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Thread: Every time i work this problem out, i get the wrong answer. If someone can help me

  1. #1
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    Exclamation Every time i work this problem out, i get the wrong answer. If someone can help me

    Consumers in a certain state can choose between three long-distance telephone services: GTT, NCJ, and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 20% of its customers to NCJ and 15% to Dash, NCJ loses 5% of its customers to GTT and 10% to Dash, and Dash loses 10% of its customers to GTT and 20% to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?


    GTT's expected market share is %. (Round to the nearest tenth as needed.)

    NCJ:

    Dash:
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  2. #2
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    Re: Every time i work this problem out, i get the wrong answer. If someone can help m

    first build up the transition matrix

    Let the state vector be the percentages of GTT, NCJ, and Dash in that order.

    $v = (G,~N,~D)$

    GTT loses 20% to NCJ and 15% to Dash, so it retains 65% of it's customers.

    Further it gains 5% from NCJ and gains 10% from Dash.

    So the top row of the transition matrix is

    $T_{1,j} = (0.65, ~0.05, ~0.1)$

    Similarly we find that the entire transition matrix is given by

    $T = \begin{pmatrix}0.65 &0.05 &0.1 \\ 0.2 &0.85 &0.2 \\ 0.15 &0.1 &0.7 \end{pmatrix}$

    we also know that the percentages for each phone company total to 1.

    At equilibrium the transition matrix acting on the state vector returns the same state vector.

    So we have the following equations to solve

    $Tv = v$

    $v\cdot (1,1,1) = 1$

    We thus see that $v$ is just the eigenvector of $T$ corresponding to eigenvalue $1$, that has been normalized so it's components sum to 1.

    Can you compute that vector?
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