Project 1
Math 314, section 1
Points: 40
Due: Tuesday, March 12

 Three automobile insurance firms compete for a fixed market of customers. 
Annual premiums are sold to these customers. We will label the companies A, B
and C.  You work for Company A, and your team of market analysts has done a
survey which draws the following conclusions: in each of the past three years,
the number of A customers switching to B is 20%, and to C is 30%. The number
of B customers switching to A is 20%, and to C is 20%. The number of C
customers switching to A is 30%, and to B is 10%. Those who do not switch
continue to use their current company's insurance for the next year. The first
part of your problem is to model this market as a Markov chain. Display the
transition matrix for the model. To illustrate the workings of the model, show
what it would predict as the market shares three years from now if currently
A, B and C owned equal shares of the market.

 The next part of your problem is as follows: your team has tested two
advertising campaigns in some smaller test markets and are confident that the
first campaign will convince 20% of the B customers who would otherwise stay
with B in a given year to switch to A. The second advertising campaign would
convince 25% of the C customers who would otherwise stay with C in a given
year to switch to A. Both campaigns have about equal costs and would not
change other customers habits.  You have to make a recommendation, based on
your experiments with various possible initial state vectors for the market.
Will these campaigns actually improve your company's market share? If so,
which one are you going to recommend to your superiors? Write up your
recommendations.  Also show the results of your calculations.  Use different
starting states and numbers of years for projections.  Try to explain your
results.  It might be helpful to examine Example 3.5 on page 132 of the text. 
It will also be helpful for calculations to look at the Maple notebook
LinalgMaple5.ms.  You may use any word processor for the report, or even a
Maple notebook; in this case, pay attention to formatting.  The end result
should look like a project report, not a homework notebook. See page 41 of
your text for some advice about project writing.  Your project should be
readable by others and not directed only to the instructor.

% Notes: improvements may be found in discussion of the key.