Question

Consumers in a certain state can choose between three long-distance telephone services: GTT, NCJ, and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 25% of its customers to NCJ and 25% to Dash, NCJ loses 20% of its customers to GTT and 20% to Dash, and Dash loses 25% of its customers to GTT and 30% 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 1

@jim_thompson5910

Answer 2

GTT's expected market share is ____%
NCJ's expected market share is _____%
Dash expected market share is _____ %

Answer 3

@amistre64

Answer 4

reminds me of a markov chain

Answer 5

Lets unpack the novel ....

Consumers in a certain state can choose between three long-distance telephone services: GTT, NCJ, and Dash.

Aggressive marketing by all three companies results in continual shift of customers among the three services.

Each year,

GTT loses 25% of its customers to NCJ and 25% to Dash,
NCJ loses 20% of its customers to GTT and 20% to Dash,
Dash loses 25% of its customers to GTT and 30% 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?

i assume its asking for the stable state?

Answer 6

yes i believe it is a makov chain

Answer 7

i believe the movement then looks like this each year? Rows define the movements, and Columns define who gets what

G N D
G .50 .25 .25
N .20 .60 .20
D .25 .30 .45

Answer 8

or do i have that backwards?

Answer 9

lets assume thats backwards, and that the columns have to add up to 1. which way would you set it up?

Answer 10

\[\begin{pmatrix}
.50&.20&.25\\
.25&.60&.30\\
.25&.20&.45
\end{pmatrix}\Huge X=\Huge X\]
then X is the stable setup

Answer 11

this should remind you of eugene vectors and values

Answer 12

what is that correct order? are u sure its not backwards?
cuz i dont think it is backwards or maybe im mistaken? im not quite sure

Answer 13

ive read too many things to be certain at this point. If a had a textbook in front of me id be able to be certain. Some sites i see say columns are 1, others say rows are 1

Do you have a textbook handy?

Answer 14

it is an online assignment

Answer 15

that is the question :p

Answer 16

even online assignments have material for resourses dont they?

i got a google book that has the columns as 1s so ill feel safe with that

Answer 17

okay

Answer 18

they have a recurssion set up with vectors x0, x1, x2, x3 .... unti xn is the stable state. but im not sure how they come up with the initial state vector x0

Answer 19

the initial state vector seems to be composed of the probability of any one person moving to either G N or D

Answer 20

arent all destinations equally likely for any random person to go to?

Answer 21

http://aix1.uottawa.ca/~jkhoury/markov.htm

this seems to be similar to your problem, with any luck we can glean something useful

Answer 22

woohoo, i got it :)

Answer 23

hmm i still dont understand how to find the market share percentage for GTT, Dash, and NCJ

Answer 24

we are multiplying the original matrix to itself, over and over again

GND^1 * GND^1 = GND^2

GND^1 * GND^2 = GND^3

GND^1 * GND^3 = GND^4

Answer 25

after a few goes at it, you should be able to see the entries evening out into a stable set

Answer 26

i plugged the matrix setup into my ti83, and i am running a few iterations at the moment

Answer 27

so what would be the result from multiplying it over and over again?

Answer 28

im getting a limit of say: 30.62%, 40.67%, and 28.71%

Answer 29

\[\begin{pmatrix}
.50&.20&.25\\
.25&.60&.30\\
.25&.20&.45
\end{pmatrix}\cdot \begin{pmatrix}
.50&.20&.25\\
.25&.60&.30\\
.25&.20&.45
\end{pmatrix}=T^1\]

\[T^1\cdot \begin{pmatrix}
.50&.20&.25\\
.25&.60&.30\\
.25&.20&.45
\end{pmatrix}=T^2\]

\[T^2\cdot \begin{pmatrix}
.50&.20&.25\\
.25&.60&.30\\
.25&.20&.45
\end{pmatrix}=T^3\]

... eventually the entries even out such that
\[T^n\approx T^{n+1}\]

Answer 30

and of course the market share is in the form of GND since that is how i setup my matrix

Answer 31

okay

Answer 32

im sure there is a faster way to work it, but im not sure what it would be. but does the process make sense now?

Answer 33

yes it makes a little more sense for me now
haha im just not sure how to get the percentage of each market share you know?

Answer 34

@amistre64

Answer 35

i can spend a few minutes and type up an html to run the matrixes

Answer 36

i just need help finding/showing me how to get the percentages

Answer 37

do you know how to multiply matrixes?

Answer 38

i forgot how to do it thats why i need someone to show me then i will be able to remember

Answer 39

lets take a simple 2x2 setup
\[\begin{pmatrix}
a&b\\
c&d\\
\end{pmatrix}\cdot \begin{pmatrix}
1&5\\
3&2\\
\end{pmatrix}=\begin{pmatrix}
1a+3b&5a+2b\\
1c+3d&5c+2d\\
\end{pmatrix}\]

Answer 40

the new matrix is constructed by the inner product of Ar1 and Bc2, meaning A row 1 and B col 2

given A = R1 , R2 and B = C1 C2

the new matrix developed from A times B is:

R1.C1 R1.C2
R2.C2 R2.C2

Answer 41

in the case of the Markov here; A and B are the same matrix ...

Answer 42

\[\begin{pmatrix}
1&5\\
3&2\\
\end{pmatrix}\cdot \begin{pmatrix}
1&5\\
3&2\\
\end{pmatrix}=\begin{pmatrix}
1(1)+3(5)&5(1)+2(5)\\
1(3)+3(2)&5(3)+2(2)\\
\end{pmatrix}\]

Answer 43

\[\begin{pmatrix}
1(1)+3(5)&5(1)+2(5)\\
1(3)+3(2)&5(3)+2(2)\\
\end{pmatrix}
\cdot
\begin{pmatrix}
1&5\\
3&2\\
\end{pmatrix}=\\
\begin{pmatrix}
1[(1(1)+3(5)]+3(5(1)+2(5)]&5[1(1)+3(5)]+2[5(1)+2(5)]\\
1[1(3)+3(2)]+3[5(3)+2(2)]&5[1(3)+3(2)]+2[5(3)+2(2)]\\
\end{pmatrix}\]

and keeps on going and going and going

Answer 44

i just wrote this up for a 3x3

Answer 45

notice that in the first question you asked, when i input:

.50 .20 .25
.25 .60 .30
.25 .20 .45

and run it for 100 times we get:

M1
0.3625 0.27 0.2975
0.35000000000000003 0.47 0.3775
0.2875 0.26 0.325


M2
0.323125 0.294 0.3055
0.38687499999999997 0.4275 0.398375
0.29 0.27849999999999997 0.29612499999999997


M3
0.31143750000000003 0.302125 0.30645625
0.39990624999999996 0.41355000000000003 0.4042375
0.28865624999999995 0.28432499999999994 0.28930625


M4
0.3078640625 0.30485375000000003 0.3064021875
0.4044 0.40895875000000004 0.40594843750000004
0.2877359374999999 0.28618749999999993 0.287649375


M5
0.306746015625 0.3057655 0.306303125
0.405926796875 0.4074449375 0.406464421875
0.28732718749999997 0.28678956249999993 0.28723245312499995


M6
0.30639016406249997 0.30606912812500003 0.30625256015625
0.40644073828125 0.40694520624999997 0.4066241703125
0.2871690976562499 0.28698566562499994 0.28712326953124995


M7
0.3062755041015625 0.30617002171875 0.30623193152343753
0.40661271328125 0.40678010546875 0.4066746230859375
0.28711178261718745 0.28704987281249994 0.287093445390625


M8
0.3062382403613282 0.30620350015625 0.30622425172656254
0.40667003877929686 0.40672553055468746 0.40669079034960937
0.28709172085937495 0.28707096928906245 0.2870849579238281


M9
0.3062260581513672 0.30621459851132815 0.3062215234141602
0.40668909961572264 0.40670748415859376 0.4066960245185547
0.28708484223291014 0.2870779173300781 0.2870824520672851


M10
0.3062220595570557 0.3062182754199024 0.30622057962761234
0.40669542697714844 0.4067015153220117 0.4066977311848584
0.28708251346579583 0.2870802092580859 0.2870816891875293


M11
0.3062207435404065 0.306219493088875 0.3062202583476602
0.4066975251152918 0.40669954082560844 0.4066982903740769
0.2870817313443017 0.2870809660855166 0.2870814512782629


M12
0.3062203096293371 0.30621989623093837 0.30622015006821124
0.4066982203575672 0.4066988875932388 0.40669847419484007
0.2870814700130957 0.2870812161758229 0.2870813757369487


M13
0.30622016638945593 0.3062200296780727 0.30622011380731085
0.4066984506258033 0.40669867146642474 0.4066985347550415
0.28708138298474073 0.2870812988555026 0.28708135143764774


M14
0.3062201190660738 0.306220073846197 0.30622010171407565
0.40669852686826824 0.4066985999560238 0.4066985547361469
0.28708135406565793 0.28708132619777926 0.28708134354977743


M15
0.3062201034231051 0.3062200884637481 0.3062200976917116
0.40669855210717676 0.40669857629449735 0.4066985613351403
0.2870813444697181 0.2870813352417546 0.2870813409731481


M16
0.30622009825041746 0.3062200933012122 0.30622009635617087
0.4066985604609978 0.40669856846516184 0.40669856351595657
0.28708134128858476 0.287081338233626 0.28708134012787256


M17
0.3062200965395545 0.306220094902045 0.30622009591324495
0.40669856322577846 0.40669856587448794 0.4066985642369785
0.287081340234667 0.28708133922346707 0.28708133984977663


M18
0.3062200959735998 0.30622009543178685 0.30622009576646236
0.40669856414075584 0.40669856501724416 0.4066985644754313
0.28708133988564444 0.28708133955096893 0.28708133975810635


M19
0.3062200957863622 0.3062200956070845 0.30622009571784403
0.4066985644435468 0.4066985647335839 0.4066985645543063
0.28708133977009104 0.2870813396593315 0.28708133972784966


M20
0.3062200957244132 0.306220095665092 0.30622009570174574
0.406698564543746 0.406698564639721 0.4066985645803997
0.2870813397318408 0.28708133969518707 0.28708133971785454


M21
0.30622009570391606 0.306220095684287 0.3062200956964165
0.4066985645769032 0.4066985646086617 0.40669856458903264
0.2870813397191808 0.2870813397070513 0.28708133971455085


M22
0.30622009569713393 0.3062200956906387 0.3062200956946526
0.4066985645878752 0.40669856459838416 0.406698564591889
0.28708133971499095 0.2870813397109771 0.2870813397134585


M23
0.3062200956948898 0.3062200956927405 0.3062200956940687
0.4066985645915059 0.4066985645949833 0.40669856459283404
0.2870813397136044 0.2870813397122762 0.2870813397130972


M24
0.3062200956941472 0.306220095693436 0.30622009569387554
0.4066985645927073 0.406698564593858 0.4066985645931468
0.28708133971314553 0.28708133971270605 0.28708133971297767


M25
0.3062200956939015 0.30622009569366615 0.3062200956938116
0.4066985645931048 0.4066985645934856 0.40669856459325027
0.2870813397129937 0.28708133971284827 0.28708133971293814


M26
0.3062200956938202 0.3062200956937423 0.30622009569379044
0.4066985645932364 0.40669856459336234 0.40669856459328446
0.2870813397129435 0.28708133971289534 0.2870813397129251


M27
0.30622009569379327 0.3062200956937675 0.30622009569378345
0.4066985645932799 0.40669856459332154 0.4066985645932958
0.2870813397129268 0.2870813397129109 0.28708133971292077


M28
0.3062200956937844 0.30622009569377584 0.30622009569378117
0.40669856459329423 0.40669856459330805 0.40669856459329956
0.2870813397129213 0.28708133971291605 0.2870813397129193


M29
0.30622009569378145 0.3062200956937786 0.3062200956937804
0.406698564593299 0.4066985645933036 0.4066985645933008
0.2870813397129195 0.28708133971291777 0.2870813397129188


M30
0.3062200956937805 0.30622009569377956 0.3062200956937801
0.4066985645933006 0.4066985645933021 0.40669856459330117
0.2870813397129189 0.2870813397129183 0.2870813397129187


M31
0.30622009569378017 0.30622009569377984 0.30622009569378006
0.40669856459330117 0.40669856459330167 0.4066985645933013
0.2870813397129187 0.2870813397129185 0.2870813397129186


M32
0.30622009569378006 0.30622009569377995 0.30622009569378
0.4066985645933013 0.4066985645933015 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M33
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330145 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M34
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.4066985645933014 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M35
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M36
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M37
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M38
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M39
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M40
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M41
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M42
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M43
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M44
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M45
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M46
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M47
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M48
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M49
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M50
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M51
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M52
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M53
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M54
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M55
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M56
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M57
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M58
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M59
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M60
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M61
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M62
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M63
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M64
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M65
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M66
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M67
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M68
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M69
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M70
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M71
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M72
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M73
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M74
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M75
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M76
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M77
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M78
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M79
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M80
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M81
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M82
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M83
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M84
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M85
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M86
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M87
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M88
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M89
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M90
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M91
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M92
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M93
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M94
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M95
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M96
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M97
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M98
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186


M99
0.30622009569378 0.30622009569378 0.30622009569378
0.40669856459330134 0.40669856459330134 0.40669856459330134
0.2870813397129186 0.2870813397129186 0.2870813397129186

Answer 46

by the time we get down to 99 iterations; the entries of each row is the same

the limit that they are approaching tells us the long run effect of the movements within the market share.

30.62%
40.67%
28.71%

Answer 47

now i did 100 to show you that these value dont seem to change that much from about the 10th iteration

Answer 48

i spose i could modify it for a 4x4 matrix .... for the second question.

Answer 49

how do i do the second question?

Answer 50

when PS^(k) = S^(k+1)

this is the same concept and procedure, and the one i first thought this post expected.

you want to find the condition such that:\[S^{(k+1)}\approx S^{(k)}\]

Answer 51

Okay so how do i find that condition