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(Markov chain model) Question 3. Three big companies, A, B, and C, share cus- tomers in one region: A has 50% share of cus- tomers, B has 30%, and C has 20%. Each wants to increase its share of customers, and to do so, each introduces a new promotion. After one year, it is learned that

(Markov chain model) Question 3. Three big companies, A, B, and C, share cus- tomers in one region: A has 50% share of cus- tomers, B has 30%, and C has 20%. Each wants to increase its share of customers, and to do so, each introduces a new promotion. After one year, it is learned that

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 47E: Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.
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I (Markov chain model) Question 3. Three big companies, A, B, and C, share cus- tomers in one region: A has 50% share of cus- tomers, B has 30%, and C has 20%. Each wants to increase its share of customers, and to do so, each introduces a new promotion. After one year, it is learned that (i) A keeps 70% of its customers and loses 20% to B and 10% to C. (ii) B keeps 60% of its customers and loses 20% to A and 20% to C. (iii) C keeps 50% of its customers and loses 30% to A and 20% to B. Questions: (a) What is the transition matrix for that model? (b) What is the share of customers of each su- permarkets after 2 years? (c) In the long term, what is the share of each companies?
Transcribed Image Text:I (Markov chain model) Question 3. Three big companies, A, B, and C, share cus- tomers in one region: A has 50% share of cus- tomers, B has 30%, and C has 20%. Each wants to increase its share of customers, and to do so, each introduces a new promotion. After one year, it is learned that (i) A keeps 70% of its customers and loses 20% to B and 10% to C. (ii) B keeps 60% of its customers and loses 20% to A and 20% to C. (iii) C keeps 50% of its customers and loses 30% to A and 20% to B. Questions: (a) What is the transition matrix for that model? (b) What is the share of customers of each su- permarkets after 2 years? (c) In the long term, what is the share of each companies?
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