2025 (Current Year) Faculty Courses Liberal arts and basic science courses Humanities and social science courses
Social Modeling A
- Academic unit or major
- Humanities and social science courses
- Instructor(s)
- Takuhei Shimogawa
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- LAH.T107
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 27, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The purpose of the lecture course is to provide a story leading to the introduction of social choice theory and to establish a mathematical understanding of one of the milestones of the theory, Kenneth Arrow's work (mainly the impossibility theorem). Based on this understanding, the lecture will discuss and detail the Nakamura Number, one of the key conclusions of the problem of what it means for a social decision rule to be "rational". Since the content of the lecture requires mathematical description and development, some parts of the lecture will be a review of mathematics (mainly set theory and related areas) as the language we use.
Course description and aims
By the end of this course, students will be able to:
1)Understand that the system of social decision-making is a mathematical object with a rich structure.
2)Analyze and calculate the decision-making framework of voting.
3)Understand the derivation of Arrow's impossibility theorem and Nakamura Number's theorem.
4)Gain some insight into the impact of Arrow's theorem and the Nakamura number on the economy, politics, law, and society.
Keywords
Cooperative Games, Solution Concept, Social Choice Theory, Voting, Simple Games, Arrow's Impossibility Theorem, Nakamura Number
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Lectures and exercises are the mainstays of the course. The cycle of the introduction of concepts and definitions, various theorems and their proofs, exercises, and the introduction of the next concept will be the same as in a standard university lecture. For the exercises, answers and explanations will be given in class, but we will also use the online portal service: Science Tokyo LMS in parallel.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Math Preparation Part 1: Sets, Mappings, Operations and Relations | Review of basic mathematics. Naive set theory as the "language". |
| Class 2 | Math Preparation Part 2: Exercises | Elementary Set Theory, Exercises and solutions. |
| Class 3 | Simple games: Formalizing the voting system | Simple games and Shapley-Shubik power index. |
| Class 4 | Arrow's Impossibility Theorem: Social Welfare Functions(Institutions) | Several ways to express the statement, "fairness" criteria. |
| Class 5 | Arrow's Impossibility Theorem: Proof | Mathematical proof |
| Class 6 | Nakamura's Theorem: Statement, and its implications | Nakamura’s number: its significance. |
| Class 7 | Nakamura's Theorem: Proof | Mathematical proof. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 60 minutes preparing for class and another 60 minutes reviewing class content afterward (including assignments) for each class.
They should do so by referring to textbooks and other course material provided by the lecturer.
Textbook(s)
Any textbook will not be used. Lecture materials will be distributed as necessary. Lecture materials and explanatory notes (including photographs of the blackboard)will be uploaded to the online portal service(Science Tokyo LMS).
Reference books, course materials, etc.
Arrow, Kenneth (1963), Social Choice and Individual Values (2nd ed.), Yale University Press
Maskin, Eric et al. , The Arrow Impossibility Theorem (Kenneth J. Arrow Lecture Series), Columbia University Press, ISBN-13: 978-0231153287
Austen-Smith, David; Banks, Jeffrey S. (1999). Positive political theory I: Collective preference. Ann Arbor: University of Michigan Press. ISBN 978-0-472-08721-1
Evaluation methods and criteria
Midterm and final exams 70%, exercises 30%.
Insights into social decision-making rules, basic mathematics, formal proofs of impossibility theorems and Nakamura's theorem, and consequences. These are required as outcomes of understanding the lecture.
Related courses
- LAH.T209 ： Social Modeling B
- LAH.T308 ： Social Modeling C
- LAH.T108 ： Decision Making A
- LAH.T208 ： Decision Making B
- LAH.T307 ： Decision Making C
Prerequisites
Courses related to set theory, logic, and interest in social sciences.
Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).
T.Shimogawa smgw[at]u.musashi.ac.jp
Office hours
Contact above email account in advance.