Exam 9: Markov Chains and the Theory of Games

Determine whether the matrix is an absorbing stochastic matrix. ​

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

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S. ​ ​

Rewrite the absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices R and S.

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

Determine whether the given matrix is stochastic. ​ ​

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. ​ ​

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

Records compiled by the admissions office at a state of university indicating the percentage of students who change their major each year are shown in the following transition matrix. Of the freshman now at the university, 20% have chosen thier major field as business, 30% in the humanities, 30% in education, and 20% in the natural sciences and other fields. Assuming that this trend continues, find the percentage of these students that will be majoring in each of the given areas in thier senior year. Hint : Find . Bus) Hum) ​ Educ. Nat) Sc. and others Business Humanities Education Nat) sci. and others

Determine whether the given matrix is an absorbing stochastic matrix. ​

Is the matrix regular? ​

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game. ​ ​ R: row __________C: column __________

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. ​

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

Consider the two-person, zero-sum matrix, strictly determined game. ​ ​ Find the saddle point(s) of the game. ​

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

A study conducted by the Urban Energy Commission in a large metropolitan area indicates the probabilities that homeowners within the area will use certain heating fuels or solar energy during the next 10 years as the major source of heat for their homes. The transition matrix representing the transition probabilities from one state to another is Elec. Gas) Oil Solar Electricity Natural gas Fuel oil Solar energy Among homeowners within the area, 35% currently use electricity, 30% use natural gas, 25% use oil, and 10% use solar energy as the major source of heat for their homes. What is the expected distribution of the homeowners that will be using each type of heating fuel or solar energy within the next decade?

Determine whether the given matrix is stochastic. ​ ​

Determine whether the matrix is regular. ​ ​