what is each company's expected martket share in the long run?

Question

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 5% of its customers to NCJ and 20% to Dash, NCJ loses 25% of its customers to GTT and 25% to Dash, and Dash loses 30% of its customers to GTT and 30% to NCJ. Assuming that these percentages remain valid over a long period of time, what it each company's expected martket share in the long run?


GTT's Expected market Share is ___ %(round to the nearest tenth as needed)


NCJ's Expected market Share is ___ %(round to the nearest tenth as needed)


Dash's Expected market Share is ___ %(round to the nearest tenth as needed)

Answer 1

You can solve this question using Markov chains, but we stick to a simpler analysis.

 

Every year,

GTT: 5% to NCJ, 20% to Dash

NCJ: 25% to GTT, 25% to Dash

Dash: 30% to GTT, 30% to NCJ

 

Now let G, N, D be the proportion of market share of GTT, NCJ, and Dash respectively.

We want

G = G + 0.25N + 0.3D - 0.05G - 0.2G = 0.75G + 0.25N + 0.3D

N = N + 0.05G + 0.3D - 0.25N - 0.25N = 0.5N + 0.05G + 0.3D

D = D + 0.2G + 0.25N - 0.3D - 0.3D = 0.4D + 0.2G + 0.25N

G + N + D = 1

 

Simplifying,

0.25G = 0.25N + 0.3D

0.5N = 0.05G + 0.3D

0.6D = 0.2G + 0.25N

G + N + D = 1

 

Intuitively, the 4 equations imply that the net change in market share for each company must be zero.

 

Solving,

G = 52.63%

N = 21.05%

D = 26.32%

 

Leave a note in the comments if you want me to explain further.