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 we give a lecture in the first half and then assign some exercise problems in the last half.
|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|
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.
K. Murota: Discrete Convex Analysis, SIAM, 2003
Evaluation based on reports and exams
No prerequisites are necessary, but enrollment in related courses is desirable.
Any time. Prior appointment by e-mail is desirable.