Why Does Party Distribution Stabilize in a Political Matrix Model?

[gamerninja213]

Homework Statement

We are given a matrix with the following data

Democrats Republicans Independents Libertarians
Democrats 0.81 0.08 0.16 0.10
Republicans 0.09 0.84 0.05 0.08
Independent 0.06 0.04 0.74 0.04
Libertarians 0.04 0.04 0.05 0.78

Homework Equations

Let x_n be the vector (D_n, R_n, I_n, L_n)^T. It represents the percentage of representatives of each party after n presidential elections and we shall call it the party distribution

in general x_n = (Pⁿ)(x₀)
Why is it that for for a significantly large n, say n approaches infinity, no matter what your initial distribution of the electorate is, it does not seem to affect the distribution in the long term.

The Attempt at a Solution

Say, you can choose any vector in R4 for x0 so long as all the entries sum up to 100 and multiplying this by

Pinf =

0.3548 0.3548 0.3548 0.3548
0.3286 0.3286 0.3286 0.3286
0.1570 0.1570 0.1570 0.1570
0.1600 0.1600 0.1600 0.1600
will always give you the following vector:

35.4809
32.8634
15.7046
15.9955Why is this?I notice this is true for any Matrix P whose first row has entries that are all the same value second row has entries that are all the same value etc.

 

[Valkarie]

This is due to the fact that the matrix P is stochastic, which means that all of its columns each sum up to 1. This implies that when you multiply any vector x0 by this matrix P, its entries will always sum up to 1 and so the distribution of the parties will remain the same. As n approaches infinity, the entries in Pn will become more and more concentrated around this fixed vector.