In Example 2, what percentage of the voters are likely to vote for the Republican candidate in some election many years from now, assuming that the election outcomes form a Markov chain?

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For computations by hand, the wrong approach is to pick some initial vector [latex] x_{0} [/latex] and compute [latex] x_{1} , . . . , x_{k} [/latex] for some large value of k. You have no way of knowing how many vectors to compute, and you cannot be sure of the limiting values of the entries in [latex] x_{k} [/latex].

The correct approach is to compute the steady-state vector and then appeal to Theorem 18.

THEOREM 18 :

If P is an n × n regular stochastic matrix, then P has a unique steady-state vector q. Further, if [latex] x_{0} [/latex] is any initial state and [latex] x_{k+1} = Px_{k} ext{for} k = 0, 1, 2, . . . , [/latex] then the Markov chain [latex] \{x_{k}\} [/latex] converges to q as [latex] k ightarrow \infty [/latex].

Given P as in Example 2, form P - I by subtracting 1 from each diagonal entry in P. Then row reduce the augmented matrix:

[latex] \left [ \begin{matrix} (P-I) & 0 \end{matrix} ight ] = \left [ \begin{matrix} -.3 & .1 & .3 & 0 \ .2 & -.2 & .3 & 0 \ .1 & .1 & -.6 & 0 \end{matrix} ight ] [/latex]

Recall from earlier work with decimals that the arithmetic is simplified by multiplying each row by 10¹.

[latex] \left [ \begin{matrix} -3 & 1 & 3 & 0 \ 2 & -2 & 3 & 0 \ 1 & 1 & -6 & 0 \end{matrix} ight ] \sim \left [ \begin{matrix} 1 & 0 & -9/4 & 0 \ 0 & 1 & -15/4 & 0 \ 0 & 0 & 0 & 0 \end{matrix} ight ] [/latex]

The general solution of (P - I)x = 0 is [latex] x_{1} = \frac{9}{4}x_{3} , x_{2} = \frac{15}{4}x_{3} [/latex], and [latex] x_{3} [/latex] is free. Choosing [latex] x_{3} = 4 [/latex], we obtain a basis for the solution space whose entries are integers, and from this we easily find the steady-state vector whose entries sum to 1:

[latex] w = \left [ \begin{matrix} 9 \ 15 \ 4 \end{matrix} ight ] , ext{and} q = \left [ \begin{matrix} 9/28 \ 15/28 \ 4/28 \end{matrix} ight ] \approx \left [ \begin{matrix} .32 \ .54 \ .14 \end{matrix} ight ] [/latex]

The entries in q describe the distribution of votes at an election to be held many years from now (assuming the stochastic matrix continues to describe the changes from one election to the next). Thus, eventually, about 54% of the vote will be for the Republican candidate.

¹Warning: Don’t multiply only P by 10. Instead, multiply the augmented matrix for equation (P - I)x = 0 by 10.

Question 4.9.6: In Example 2, what percentage of the voters are likely to vo...

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