MATH 188 --- Social Choice and Decision Making
Fall 2007
Professor Francis Su
Olin 1269, 607-3616
my-last-name@math.hmc.edu

People make decisions everyday --- from small ones such as what to eat or wear, to large ones such as which job to take or whether to buy a house. Groups of people also make decisions, such as which president to elect, or what policies to adopt, what business strategies to pursue. Mathematics has much to offer the decision-maker --- both in the analysis of strategies and suggestions for a course of action.

In this course, we will focus on the modeling of individual and group decisions using mathematical techniques from "game theory", the area of mathematics that was pioneered in the 1950's by John Nash and others, but now has applications to a wide variety of disciplines: economics, biology, computer science.

Topics will include: basic concepts of game theory and social choice theory, representations of games, Nash equilibria, utility theory, non-cooperative games, cooperative games, voting games, paradoxes, Arrow's impossibility theorem, Shapley value, power indices, "fair division" problems, and applications.

This course has substantial overlap with Math 155, so Math 188 and Math 155 cannot both be counted for credit. This course counts towards the Integrative Experience requirement at HMC.

Lectures: MW 2:30-4:00pm, Location: Beckman B126

Grader: Ahmed Oweis.

Office Hours:

  • Mondays 4 - 5 pm: (Ahmed Oweis), Math Library (except Sept. 10)
  • Tuesdays 1:15 - 2:30pm: (Prof. Su) my office: Olin 1269, or by appt.

Prerequisites: Linear algebra (Math 12) and Probability (Math 62).
Required Texts:

  • Phil Straffin, Game Theory and Strategy.
  • Alan Taylor, Mathematics and Politics.
Optional Text:

  • R. Duncan Luce and Howard Raiffa, Games and Decisions.
Course Page: http://www.math.hmc.edu/~su/math188/

Lectures: Attendance and doing the assigned reading will be essential for success in the course.
Homework: There will be weekly homeworks, announced on the course homepage. You may discuss homework problems with others, but you must write them up individually. Identical homeworks will not receive credit. No late homeworks allowed.
Exams: 1 midterm and a final exam.
Grading: Homework, Midterm, Final Project, and Final Exam are each worth 25%.

The HMC Mathematics Conference this year is on Public Sector Operations Research. This will be a great opportunity to hear several great talks on applications of game theory to real-world problems. You will be required to attend one of the talks on Fri Sept. 28 and Sat Sept.29. See the conference webpage for more details. (You do not need to register for the conference if you are attending only one talk.)

Writing Project: As this course meets the Integrative Experience requirement for HMC students, there will be a paper and an oral presentation involving a critical analysis of the relationship of the mathematical methods of game theory to contemporary society. In your paper and presentation, you'll model a real-life situation of your choice using the techniques of decision analysis and game theory, or choose an existing model in the literature to critique. Some examples could include:

  • I. Choose a past historical event or situation, model the agents and strategies involved, and use your model to explain why events transpired the way they did.
    • Ex. Division of post-war Germany
    • Ex. Intervention of 3rd party candidates in nat'l elections
    • Ex. Predator-prey relationships in biological ecosystems
  • II. Choose a current situation or event, model the agents and strategies involved, and then use your model to (a) suggest a course of action or (b) make a prediction about what will happen
    • Ex. marketing strategies for a new business
    • Ex. the Palestinian conflict
    • Ex. strategies to contain the spread of AIDS
  • III. Choose a current social choice problem and understand it in terms of ideas from cooperative game theory.
    • Ex. coalition formation in an organization
    • Ex. analysis of the power structure in a business or organization
    • Ex. resource allocation and "fair division".
Your paper should also contain a critical analysis of the methods used. For instance, does mathematics provide a good model for decision making in the context you describe? What's missing in the model? As people often make decisions based on such models, what would be appropriate or inappropriate uses of this model?

Writing Center: the friendly folk there can help you with your writing! The Writing Center provides a good opportunity for you to get feedback on your work at each stage of the writing process, from working out ideas to polishing a final draft. This is a resource that can be helpful to all writers, from novices to experts. The center is open Sunday through Thursday evenings from 7-11 P.M., and is located in TG 106. You may schedule an appointment through their website, http://www.hmc.edu/writingcenter/, or you may simply drop in during normal hours. If you'd like an appointment outside of normal hours, contact the director, Wendy Menefee-Libey, at menefee@hmc.edu. You are likely to find your writing center visit more valuable if you come earlier than the night before the paper is due.

HW Assignments
HW #1 (due Wed 9/12) READ YES, PLEASE READ
Straffin: Preface, Chapters 1, 2, 7 and
Taylor: Preface, Chapter 1.

DO Straffin Chapter 7 (3, 4) and:
Problem A. Find a short news article involving some sort of interaction or decision problem. Xerox it (or print it) and give a reference for the original source. On a separate sheet, do the following: (1) summarize the decision problem in the article, (2) then identify each of the following: (a) the players, (b) some possible strategies for each player, (c) some possible aspects of conflict or cooperation between players.

Remember to write clearly and in complete sentences and explain your reasoning carefully. Good communication is important in this class. You should follow the Math Department's Homework Guidelines.

HW #2 (due Wed 9/19) READ Taylor: Preface, Chapter 1, AND Straffin, Chapter 7.
"R" means READ and THINK about, but do not do the problem.
DO Straffin Chap.2 ( 1, 2, 3, R6, 7 ) and Chap.7 ( 5, 6abc, R6d )
AND Taylor Chap. 1 ( 1 ).
HW #3 (due Wed 9/26) READ Straffin, Chapter 3.
DO Handout ( 22, 23, 24 ) and Straffin Chap.2 ( 4, R5 ) Chap.3 ( 1, 2 )

Remember to write clearly and in complete sentences and explain your reasoning carefully. Good communication is important in this class.

Remember to attend one of the The HMC Mathematics Conference conference talks on Public Sector Operations Research, this weekend on Fri Sept. 28 and Sat Sept.29. This will be a great opportunity to hear several great talks on applications of game theory to real-world problems. See the conference webpage for more details.

You are required to attend one talk. If you attend a second talk, you will receive a bonus credit in the form of the lowest homework score dropped. Please write up a paragraph for each talk you attend, and hand these in separately from your Homework next Wednesday. (p.s. You do not need to register for the conference if you are attending only one or two talks.)

HW #4 (due Wed 10/3) READ Straffin, Chapter 4, 5, 6, 9.
DO Straffin Chap. 3 ( 3, 4, 5, R7, R9 ) Chap.4 ( 2 ) Chap.7 ( 6d ).
Also HAND-IN the one-paragraph write-ups from the Mathematics Conference talk(s) you attended.

On the problems beyond Chapter 5, you may use Tom Ferguson's game solver if you cannot reduce the matrix easily. The problems in Chapter 3 should be done by hand (reduce by deleting dominated strategies when possible), though you can check your answer with the game solver.

HW #5 (due Wed 10/10) READ Straffin: Chapters 11, 12 AND Taylor: Chapter 2.
DO Straffin 6 ( 2), 9 ( 1, 2, 3, 4, 5abc ),
AND Taylor Chap.2 ( 14, R17 ).
HW #6 (due Wed 10/17) READ Straffin Chapters 10, 16 and Taylor Chapter 2. Also skim Straffin Chapter 13-15 and read what interests you.
DO Straffin 10 ( 3 ), 12 ( 4, R6 ),
AND Taylor 2 ( 12, 13[pure strategies only], 15, 16, R18). (You can do 15b by typing "Tosca: Synopsis" in Google.)

Begin reviewing for your midterm. The Midterm will be handed out Wed 10/17, and will be due Fri 10/26 (so you don't have to do it over Fall break. There will be no HW over Fall Break.)

EXAM (due Fri 10/26) Note that the exam is due Friday 10/26 at NOON at my office (Olin 1269). You may slip the exam under my door. There is no homework due over Fall Break.
HW #7 (due Wed 10/31) READ Straffin: Chapters 17-19, Taylor: Chapter 3.
DO Handout ( 7, 9, 11, 12, 13, 15 ) [do "Y only" for problems 11,12,13]
AND Problem B: Talk with 2 other people in this class about possible final project ideas. Discuss players, strategies, and the form of the model. Record the names of those 2 people in your homework.
HW #8 (due Wed 11/7) READ Straffin: Chapters 23, 25, 26, 27 and Taylor: Chapter 4.
Do the following: get a start on NEXT week's assignment (due next week)
AND:

Flesh out some more of your ideas on your final project. In particular, use the guidelines described in Writing Project above,and write-up a one page document indicating (in separate parts)
(a) whether your project is of type I, II, III above
(b) who the players are
(c) what aspect of the situation you hope to capture in your model (cooperative/non-cooperative, decision-making aspects, possible strategies or worth function?) and
(d) what you intend to find out.

You can be quite general at this point, and this proposal does not commit you should you change your mind on the topic. Keeping the model smaller is better, and do spend some time thinking about the social or practical implications.

Please write in complete sentences, and do not write more than a page. Discuss this document with 2 other people in this course (at least one of which must be different from the last assignment) and hand the document in, along with names of people you discussed the document with.

HW #9 (due Wed 11/14) READ Straffin: Chapter 28 and Taylor: Chapter 5.
DO Straffin Chap. 23 ( 3 ) 25 ( 1, 6 ) AND Taylor Chap.4 ( 6, 10).

AND continue to work on your final papers. A paper draft will be due 11/21.

HW #10 (due Wed 11/21) READ Taylor: Chapters 6, 9 and SKIM Taylor Chapter 10.
Work on your final papers, and HAND IN 3 COPIES of a PAPER DRAFT on 11/21. Be sure to use the Writing Guide and staple a copy of the rubric to the top of each draft.

Also be working on your presentations for after Thanksgiving. Dates for presentations were assigned earlier.