# Calculate market share based on probability

I am trying to estimate the distribution of market share based on current market share and incorporating switching probability.

Consider 2 Coffee chains - Starbucks and Pret.

Each day there is 20% chance that a customer switches from Starbucks to Pret A Manger and 10% chance that a customer switches from Pret to Starbucks.

Current Market share = [.4,.6]
Switching probability = [[.8,.2][.1,.9]]


How do I calculate the final market share distribution?

• The question is unanswerable unless you make additional assumptions about all other ways in which customers can be lost or gained, such as there is no growth in the market and no other competitors. Also, by "20% chance" shall we presume this applies to all customers independently and doesn't just mean that there's a 20% chance a single customer switches (and therefore an 80% chance that no customers switch)? – whuber Mar 2 '18 at 17:47
• If this is homework, add the self-study tag and read our policy on homework questions. – Kodiologist Mar 2 '18 at 18:13
• @whuber Let's relax those assumptions. You're right there are probably other observables and unobservable variables that we're not accounting for. And 20% probability is an average across all customers. – keval Mar 2 '18 at 18:22
• If there are other sources of changes in customers, then you haven't provided sufficient information to answer the question. There might not even exist any "final" market share. – whuber Mar 2 '18 at 19:05
• After looking at @CarlosCampos solution, I agree, don't really need numbers after all (+1 to answer) – MikeP Mar 8 '18 at 16:29

It can be modeled by Markov Process if you assume decision (switch or not from one chain to the other) depends only on previous state. You have two states: $(1,0)$ (Starbucks) or $(0,1)$ (Pret), and at the begining (day 0) you have $P((1,0)) = 0.6$ and $P((0,1)) = 0.4$ (current share). Transition matrix for both states is:
$M=\begin{bmatrix}0.8&0.2\\0.1&0.9\end{bmatrix}$
This matrix describes the probability of switching or not from one state to the other. Now, with this matrix you can compute the long term share, by powering to $\infty$:
$M^{\infty} = \lim_{n \rightarrow \infty} M^n = \begin{bmatrix}1/3 & 2/3 \\1/3 & 2/3\end{bmatrix}$
This matrix points out that at the end, $1/3$ of total costumers will be Starbucks, and $2/3$ Pret, independently on the current share.