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14.2 Matrix of Transition Probabilities

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:

Let P_{i j} = conditional probability of being in state j in the future given the current state of i

$Let P_{i j} = conditional probability of being in state j in the future given the current state of i$

For example, $P_{12}$ $P_{12}$ is the probability of being in state 2 in the future given the event was in state 1 in the period before:

Let $P =$ $P =$ matrix of transition probabilities

P = [\begin{array}{l} P_{11} & P_{12} & P_{13} & \dots & P_{1 n} \\ P_{21} & P_{22} & P_{23} & \dots & P_{2 n} \\ ⋮ & ⋮ \\ P_{m 1} & P_{m n} \end{array}]

$P = [\begin{array}{l} P_{11} & P_{12} & P_{13} & \dots & P_{1 n} \\ P_{21} & P_{22} & P_{23} & \dots & P_{2 n} \\ ⋮ & ⋮ \\ P_{m 1} & P_{m n} \end{array}]$ (14-2)

Individual $P_{i j}$ $P_{i j}$ values are usually determined empirically. For example, if we have observed over time that 10% of the people currently shopping at store 1 (or state 1) will be shopping at store 2 (state 2) next period, then we know that $P_{12} = 0.1 or 10, .$ $P_{12} = 0.1 or 10, .$

Transition Probabilities for Grocery Store Example

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:

P = [\begin{array}{l} 0.8 & 0.1 & 0.1 \\ 0.1 & 0.7 & 0.2 \\ 0.2 & 0.2 & 0.6 \end{array}]

$P = [\begin{array}{l} 0.8 & 0.1 & 0.1 \\ 0.1 & 0.7 & 0.2 \\ 0.2 & 0.2 & 0.6 \end{array}]$

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

\begin{array}{l} 0.8 & = & P_{11} & = & probability of being in state 1 after being in state 1 the preceding period \\ 0.1 & = & P_{12} & = & probability of being in state 2 after being in state 1 the preceding period \\ 0.1 & = & P_{13} & = & probability of being in state 3 after being in state 1 the preceding period \end{array}

$\begin{array}{l} 0.8 & = & P_{11} & = & probability of being in state 1 after being in state 1 the preceding period \\ 0.1 & = & P_{12} & = & probability of being in state 2 after being in state 1 the preceding period \\ 0.1 & = & P_{13} & = & probability of being in state 3 after being in state 1 the preceding period \end{array}$

Row 2

\begin{array}{l} 0.1 & = & P_{21} & = & probability of being in state 1 after being in state 2 the preceding period \\ 0.7 & = & P_{22} & = & probability of being in state 2 after being in state 2 the preceding period \\ 0.2 & = & P_{23} & = & probability of being in state 3 after being in state 2 the preceding period \end{array}

$\begin{array}{l} 0.1 & = & P_{21} & = & probability of being in state 1 after being in state 2 the preceding period \\ 0.7 & = & P_{22} & = & probability of being in state 2 after being in state 2 the preceding period \\ 0.2 & = & P_{23} & = & probability of being in state 3 after being in state 2 the preceding period \end{array}$

Row 3

\begin{array}{l} 0.2 & = & P_{31} & = & probability of being in state 1 after being in state 3 the preceding period \\ 0.2 & = & P_{32} & = & probability of being in state 2 after being in state 3 the preceding period \\ 0.6 & = & P_{33} & = & probability of being in state 3 after being in state 3 the preceding period \end{array}

$\begin{array}{l} 0.2 & = & P_{31} & = & probability of being in state 1 after being in state 3 the preceding period \\ 0.2 & = & P_{32} & = & probability of being in state 2 after being in state 3 the preceding period \\ 0.6 & = & P_{33} & = & probability of being in state 3 after being in state 3 the preceding period \end{array}$

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.

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Table of Contents for 14.2 Matrix of Transition Probabilities

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Table of Contents for
14.2 Matrix of Transition Probabilities