Mathematical Modeling: Math 456, Spring 2013

Class Meeting : TuTh 1:00--2:15, Goessmann Lab Addtn 151

Instructor : Luc Rey-Bellet

Office :  1423 J LGRT
Phone :  545-6020
E-Mail :   luc <at> math.umass.edu
Homepage:   http://www.math.umass.edu/~lr7q
Office Hours :   Tu 11:00--12:00,   Th 3:00--4:00,   or by appointment.

Teaching Assistant :  Christopher Rocheleau

Office :  1235 L LGRT
Phone :   545-4773
E-Mail :   rocheleau <at> math.umass.edu
Office Hours :   MW 4:00--5:30

Syllabus: This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate problems from "real-life" into a mathematical model and how to use mathematics to solve the problem. The domain if applicability of mathematics is huge and covers much of the natural sciences, but mathematics plays a central role in modern economics, and increasingly in social sciences. In this class we will pick a number of topics from games and gambling, economics, social sciences, but also magic (card tricks) and some biology (evolutionary theory) to illustrate how mathematics can be useful to analyze concrete problems. From the mathematical point of view we will use elementary tools from probability, game theory, information theory, and optimization. The prerequisite for this class is a one year sequence of calculus. We will use throughout the class very elementary notions from probability (discrete math), linear algebra, and differential equations. All the necessary mathematics will be introduced from scratch and motivated by examples. Among the problems to be discussed are

• The Monty Hall problem: Behind one of three doors is a prize hidden. After you pick one of the three doors, the game show host opens one of the remaining door and reveals no prize. You are given the choice to keep your door or switch. What do you do?
  
  
• The Gambler's ruin problem. You walk into a casino with $100 and decide to play roulette (say you will bet on red). Your goal is to make at least $1000 or of course get wiped out. What is the optimal strategy?
  
  
• Why are duopolies better than monopolies?
  
  
• Why are there as many males as females in most animals populations?
  
  
• Why does it seem that if an accident happens (say a man falls on the rail track) the more people are around the less likely it is that somebody helps?
  
  
• How to compute the number π using a random number generator?
  
  
• Can mathematics help you to make card tricks?
  
  
• Many more .......

Classnotes:: As the class progresses I will post regularly some handouts. You should use as a COMPLEMENT of your class notes, not as a substitute. They will provide a summary of the class and a reference for definitions and main results.

• Week 1: Probability models and counting
  
  
• Week 2: Conditional probability and Bayes formula
  
  
• Week 3: Part 1: Information Cascades and Herding. I borrowed the material directly from the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World, by David Easley and John Kleinberg (preprint version freely available here). The chapter on information cascades is Chapter 16 and can be found there
  
  Week 3: Part 2: Difference equations.
  
  
• Week 4: Gambler's ruin and bold play.
  
  
• Week 5: Expected value and betting systems.
  
  
• Week 6: The Law of large numbers and Kelly's criterion.
  
  
• Week 7:
  
  
• Week 8: The basics of game theory.
  
  
• Week 9: Part 1 Monopolies and duopolies.
  
  

Textbooks and references: We are not following any particular textbook but we are borrowing material from many sources.

General mathematical references: Two references for background material for the class.

• Introduction to Probability by Charles M. Grinstead and J. Laurie Snell. American Mathematical Society.
  
  This is a very good introduction to probability and you can download a free copy (it is all legal) HERE.
  
  
• Game theory by James N. Webb. Springer Undergraduate Series.
  
  This is an introductory text to game theory, well-suited to the class. See in particular the section called "Interactions". UMass library owns an electronic copy which can be found HERE (UMass NET ID needed).

References on special mathematical topics: Here we have books on various topics of mathematical modeling which I have found useful in preparing this class. The mathematical level varies enormously.

• Naive decision making by T.W. Körner. Cambridge University Press.
  A nice book on mathematical modeling for social sciences in the spirit of our course.
  
  
• Networks, Crowds, and Markets: Reasoning about a Highly Connected World, by David Easley and John Kleinberg. Cambridge University Press.
  This book is at a very elementary level. A preprint version of the book is freely available HERE
  
  
• The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser by Jason Rosenhaus. Oxford University Press.
  This mathematics books is entirely devoted to the Monty Hall problems and all its refinment. It is very good and you could use it for a project.
  
  
• The Mathematics of Games and Gambling by Edward Packel. Mathematical Association of America.
  This books covers much of the same material as our course, but it at a more elementary mathematical level. Very well-written.
  
  
• The doctrine of chances: Probabilistic aspects of gambling by Stewart N. Ethier. Springer
  This is an advanced book on the probability behind gambling. Many games are analyzed in details. Awesome but hard. UMass library owns an electronic copy which can be found HERE (UMass NET ID needed).
  
  
• The website www.gametheory.net, although a bit dated, contains a lot of useful information. In particular there are links to many useful applets (game solvers, Monty Hall problems..).
  
  

Popular literature: Here we have a series books, many containing not much mathematics but relating mathematical ideas and concepts to many problems in real life.

• The signal and the noise by Nate Silver
  An excellent book on the applications of Bayes formula to a range of topics. Recommended.
  
  
• Beat the dealer Edward O. Thorp.
  How does card counting at Blackjack can tilt the odds in your favor.
  
  
• Fortune's Formula by William Poundstone.
  A quite entertaining book about betting systems. Recommended.

Grading and assignments: Your grade will based on your group project (%50), homework (%40), and class participation (%10).

Homwework: Homework will be assigned regularly throughout the class. The homework will be graded. I expect you to do your homework regularly and carefully to assimilate the material. Your homework counts as %40 percent of your grade.

• Homework 1: Do the problems in Week 1 and Week 2 class notes. Due on February 12.
  
• Homework 2: Do the problems in Week 3.2 and Week 4 class notes. Due on February 28.
  
• Homework 3: Do the problems in Week 5 and Week 6 class notes. Due on March 14.
  
• Homework 4: Do the problems in Week 8 and Week 9.1 class notes. Due on April 11.
  
• Homework 5:
  

Group project: The class will be divided in groups of three students. Every group will select a subject in consultation with the instructora. You have a lot of freedom to choose your subject as long as it is related to mathematical modeling. The references above can be a good starting point to find a project. In particular game theory is used in a very wide variety of contexts and will give you plenty of options, depending on your tastes and backgrounds.

The group project as 2 parts:

• An in-class presentation of around 15 minutes (+ a few minutes for questions). This presentation is done as a group. The main goal of the presentation is to explain our project to your fellow students. Make sure you spend enough time to explain the problem and give a quick summary of your findings.
• The second part is a paper which is written individually by each member of the group. I DO NOT want three times the same paper. If, in your group, you have divided the work and considered different aspects, your paper may certainly reflect this, but each paper should present the whole project in its entirety.
  There is no lower or upper bound on the length of the paper since your topic may request more or less space. But being precise and concise are two necessary attributes of well-written mathematics.
  Your paper should adress the following points.
  • Motivation for your choice of topic.
  • Modeling issues. How hard is it to translate your model in mathematics?
  • What are your sources? Gives references to the literature.
  • A clean presentation of the mathematics involved. Make your results clear. If the computations are long, present your main finding and put the computation in an appendix.
  • A summary of your findings and open questions.
  • If in your presentation in class you use slides, please attach them as an appendix (or better send me the file).