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.
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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
In each class the instructor gives a lecture. At the end of the lecture, the instructor presents some problems for exercise.
Course schedule | Required learning | |
---|---|---|
Class 1 | overview of the lecture | Details will be given in each lecture. |
Class 2 | Review of shortest path problem (1) | |
Class 3 | Review of shortest path problem (2) | |
Class 4 | Matching problem on a bipartite graph (1) | |
Class 5 | Matching problem on a bipartite graph (2) | |
Class 6 | optimality condition for maximum matching | |
Class 7 | Maximum-weight matching problem | |
Class 8 | relationship between maximum-weight matching and equilibrium allocation | |
Class 9 | algorithm for computing an equilibrium approximately | |
Class 10 | algorithm for computing an equilibrium exactly | |
Class 11 | multi-demand model and equilibrium | |
Class 12 | gross substitutes property for valuation functions | |
Class 13 | algorithm for computing an equilibrium approximately | |
Class 14 | algorithm for computing an equilibrium exactly |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
None.
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
Evaluation based on reports and exams
Knowledge about the theory of combinatorial optimization is required.
In particular, students should have knowledge about the optimality conditions and algorithms
of the minimum spanning tree problem, the maximum (weight) matching problem, and the network flow problem.
Any time. Prior appointment by e-mail is required.