Political Party Switching in [Country Name]: Chart & Analysis

[jkeatin]

In a certain country, there are three political parties: The conservative Butter-side-up party
(the Uppers), the radical Butter-side-down party (the Downers), and the progressive Pitapocket
party (the Pocketeers). Recent polls show that each year, 70% of Uppers remain
Uppers, 80% of Downers remain Downers, 40% of Pocketeers remain Pocketeers. Everyone
else switches, and they divide equally among the other two parties.
(a) Write the transition matrix representing the party-switching process.
(b) In the long run, how will the people be distributed among the political parties?

 

[Vid]

What have you tried?

 

[jkeatin]

.7 .15 .15
.8 .2 .2
.4 .3 .3


is that say matrix A?

 

[HallsofIvy]

No, that's not correct. Assuming (A, B, C) means that one year A% uppers, B% downers, C% pitas, then you must have
$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a{_33}\end{array}\right]\left[\begin{array}{c}1.0 \\ 0.0 \\ 0.0\end{array}\right]= \left[\begin{array}{c} 0.7 \\ 0.15 \\ 0.15\end{array}\right]$$

$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\left[\begin{array}{c}0.0 \\ 1.0 \\ 0.0\end{array}\right]= \left[\begin{array}{c} 0.1 \\ 0.8 \\ 0.1\end{array}\right]$$
and
$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\left[\begin{array}{c}0.0 \\ 0.0 \\ 1.0\end{array}\right]= \left[\begin{array}{c} 0.3 \\ 0.3 \\ 0.4\end{array}\right]$$

You can determine the "a"s from that.

 

[jkeatin]

.7 .1 .3
.15 .8 .3
.15 .1 .4





then do i just reduce the matrix equation and find the values?

 

[jkeatin]

or do i find the eigenvalues and eigenvectors?

 

[Mark44]

or do i find the eigenvalues and eigenvectors?

Have you tried what Halls suggested?

 

[HallsofIvy]

.7 .1 .3
.15 .8 .3
.15 .1 .4





then do i just reduce the matrix equation and find the values?

or do i find the eigenvalues and eigenvectors?

C'mon now, you are not coming to this having no idea what you are doing are you? WHY would you "reduce the matrix equation" (what matrix equation) or "find the eigenvalues and eigenvectors"? What would either of those tell you?

Intrepret "in the long run as meaning "after many years". After n years, You will have A_n X where X is the initial distribution. How can you most easily find Aⁿ for large n?

Have you tried what Halls suggested?

He did- in post #5.

 

[jkeatin]

I took .7 , .8 , and .4 and subtracted 1 from each, then i got x1=6t x2=3t and x3=t
so 60% uppers, 30% downers 10% pocketeers, is that close or am i just way off

 

[HallsofIvy]

What reasoning did you use to arrive at that answer?