Markov Model Mania (or Will You Stay in Business?)

We are coming to the end of the year so it is a good time to reflect on where we have been and start thinking about next year. And to make the thinking definite, let’s suppose that you estimate that 85% of your customers will stick with you next year and 10% of your competitor’s customers will switch and become your customers. Is it time to open the champagne and give everyone a big holiday bonus???

And just to make the math easy, let’s just suppose that you presently have 1000 customers and your competitors have a total of 1000 customers as well. 

The math is really easy. The number of customers you are going to have next year is:

markov5 So far this is looking like you many want to keep the champagne on ice. Suppose this trend continues, where will you be after two years?

markov6 Hmmm…this picture just keeps getting more disturbing.

You have lost customers two years in a row despite retaining 85% of your present customers and despite winning 10% new customers each year. What's going on?

Will you keep losing customers until you lose them all, or will you reach some steady state value? Both excellent questions and both can be easily answered by modeling your customer churn as a Markov process.

Markov model candidates cover processes that fulfill the following four criteria:

  1. There are a finite number of states.
  2. The transition probabilities remain constant.
  3. It is possible to transition between states.
  4. The system does not simply cycle through the states.

Our customer market share process fits this description. There are only two states, customers and non-customers. It is given that the transition probabilities are fixed. Customers can become non-customers and vice versa, and finally all customers don’t simply become non-customers and all non-customers don’t simply become customers (that is, they there isn’t a simple cycle through the states).

The figure that follows illustrates the dynamics of our process. The system has two states. There are customers and there are non-customers. You retain 85% of the customers and gain 10% of the new customers each cycle. 

Markov1 If we used matrices to represent the mathematics that we are doing, the calculation to determine the number of customers after one year would look like this:

markov7And the calculation to determine the number of customers after two years would look like this:

markov8 And, if there is an equilibrium to this process, then for some number of customers K (and some number of non-customers L), after a churn cycle, we should end up with the same number of customers and non-customers that we started with. That is we simply need to solve the following equation:

markov10Which means that:

markov2 And since we know that:

Markov3 It follows that:

markov4 That is, the long-range equilibrium is that you will end up with 800 customers or 40% of the total market share. This is a remarkable result.

Just think about it. As long as you have some customers to start with, you will end up with 800 customers. And despite keeping 85% of your customers year over year, you are only going to end up with 40% market share.

If you wanted to gain market share you could have a year-end sale or some other promotion. While that will give you a brief injection of new customers, if the transition probabilities go back to where they were, the dynamics of the process are once again going to drive towards 800 customers. 

So you could, of course, just add more promotions throughout the year (President’s Day, Labor Day, Memorial Day, Arbor Day etc.) and that would temporarily inject customers who would then churn to your competition. Or you could work on changing those transition probabilities so you keep more than 85% of your existing customers and increase your long-term overall market share.