The concept that allows us to get from a current state, such as market shares, to a future state is the matrix of transition probabilities. This is a matrix of conditional probabilities of being in a future state given a current state. The following definition is helpful:
For example,
Let
Individual
We used historical data with the three grocery stores to determine what percentage of the customers would switch each month. We put these transitional probabilities into the following matrix:
Recall that American Food Store represents state 1, Food Mart is state 2, and Atlas Foods is state 3. The meaning of these probabilities can be expressed in terms of the various states as follows:
Row 1
Row 2
Row 3
Note that the three probabilities in the top row sum to 1. The probabilities for any row in a matrix of transition probabilities will also sum to 1.
After the state probabilities have been determined along with the matrix of transition probabilities, it is possible to predict future state probabilities.