Math 188: Social Choice and Decision Making

Francis Su > Math 188: Social Choice and Decision Making

Term: Summer 2021, May 24 – June 11.

Course meets: Virtually, Mon-Tue-Thu-Fri, 8:45-noon (Pacific time). The private Zoom link was sent by e-mail, and is also available on the course Sakai page.

Office Hours: By appointment, usually at 1:30pm.

People make decisions everyday — 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.

Prerequisites

Math 55 (or some experience with discrete math) is recommended, but not required.

Texts

E-books are available through the Claremont Colleges Library at these links: Straffin, Taylor-Pacelli.

Coursework

As a 3-week summer course, there will be daily homework, which in the latter part of the course will include time to work on final projects. There will also be an exam at the beginning of week 3.

Homework

Homeworks will be graded based on clarity of ideas and communication. You’ll note that Straffin has all answers at the back of the text, but these are rather minimal. (On the Honor Code, do NOT look at them until AFTER you have attempted each problem.) You’ll need to explain how you arrived at your answers to get full credit. Good communication is important in this class; be sure to read my Guidelines for Good Mathematical Writing.

All homework will be submitted through Gradescope. Access this through the Sakai course page. On the left tab, you’ll see a link for Gradescope. You will need to upload by scanning your homework, and tagging which portion is associated with specific problems. Here is some guidance on how to do that: https://www.gradescope.com/get_started#student-submission.

Exam

There will be one take-home exam due at the start of Week 3.

Final Project

There will be a final project involving a critical analysis of the relationship of the mathematical methods of game theory to contemporary society. This will be a group project (2 to 3 students). You will create a video presentation of up to 10 minutes.

Grading

Homework, Exam, Final Project are each worth 25%, with highest component worth 25%. The grader for the course is Allen Wu.

READ
Straffin: Preface and Chapters 1, 2, and 7.
Taylor-Pacelli: Preface and Sections 6.1-6.3.

Note: Problems marked with an R means “READ the problem and think about it, but do not hand it in.”

DO
Straffin Chapter 7 (3, 4, 5, 6abc, R6d) and

Problem A. Find a short news article involving some sort of interaction or
decision problem. Then, in just a few brief sentences:

(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.
(3) Include a scan of the article with your assignment.

Remember to write clearly and in complete sentences and explain your reasoning carefully. Box answers where appropriate. Good communication is important in this class; be sure to read my Guidelines for Good Mathematical Writing.

READ
Taylor-Pacelli: Sections 4.1-4.3, AND
Straffin: Chapter 3.

“R” means READ and THINK about, but do not do the problem.

DO
Straffin Chap.2 ( 1, 2, 3, R6, 7 ) AND Taylor Chap. 4 ( 6 )

READ
Straffin: Chapters 4, 5, 6.
Also, take a moment to reach out to someone else in the class, and have a chat with them about anything that strikes you as interesting from the class material.

DO
Straffin Chap. 3 ( 1, 2, 3, 4, R5, 7 ).

READ
Straffin Chapter 4, 5, 6, 9.
Taylor-Pacelli Chapter 10.
Also, take a moment to reach out to someone else in the class, and have a chat with them about anything that strikes you as interesting from the class material.

DO
Straffin
Chap. 3 ( 5, R9 )
Chap.4 ( R2 )
Chap. 7 ( 6d )
Chap 9 ( R1, 2, 3, 5 )

READ
Taylor-Pacelli Chapter 4
Straffin Chapters 10, 11, 12.

DO
Straffin Chapter 10 ( R3 ), Chap 11 ( 1 ), Chap 12 ( 4, 6 ) and
Taylor-Pacelli Chapter 4 ( 13, 15, R23 ).

READ
Straffin Chapters 16-17.

DO
On this Handout ( R21, R22, R23, especially R24 )
and on this Handout ( 7, R9, 11Y, 12Y, 15)
and
Problem B. Meet with at least one other person in this class to discuss ideas that might be interesting to model using game theory. (See the last HW tab below under 6/10 and 6/11 for the final project guidelines.) On this homework, write down a few sentences about what you discussed. (Let me know if you need help connecting with someone else in the class.)

READ
Straffin Chapter 19, 23, 25 and Taylor-Pacelli Chapter 2, 3.

DO
Straffin Chap. 23 ( 2, 3abc, 3def ) 25 ( R1, 6 ) AND
Problem C. Identify a group for your final project. For 5 points on this homework ,state who you are working with (if anyone) and state any preliminary thoughts you have about a project topic, in a couple of sentences.

REVIEW
Concepts from the course in the last 2 weeks. You may also look at the take-home exam (on Gradescope) and get started on that if you wish (due Tuesday).

DO
Taylor-Pacelli Chapter 2 ( 2, R3, 10 ) AND
Problem D. Meet with your project group (if you have one). If you have not yet picked a topic, discuss 3 ideas you have for a project (see HW for 6/10 and 6/11 below for guidelines on the final project). Summarize the pros and cons of pursuing each of those ideas for a final project (5 points each, 15 points total). If you have already settled on a topic, describe: the topic idea, the way you expect game theory enters into analyzing that topic, and develop a plan for work on your project presentation (5 points each, 15 points total).

Look on Gradescope. The exam will be available Friday June 4, and is due Tuesday at 5pm on June 8.

Your final project will involve a critical analysis of the relationship of the mathematical methods of game theory to contemporary society. This will be a group project (2 to 3 students). You will create a video presentation of up to 10 minutes.

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 presentation 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 are the limitations of the model? As people often make decisions based on such models, what would be appropriate or inappropriate uses of this model?

Your project video will be judged on the basis of :
(i) clarity of explanation: can the video be understood by another student in the course?
(ii) game theoretic connection: topic identifies where game theory is used, and is mathematically correct
(iii) insightfulness: your project identifies clearly what you are bringing beyond a mere description of the topic. This can be achieved by demonstrating originality in choosing a model, offering your own critique of an existing model, or offering a new perspective or insight.

You can submit the video to the Google Drive folder that was linked in an e-mail sent June 5. It’s also accessible through the Course Videos folder where my lectures have been posted (linked from the Sakai page).

You can work ahead, but some assignments may change.