy. ​ Find the expected payoff to the row player if R uses the maximin strategy 40% of the time and chooses each of the other two rows 30% of the time, while C uses the minimax strategy 50% of the time and chooses each of the other columns 25% of the time. ​ Which of these strategies favors the row player?

Find (the probability distribution of the system after two observations) for the given distribution vector and the given transition matrix . ​

Consider the two-person, zero-sum matrix, strictly determined game. ​ ​ Find the saddle point(s) of the game. ​ Find the optimal strategy for each player. ​ R: row __________C: column __________ ​ Find the value of the game. ​ Determine whether the game favors one player over the other.