Exam 9: Markov Chains and the Theory of Games

A psychologist conducts an experiment in which a mouse is placed in a T-maze, where it has a choice at the T-junction of turning left and receiving a reward (cheese) or turning right and receiving a mild shock. At the end of each trial a record is kept of the mouse's response. It is observed that the mouse is as likely to turn left (state 1) as right (state 2) during the first trial. In subsequent trials, however, the observation is made that if the mouse had turned left in the previous trial, then the probability that it will turn left in the next trial is .6, whereas the probability that it will turn right is .4. If the mouse had turned right in the previous trial, then the probability that it will turn right in the next trial is .2, whereas the probability that it will turn left is .8. In the long run, what percentage of the time will the mouse turn left at the T-junction? Please round the answer to one decimal place. ​ __________ %

Determine whether the given matrix is stochastic. ​

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

From data collected by the Association of Realtors of a certain city, the following transition matrix was obtained. The matrix describes the buying pattern of home buyers who buy single-family homes (S) or condominiums (C) . Currently, 75% of the homeowners live in single-family homes, whereas 25% live in condominiums. If this trend continues, what will be the percent of homeowners in this city who will own single-family homes and condominiums 2 years from now? In the long run? Please round the answers to one decimal place. Percentage of single-family homes owners 2 years from now is __________% Percentage of condominiums owners 2 years from now is __________% Percentage of single-family homes owners the long run is __________% Percentage of condominiums owners over the long run is __________%

In a study of the domestic market share of the three major automobile manufacturers , , and in a certain country, it was found that of the customers who bought a car manufactured by , 75% would again buy a car manufactured by , 10% would buy a car manufactured by , and 15% would buy a car manufactured by . Of the customers who bought a car manufactured by , 90% would again buy a car manufactured by , whereas 5% each would buy cars manufactured by and , respectively. Finally, of the customers who bought a car manufactured by , 75% would again buy a car manufactured by , 5% would buy a car manufactured by , and 20% would buy a car manufactured by . Assuming that these sentiments reflect the buying habits of customers in the future model years, determine the market share that will be held by each manufacturer in the long run. ​ Please round your answers to the nearest tenth of a percent, if necessary. ​ __________ % manufactured by ​ __________ % manufactured by ​ __________ % manufactured by

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

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. ​ ​

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

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

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 30% of those who are now commuting to work via automobile will switch to public transportation, and 70% 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. Let denote the state "use the public transportation system" and denote the state "commute via automobile". Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters. Find the initial distribution vector for this Markov chain. What percentage of the commuters are expected to use public transportation 6 months from now? __________ %

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. Give your answer to two decimal places, if necessary. ​

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

A television poll was conducted among regular viewers of the national news in a certain region where the three national networks share the same time slot for the evening news. Results of the poll indicate that 40% of the viewers watch the ABC evening news, 40% watch the CBS evening news, and 20% watch the NBC evening news. Furthermore, it was found that of those viewers who watched the ABC evening news during 1 week, 75% would again watch the ABC evening news during the next week, 10% would watch the CBS news, and 15% would watch the NBC news. Of those viewers who watched the CBS evening news during 1 week, 75% would again watch the CBS evening news during the next week, 5% would watch the ABC news, and 20% would watch the NBC news. Of those viewers who watched the NBC evening news during 1 week, 85% would again watch the NBC news during the next week, 10% would watch ABC, and 5% would watch CBS. What share of the audience will each network command in the long run? ​

The registrar of Computronics Institute has compiled the following statistics on the progress of the school's students in their 2-yr computer programming course leading to an associate degree: Of beginning students in a particular year, 80% successfully complete their first year of study and move on to the second year, whereas 20% drop out of the program; of second-year students in a particular year, 85% go on to graduate at the end of the year, whereas 15% drop out of the program. Determine the probability that a beginning student enrolled in the program will complete the course successfully. ​

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.

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

Because of the continued successful implementation of an urban renewal program, it is expected that each year 2% of the population currently residing in the city will move to the suburbs and 4% of the population currently residing in the suburbs will move into the city. If the initial probability distribution is ​ ​ What will be the population distribution of the city after 1 year? After 2 years? Assume that the total population of the metropolitan area remains constant. ​

Within a large metropolitian area, 20% of the commuters currently use the public transportation system, whereas the remaining 80% commute via automobile. The city recently revitalized and expanded its public transportation system. It is expected that 6 months from now, 20% of those who are now commuting to work via autombile will switch to public transportaion, and 80% will continue to commute via automobile. At the same time, it is expected that 70% of those now using public transportation will commute via automobile, and 30% will continue to use public transportation. In the long run, what percentage of commuters will be using public transportation? ​

Morris Polling conducted a poll 6 months before an election in a state in which a Democrat and a Republican were running for governor and found that 70% of the voters intended to vote for the Republican and 30% intended to vote for the Democrat. In a poll conducted 3 months later, it was found that 60% of those who had earlier stated a preference for the Republican candidate still maintained that preference, whereas 40% of these voters now preferred the Democratic candidate. Of those who had earlier stated a preference for the Democrat, 70% still maintained that preference, whereas 30% now preferred the Republican candidate. ​ If the election were held at this time, who would win? ​ Assuming that this trend continues, which candidate is expected to win the election?

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