Problem Statement
Subscribers to two music streaming platforms, SoundWave ($S$) and MelodyHub ($M$), switch services each month according to the transition probability matrix:
$$\mathbf{T} = \begin{pmatrix} 0.85 & 0.10 \\ 0.15 & 0.90 \end{pmatrix}$$
where column 1 represents current SoundWave users and column 2 represents current MelodyHub users.
**(a)** [3 marks] If initial market shares are $60\%$ SoundWave and $40\%$ MelodyHub, find the market share after 3 months.
**(b)** [5 marks] Find the long-term steady-state market share for each service.
Verified Solution & Marking Scheme
Part (a) [M1]
Initial state vector $\mathbf{s}_0 = \begin{pmatrix} 0.60 \\ 0.40 \end{pmatrix}$. State after 3 months is $\mathbf{s}_3 = \mathbf{T}^3 \mathbf{s}_0$
Part (a) [M1]
Calculate $\mathbf{T}^3$ using GDC: $\begin{pmatrix} 0.85 & 0.10 \\ 0.15 & 0.90 \end{pmatrix}^3 = \begin{pmatrix} 0.655 & 0.230 \\ 0.345 & 0.770 \end{pmatrix}$
Part (a) [A1]
$\mathbf{s}_3 = \begin{pmatrix} 0.655(0.60) + 0.230(0.40) \\ 0.345(0.60) + 0.770(0.40) \end{pmatrix} = \begin{pmatrix} 0.485 \\ 0.515 \end{pmatrix}$. SoundWave: $48.5\%$, MelodyHub: $51.5\%$
Part (b) [M1]
Set up steady-state equation: $\mathbf{T}\mathbf{s} = \mathbf{s}$, where $\mathbf{s} = \begin{pmatrix} p \\ 1 - p \end{pmatrix}$
Part (b) [A1]
$0.85p + 0.10(1 - p) = p$
Part (b) [M1]
$0.85p + 0.10 - 0.10p = p \implies 0.75p + 0.10 = p \implies 0.25p = 0.10$
Part (b) [A1]
$p = \frac{0.10}{0.25} = 0.40$
Part (b) [A1]
Steady-state: SoundWave has $40\%$ market share, MelodyHub has $60\%$ market share.