Estimating results of web business changes using Markov chains

   A Markov chain is a special type of discrete-time stochastic process. It can be used as a method for estimating parameters in many economic and financial applications. It can be also used in modeling of new web business processes. Here is one example.

  Let's take a web site that is providing some service for free. After acquiring a lot of users, the webmaster is planning to split this service at 2 levels: one is free with basic features, and one with some extra features for a small fee. In situations such as this, it is very useful to know how many users will be willing to switch to level 2 and how many will want to use the system for free. A Markov chain can help to estimate the numbers. The initial data for solving this problem can be obtained from the past history of this service or by conducting a survey of users or in some other way.

  
To solve this problem, we need a transition probability matrix. Each element of this matrix there is the probability of switching from one state to another or from one state to this state. In our case, we have 2 states: a free service and a paid service. How can these probabilities be obtained? Here is one example. Let's say that currently 80% of the customers are using only free features, and the other 20% are using all of the features. Then 80% will not switch to the second level service. Also, let's say that from a survey we know that 10% of those who are using extra features will not want to pay the fee.

  The transition probability matrix P, in this case, will be:

0.8    0.2
0.1    0.9

   The next step is to find so called steady-state probabilities. They will reveal the probability that the process will be in a given state after a very long time, i.e., in the long run. In our example, it means the probability of switching from one level of service to the other or staying in the same level.

   To obtain these probabilities, we need to solve the following system of equations.

(p0, p1)=(p0,p1)P
p0+p1=1

   For our problem, substituting transition probability matrix P in the first eq., we have

p0*0.8+p1*0.1=p0
p0*0.2+p1*0.9=p1
p0+p1=1

  Substituting p1=1-p0 in the first eq., we can find p0 and then p1.

And, finally, the solution of the system will be p0=1/3 p1=2/3

  The qty p0 can be interpreted as the long-run fraction of users that will stay with the free service, and p1 can be interpreted as the long-run fraction of users that will use this service for a fee.

  Thus, a Markov chain and steady-state probabilities can be used in the modeling of web site processes. Specifically, we saw how they can help to estimate results in terms of adding or expanding a new business, service, site, product, and so on.

References
Barry L. Nelson (1995) "Stochastic Modeling Analysis & Simulation"
R. Bhar, Shigeyuki Hamori (2004) "Hidden Markov Models : Applications to Financial Economics"