2001 Term Project
Due: Friday, December 14, 2001 @ 17:00
What you need to turn in to complete this project:
- A précis of your work—include a description of the algorithm (s) you use and the structure of your program. This is the place to discuss any difficulties you might have encountered.
- Your program, with LOTS of comments and dcoumentation.
- Output from representative runs.
- Your results, in tabular form, including any (optional) graphical representations you might wish to include. Since this is Monte Carlo, estimate the statistical uncertainty of your answers.
The problem:
Pretend you are a criminal who must choose among different types of crimes. The justice system provides a probability qk that if you commit the k'th type of crime you will be caught and punished. Each kind of crime has a distribution of sentence length,
pk(J)=4Jexp[–2J/j
k]/(jk)2
where jk is the average sentence for each kind of crime. That is, if you are caught for crime k you will receive a sentence of length J years with probability pk(J)dJ. On the other hand, you receive a payment mk for each successful crime.
Capture and sentencing parameters
Category of crime
|
q
|
j
(years) |
m
(dollars) |
| larceny |
0.05 |
2 |
50 |
| robbery |
0.1 |
4 |
100 |
| burglary |
0.15 |
6 |
500 |
| arson |
0.25 |
10 |
1000 |
In a time dt you can commit Rkdt crimes of type k, gaining profit
dPk=(1-qk)mk
Rkdt.
Of course you could go to jail, in which case your personal clock is advanced by J years.
Your task is to find values of Rk that will yield you at least $16,000 total profit per year and that will maximize your lifetime earnings (by minimizing the total time you spend in jail).
What would your answer be if you had a legitimate income of $30,000/yr?
Suggestions
After some discussions with students who asked about the project, I decided I had better offer some suggestions as to how to proceed.
- First write a subroutine to trace a single criminal's working life. Take his (her?) career to begin at age 16 and to end at age 65, giving 50 years of gainful employment in all. The input is a set of 4 (positive!) rates Rk for the 4 categories of crime.
- Include that subroutine in a larger one that
- creates and follows —say— a population of 100 criminals, and
- computes population averages of their total income and total time spent in jail.
- Thus the average income <I> and average (total) time in jail <Jtot> are functions of the vector R. You can now use a multivariable minimizer to maximize the ratio <I>/<Jtot> .