A political scientist studying the thinking of voters classifies them as either

Question

A political scientist studying the thinking of voters classifies them as either conservative ,liberal , or moderate .Based on past ecperience , the following percentages on how a voter's political tendency changes in a one-year period have been determined : 80% of all conservative voters will remain conservative one year later , while 5% will become liberal and 15% will become moderate . 80% of all liberal voters will remain liberal one year later ,while 5% will become conservative and 15% will become moderate . 90% of all moderate voters will remain moderate one year later , while 5% will become conservative and 5% will become liberal .

using the political scientist's data , predict the distribution of the voting populace in the long run .

please solve in linear algebra ways by seting up matrix . thanks ...

Explanation / Answer

This is known as a Markov chain. What I will be next year only depends on what I am this year (not what I was last year). We will construct a matrix called a transition probability matrix. There are three states, C, L, and M. The top will be the party affiliations next year and the left side will be party affiliations this year.
C is state 1
L is state 2
M is state 3
The top row is if I am conservative this year. What I will be next year is the top row of the matrix. I will be transitioning into the new states according to 0.8 0.05 0.15
The middle row is if I am liberal this year. The transistion probabilities are 0.05 0.8 0.15
The bottom row is if I am a moderate this year I will be conservative next year with a transition probability of 0.05, liberal also with a 5% chance and moderate again with 0.9.
Looking at it, we see very symmetrical numbers. Moderates will get population from the conservatives and from the liberals (15% from each each year) and only lose population at 5% to each, so the long-term distribution should be same amount of liberals and conservatives and more moderates than the others.
Here is the matrix
   C    L    M
C 0.8 0.05 0.15
L 0.05 0.8 0.15
M 0.05 0.05 0.9

Here is the matrix after many multiplications
   C     L    M
C 0.2 0.2 0.6
L 0.2 0.2 0.6
M 0.2 0.2 0.6

So, in the long run 20% will be conservative, 20% will be liberal, and 60% will be moderate. Note that each column contains the same number three times. This means that it doesn't matter what party you started in; after many years your party affiliation only depends on the transition probablities. If, for example, 100% of the people started out liberal, after many years 20% would be conservative, 20% would be liberal, and 60% would be moderate.

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