Question

Consumers in a certain state choose between three long distance telephone services. GTT, NCJ, and Dash. Aggressive marketing by all three companies results in a continual shift of customers among the three services. Each year, GTT loses 25% of its customers to NCJ and 20% to Dash, NCJ loses 5% of its customers to GTT and 20% to Dash, and Dash loses 15% of its customers to GTT and 5 % to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?

Please help me solve this.

Answer 1

Every year,
GTT: 5% to NCJ, 20% to Dash
NCJ: 25% to GTT, 25% to Dash
Dash: 30% to GTT, 30% to NCJ

Now let G, N, D be the proportion of market share of GTT, NCJ, and Dash respectively.
We want
G = G + 0.25N + 0.3D - 0.05G - 0.2G = 0.75G + 0.25N + 0.3D
N = N + 0.05G + 0.3D - 0.25N - 0.25N = 0.5N + 0.05G + 0.3D
D = D + 0.2G + 0.25N - 0.3D - 0.3D = 0.4D + 0.2G + 0.25N
G + N + D = 1

Simplifying,
0.25G = 0.25N + 0.3D
0.5N = 0.05G + 0.3D
0.6D = 0.2G + 0.25N
G + N + D = 1

Intuitively, the 4 equations imply that the net change in market share for each company must be zero.

Solving,
G = 52.63%
N = 21.05%
D = 26.32%

Answer 2

Does this help?

Answer 3

these numbers are different in my problem. that is the wrong answer

Answer 4

@mathmate

Answer 5

@Hero @Mehek14

Answer 6

@Kainui

Answer 7

What course is this? This looks like a markov matrix, and the long-term behavior is given by the eigenvector associated with lambda = 1

Answer 8

you should get 20% , 30% , 50% for G, N and D

Answer 9

@phi
that was the answer!!!!! i actually have the same problem with different numbers if you can help me again....
Consumers in a certain state choose between three long distance telephone services. GTT, NCJ, and Dash. Aggressive marketing by all three companies results in a continual shift of customers among the three services. Each year, GTT loses 10% of its customers to NCJ and 25% to Dash, NCJ loses 10% of its customers to GTT and 30% to Dash, and Dash loses 30% of its customers to GTT and 15 % to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?

Answer 10

you should first find the markov matrix, so that
\[\left[\begin{matrix}G \\ N \\ D\end{matrix}\right]=A\left[\begin{matrix}G \\ N \\ D\end{matrix}\right]\]

Answer 11

can you do that?

Answer 12

no I'm lost with this. i have asked my professor but haven't heard from him

Answer 13

how much does G lose (in total) to its competitors?

Answer 14

35%?

Answer 15

yes, so if it gained nothing from the other 2, how much would it have the next year ?

Answer 16

in other words, what percent does it keep?

Answer 17

65%

Answer 18

yes, so it would get
G_new= 0.65 G_old

Answer 19

\[\left[\begin{matrix}G \\ N \\ D\end{matrix}\right]=\left[\begin{matrix}0.65 & ?& ? \\ ? & ?&?\\ ? & ?&?\end{matrix}\right]\left[\begin{matrix}G \\ N \\ D\end{matrix}\right]\]

Answer 20

do you know the G on the left side is calculated by multiplying the top row of the matrix times the vector on the right?

Answer 21

.65 .10 .30
.10 .60 .15
.25 .30 .55

Answer 22

i didnt know that. i don't know how to find the vector on the right

Answer 23

as a check, the columns should sum to 1
and they do.

next, we have to find the eigenvector for eigenvalue 1
do you know how to do that ?

Answer 24

would it be
-.35 .10 .30
.10 -.40 .15
.25 .30 -.45

Answer 25

yes, that is a good start
A- I (in this case)

now find reduced row echelon form

Answer 26

this is where i got stuck the last time ... how do i find the form

Answer 27

step 1: use elimination to zero out the first column
can you do that ?

Answer 28

do i divide the first row by 1/65?

Answer 29

-.35 .10 .30
.10 -.40 .15
.25 .30 -.45

you multiply the first row by 0.10/0.35 and add the resultant to the 2nd row

Answer 30

.025?
i added it to the entire row

Answer 31

the idea is 0.10/0.35 * -0.35 = -0.10
and when you add that to the 2nd row, the first entry is zeroed out

Answer 32

0.10/0.35 is 0.28571
when you multiply the first row by that you get
-0.100000 0.028571 0.085714

when you add that row to the 2nd row you get
0.00000 -0.37143 0.23571

Answer 33

ok i understand now.... what is the next step

Answer 34

you use the same idea to zero the first entry in the last row

Answer 35

how did you get -0.37143 0.23571 in the second row?

Answer 36

starting at the beginning

multiply the first row by 0.10/0.35
you get
-0.100000 0.028571 0.085714

Answer 37

you then add that to the 2nd row
-0.100000 0.028571 0.085714
.10 -.40 .15

Answer 38

ok then i know you add -0.100000 to the first one in the second row giving you 0

Answer 39

you get a new 2nd row
0.00000 -0.37143 0.23571

Answer 40

ok .. i will try the 3rd row now

Answer 41

-0.100000 0.028571 0.085714
0.00000 -0.37143 0.23571
.15 0.328571 -.364286

Answer 42

is that correct?

Answer 43

you leave the first row as it was. The -0.100000 0.028571 0.085714 is only a "side calculation"

in other words, after we zero out the first entry in the 2nd row
*only* the 2nd row in the matrix changes.

when you do the 3rd row, you still use the *original* first row.

Answer 44

here is what you have after doing the first row, but before doing the 3rd row
-.35 .10 .30
0.00000 -0.37143 0.23571
.25 .30 -.45

Answer 45

*after doing the 2nd row, but before doing the 3rd row

Answer 46

yes i forgot you multiply ... i was adding
what is the next step

Answer 47

zero out the 0.25 in the 3rd row
you do that by multiplying the first row by 0.25/0.35 to get a *temporary row"
that you add to the 3rd row to get a *new third row*

Answer 48

is the third row ....
0 .37142857 -0.23571429

Answer 49

yes. good news, it looks close to the 2nd row

Answer 50

ok what is the second row

Answer 51

what is the next step***

Answer 52

-.35 .10 .30
0.00000 -0.37143 0.23571
.25 .30 -.45

becomes
-.35 .10 .30
0.00000 -0.37143 0.23571
0 .37142857 -0.23571429

Answer 53

yes i have that

Answer 54

we go to the next column over, and down 1 row
the new "pivot" is the -0.37143
multiply the 2nd row by 1 (i.e. do nothing)
and add it to the 3rd row. This will eliminate the entry below the pivot

Answer 55

you will get
-.35 .10 .30
0.00000 -0.37143 0.23571
0.000 0.00000 0.0000

Answer 56

next, we want the pivots to be 1
the first pivot is the -0.35 in location (1,1)
to make it one, we divide the 1st row (all entries) by -0.35 to get a *new first row*

Answer 57

1 .28571427 .85714286
0.00000 -0.37143 0.23571
0.000 0.00000 0.0000

Answer 58

is that correct?

Answer 59

except for the signs. they should be negative for the other 2 entries, right ?

Answer 60

correct i was diving by .35 not -.35

Answer 61

dividing*

Answer 62

1 - 0.28571427 - 0.85714286
0.00000 -0.37143 0.23571
0.000 0.00000 0.0000

now make the 2nd pivot 1. that is, divide the 2nd row by -0.37143 to get a new 2nd row

Answer 63

1 - 0.28571427 - 0.85714286
-0.37143 1 -0.63460141
0.000 0.00000 0.0000

Answer 64

the first entry in the 2nd row stays 0

Answer 65

oh ok

Answer 66

1 - 0.28571427 - 0.85714286
0 1 -0.63460141
0.000 0.00000 0.0000

Answer 67

finally, we want the pivot columns to be all zeros (except for the pivot)
in other words, we want to "clear out" the -0.28... above the 2nd pivot
to do that, multiply the 2nd row by 0.2857... and add that to the first row to get a *new 1st row* (the 2nd row stays as is)

Answer 68

0.2857 0 0.67582818
0 1 -0.63460141
0.000 0.00000 0.0000

Answer 69

you did something wrong.
the 2nd row (temp result) should be
0 0.28571427 -0.1813...

now add that to the 1st row
1 - 0.28571427 - 0.85714286

Answer 70

oh ok

Answer 71

what is the matrix ?

Answer 72

1 0 - 1.03544286
-0.37143 1 -0.63460141
0.000 0.00000 0.0000

Answer 73

sorry this....
1 0 - 1.03544286
0 1 -0.63460141
0.000 0.00000 0.0000

Answer 74

better!

Answer 75

now we "read off" the eigenvector
or
we solve for the eigenvector

Answer 76

how?

Answer 77

do you know how to find vectors in the "null space" of a matrix? (or have you heard of this?)

Answer 78

sorry i have not

Answer 79

not to get to far off track: the eigenvector equation is
\[ A x = \lambda x \\ Ax - \lambda x = 0 \\ \left( A - \lambda I\right) x = 0 \]
and x is a vector that gives 0 (i.e. a vector of zeros) when we multiply by the matrix
\( \left( A - \lambda I\right) \)

Answer 80

all of which means reduced row echelon form matrix we found times the the eigenvector x gives 0

\[\left[\begin{matrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 0\end{matrix}\right]\left[\begin{matrix}x \\ y \\z\end{matrix}\right]=\left[\begin{matrix}0\\0 \\0\end{matrix}\right]\]

Answer 81

the z (associated with the row of zeros) is a free variable. People usually pick it to have a value of 1
now solve for x and y
(I used a and b instead of the long decimals)

Answer 82

remember the matrix is short-hand for a system of equations
in this case

0*z = 0 (let z=1 , though any value would work, 1 is a nice choice)
0x + 1y + b*z= 0
1 x + 0*y +a*z = 0
solve for y and x

Answer 83

you get x= -a, y=-b, z = 1
in other words, the vector is
\[\left[\begin{matrix}-a \\ -b \\ 1\end{matrix}\right]\]

Answer 84

where in our case, a and b are the entries in the last column of the reduced row echelon matrix

Answer 85

ok sorry I'm trying to do it on my end .... I'm lost

Answer 86

the last step is we want the entries in the vector to sum up to 1
so divide the entries by the sum

Answer 87

what step ?

Answer 88

so x= -1.03544286
y= -.63460141
z= 1

Answer 89

almost.
the equation (for x for example) is
x -1.035*z = 0
where we let z= 1 we get
x - 1.035 = 0

Answer 90

in other words, we negate the two entries in the 3rd column when we use them in the eigenvector

Answer 91

ok so how do we divide by the sum

Answer 92

my assignment is do in a few so I'm trying to understand this quicker

Answer 93

add up the entries [ 1.03 0.63 1] (you can use more decimals)
then divide each entry by that value

Answer 94

and, if you like, multiply that vector by 100, so the values represent percent

Answer 95

isn't the value 1? if we divide by 1 we will get the same answer??

Answer 96

1.03846
0.63462
1.00000

does not add up to 1
but we want it to.
so divide each entry by the sum 2.6731

Answer 97

it's the same idea as changing [ 1 2] into ⅓ ⅔