Mathematical Modeling: Math 456-01, Spring 2019

Class Meeting : TuTh 1:00PM--2:15PM, LGRT 143

Instructor : Luc Rey-Bellet

Office :  1423 K LGRT
Phone :  545-6020
E-Mail :   luc@math.umass.edu
Homepage:   http://www.math.umass.edu/~lr7q
Office Hours :   Tu 3:00PM--4:00PM,   Th 10:30AM-12:00AM   or by appointment.

Teaching Assistant :  ByeongHo Ban

Office:  LGRT 1427  
E-Mail :   ban@math.umass.edu
Office Hours :   Tu 9:00 Am--10:00 AM, Th 9:00 AM--10:00 AM  

Deadlines, Homeworks, Classnotes (UPDATED OFTEN!) :

• Thursday January 31 : Homework 1
  
• Thursday February 7: Paper copy of your Reflective Essay due in class.
  
• Tuesday February 12: Email me your choice of book for your essay at luc@math.umass.edu
• Thursday February 14 : Homework 2
  
• Thursday February 21 : Homework 3
  
• Tuesday March 5 : Homework 4
  
• Thursday March 7 : Paper copy of your Book Essay due in class.
  
• Tuesday March 26 : Homework 5
  
• Monday A peril 1 : Provide me with the tentative topic of your group project.
  
• Tuesday April 23 : Homework 6
  

• Chapter 1: Probability models and counting
  
  
• Chapter 2: Conditional probability and Bayes formula
  
  
• Chapter 3: Linear difference equations.
  
  
• Chapter 4: Gambler's ruin and bold play
  
  
• Chapter 5: Expected value and betting systems.
  
  
• Chapter 6: Variance and the law of large numbers
  
  
• Chapter 7: Kelly betting systems.
  
  
• Chapter 8: Stable matching and college admission.
  
  
• Chapter 9: Zero-sum games.
  
  
• Chapter 10 Nash equlibria for general games
  
  
• Chapter 11 Monopolies and duopolies.
  
  

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 of 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 good working knowledge of calculus and linear algebra. We will use throughout the class elementary notions from probability (discrete math), and some differential equations, but Math 331 and Stat 515 are not formal prerequisites for the class. 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 in you are faced with unfair bets?
  
  
• 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?
  
  
• What are strategies to allocate your money among various investments?
  
  
• 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.

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. You could look at it for projects ideas!
  
  
• 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 refinements. It is very good and you could use it for a project.
  
  
• The Mathematics of Games. An introduction to probability by David G. Taylor. CRC Press
  This books covers much of the same material as our course, as far as the gambling part is concerned. Recommended.
  
  
• The Mathematics of Games and Gambling by Edward Packel. Mathematical Association of America.
  This books covers much of the same material as our course, as far as the gambling part is concerned. But maybe a more elementary mathematical level.
  
  
• 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 wizardofodds.com., is a very good website about many games of chance.
  
  
• 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.
  
  
• 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, casinos, Wall street.
  
  
• The theory that would not die by Sharon Bertsch Mcgrayne
  All about Bayes rule and its applications.
  
  
• The Black Swan: The Impact of the Highly Improbable by Nassim Taleb
  Why events which have very small probability matter.
  
  
• Gaming the vote by William Poundstone.
  A book on whether elections are fair and how to design fair elections.
  
  
• The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall
  How math came to be used in Finance.
  
  
• Prisonner's dilemma by William Poundstone.
  A somewhat older book on the birth and applications of game theory.
  
  
• Rock-Paper-Scissors by Len Fisher.
  A book on game theory and tis multiple applications.
  
  
• Jane Austen, Game theorist by Michael Suk-Young Chwe
  For the literary inclined.
  
  
• The drunkard's walk. How randomness rules our lives by Leonard Mlodinov
  For the probability inclined.
  
  
• Chances are.... Adventures in Probability by Leonard Mlodinov
  For the probability inclined.
  
  
• The selfish Gene by Richard Dawkins.
  A classic book on, among other things, why game theory matters in biology.
  
  
• The Information: A history, a theory, a flood by James Gleick.
  A book on information theory. The mathematical concept of "information" plays an increasing role in many applications.
  
  
• The survival game, by David P. Barash.
  A book on the use of game theory in biology.
  
  
• Who Gets What ? and Why: The New Economics of Matchmaking and Market Design by Alvin Roth
  Stable matching in action.
• Several more to come.....
  
  

Grading and assignments: Your grade will based on your reflective essay (%20), book review (%20) homework (%30), and final project (%30).

Reflective essay:
Each student will write a 5-page, double-spaced, reflective essay on their experiences as a mathematics major at UMass. As you farm your essay consider the path that led you to be a mathematics major, your experience taking mathematics classes, Gen Ed. courses, and other research/extracurricular activities. You may choose to comment specifically on some of the questions posed here:

• Have your career goals and expectations changed since first entering UMass?
  
• What are the most important skills and knowledge you have learned by majoring in mathematics?
  
• What new skills and knowledge have you acquired in the last 4 years that you did not anticipate?
  
• How have learning experiences in the Gen Ed courses connected to other areas? Explore connections to interests and hobbies, interactions with friends and family, and your professional interests and the mathematics major.
  
• Considering your college education as a whole, how has your experience helped to shape you as a learner and as a member of the mathematics community? What did you find most useful in your college education? What was least useful? What are your strengths and weaknesses as a learner? Would you recommend this major to a friend or relative?
  

Book review. You should pick a book (popular literature, non-fiction) which explores the connections between mathematics and real-life applications. Most such books contains little actual mathematics but can be very informative on mathematical ideas shape social and economical ideas. You can either pick a book from the list above or suggest your own book (i will need to approve it then). Write a maximum 10-pages, double-spaced, essay on the books addressing the following points

• Explain why you chose that particular book?
• Summarize the most important things you learned from the books. Was it well-written and/or informative? How did it resonate with what you have learned as a math major.
  
• Can you imagine using any knowledge gained from the book in your future studies, your eventual professional career, and/or your project in this class?
  

Homework: 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 %30 percent of your grade.

Group project: The class will be divided in groups of three students. Every group will select a subject in consultation with the instructor. I am including here a list of project you could consider. But you have some lot of freedom to choose your subject as long as it is related to mathematical modeling. And I am very much open to suggestions. 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.

Project suggestions:

• The Monty Hall problem and all its variants (see the book in the bibliography)
  
• How to use Bayes rule to understand how to bet at Texas hold'em poker, and/or Poker and game theory
  
• Blackjack, computing odds, counting cards, and so on
  
• The Math behind march madness. How to compute the odds?
  
• Explore the mathematics and economics of Fantasy Football companies
  
• Understand how betting markets work to predict elections, oscar winners, and everything else.
  
• How to use game theory to match students with their high school choice in New York city (You can start with a article on this in the New York Times).
  
• Cool applications of game theory. You can find a list and lots of references HERE
• Explain how voting systems
  
• How to cut a cake?
  
• Understanding and solving Nim, Morra, Connect Four, Hex, and other "simple" game
  
• Study the Iterated prisonner's dilemna
  
• Information cascades (start from the wikipedia page)
  

• 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).