2017 Advanced Topics in Mathematical Economics

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Academic unit or major
Graduate major in Industrial Engineering and Economics
Shioura Akiyoshi 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(西9号館, 508号室)  Fri3-4(西9号館, 508号室)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

We discuss an auction model with many indivisible (discrete) goods. It is known that an optimal allocation of goods as well as equilibrium prices can be computed by algorithms (protocols) called iterative auctions. In this lecture, we review various iterative auctions and investigate them from the viewpoint of discrete optimization. In particular, we explain the concept of gross-substitutes valuation, which plays a crucial role in the auction, and show the connectin with discrete concavity.

This lecture aims to enable students to understand the power of theoretical results in discrete optimization in application to auction theory in economics.

Student learning outcomes

By the end of this course, students will be able to do the following:
(1) explain the auction model with indivisible goods,
(2) understand the concept of gross-substitutes condition and its properties,
(3) explain how iterative auctions find equilbrium prices,
(4) understand the connection between iterative auctions and optimization algorithms.


auction, discrete optimization, equilibrium, algorithm

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

In each class we give a lecture in the first half and then assign some exercise problems in the last half.

Course schedule/Required learning

  Course schedule Required learning
Class 1 overview of the lecture Details will be given in each lecture.
Class 2 auction models and gross substitutes condition
Class 3 gross substitutes condition and discrete concavity
Class 4 maximization of M-concave function
Class 5 relationship between M-convexity and L-convexity
Class 6 minimization of L-convex function
Class 7 duality theorems for discrete convex functions
Class 8 iterative auction and Lyapunov function
Class 9 Lyapunov function and L-convex function
Class 10 analysis of algorithms for L-convex function minimization
Class 11 application of algorithms for L-convex function minimization (1)
Class 12 application of algorithms for L-convex function minimization (2)
Class 13 application to unit-demand auction (1)
Class 14 application to unit-demand auction (2)
Class 15 summary of the lecture



Reference books, course materials, etc.

Related Paper:
K. Murota, A. Shioura, and Z. Yang: Time bounds for iterative auctions: a unified approach by discrete convex analysis, Technical Report METR 2014-39, University of Tokyo, December 2014.

Related Book:
K. Murota: Discrete Convex Analysis, SIAM, 2003

Assessment criteria and methods

Evaluation based on reports and exams

Related courses

  • IEE.B337 : Mathematical Economics
  • IEE.A206 : Operations Research
  • IEE.A330 : Advanced Operations Research
  • IEE.B401 : Advanced Microeconomics
  • IEE.B403 : Advanced Noncooperative Game Theory

Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in related courses is desirable.

Contact information (e-mail and phone)    Notice : Please replace from "[at]" to "@"(half-width character).


Office hours

Any time. Prior appointment by e-mail is desirable.

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