Exam 9: Markov Chains and the Theory of Games

The transition matrix for a Markov process is given by State 1 2 Given that the outcome state 2 has occurred, what is the probability that the next outcome of the experiment will be state 1?

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

Consider the two-person, zero-sum matrix, strictly determined game. ​ ​ Find the value of the game. ​

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

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

Find X₂ (the probability distribution of the system after two observations) for the distribution vector X₀ and the transition matrix T. ​

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

Within a large metropolitan area, 20% of the commuters currently use the public transportation system, whereas the remaining 80% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 40% of those who are now commuting to work via automobile will switch to public transportation, and 60% will continue to commute via automobile. At the same time, it is expected that 20% of those now using public transportation will commute via automobile and 80% will continue to use public transportation. ​ What percentage of the commuters are expected to use public transportation 6 months from now? ​

Consider the coin-matching game played by Richie and Chuck with the payoff matrix ​ ​ Find the optimal strategies for Richie and Chuck. ​ ​ ​ Find the value of the game. ​ Does it favor one player over the other?

Determine whether the given matrix is stochastic. ​ ​

Diane has decided to play the following game of chance. She places a $1 bet on each repeated play of the game in which the probability of her winning $1 is 0.5. She has further decided to continue playing the game until she has either accumulated a total of $3 or has lost all her money. What is the probability that Diane will eventually leave the game a winner if she started with a capital of $1? What is the probability that Diane will eventually leave the game a winner if she started with a capital of $2? Write your answer as a decimal rounded to two decimal places.

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

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

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

Within a large metropolitan area, 15% of the commuters currently use the public transportation system, whereas the remaining 85% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 10% of those who are now commuting to work via automobile will switch to public transportation, and 90% will continue to commute via automobile. At the same time, it is expected that 40% of those now using public transportation will commute via automobile, and 60% will continue to use public transportation. In the long run, what percent of the commuters will be using public transportation? ​ __________ %

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

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

Determine whether the given matrix is stochastic.

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