## Modelling Customer Behaviour in the Long Term

A market is controlled by two companies, A and B. During any given month, 10% of the customers of each company switch brands. In a certain month company A has 300 customers and company B has 200. We want to predict the number of customers for each company in the following month.
We can represent the number of customers each month by a vector. In the given month the vector is
$\mathbf{v}= \begin{pmatrix}300\\200\end{pmatrix}$
.
We can represent the change from month to moth by a transition matrix. The matrix must take into account that 10% of each customers change loyalty each month, so that if
$A, \: B$
are the number of customers of each company in the given month, the number of customers of A in the next month will be
$0.9A+0.1B$
and the number of customers of company B will be
$0.1A+0.9B$
.
The transition matrix is
$T=\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right)$
. In the following month the customer vector will be
$\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right) \begin{pmatrix}300\\200\end{pmatrix}= \begin{pmatrix}290\\210\end{pmatrix}$
.
The number of customers of each company may become stable. In the long term
$T \mathbf{v}=\mathbf{v} \rightarrow (M-I) \mathbf{v}=\mathbf{0}$
.
Hence
$(M-I) \mathbf{v}=(\left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right)-\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) ) \begin{pmatrix}A\\B\end{pmatrix} =\left( \begin{array}{cc} -0.1 & 0.1 \\ 0.1 & 0.1 \end{array} \right) \begin{pmatrix}A\\B\end{pmatrix} = \begin{pmatrix}-0.1A+0.1B\\0.1A-0.1B\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}$
.
Then
$-0.1A+0.1B=0.1A-0.1B=0 \rightarrow A=B$
but since
$A+B=500$
we must take
$A=B=250$
.
In the long term the market will split evenly. Models like this are called Markov chains.