|Lectures:||Wednesday, 14:15-16:00 (Zoom)|
|First lecture:||November 11, 2020|
|Tutorials:||Monday, 12:15-14:00 (roughly bi-weekly; also on Zoom)|
There will be (mandatory) exercise sets that have to be handed in and will be graded by us.
|Oral Exam:||February 23-24, 2021. (Re-exam: March 10, 2021)|
This course requires basic knowledge regarding
Futhermore, having some background in combinatorial optimization (e.g., the course Optimization) is usful, but not required. Finally, we assume mathematical maturity, i.e., you should be comfortable with writing mathematical proofs.
In this course we will cover topics in the areas of Algorithmic Game Theory and Computational Economics, which can be placed at the intersection of economics and theoretical computer science. The course consists of two parts.
Game theory is concerned with the study of mathematical models of strategic interaction between players. One of its core aspects is the study of equilibrium situations: `Stable' states of the model in which no player has an incentive to switch strategies. Roughly twenty years ago, computer scientists became interested in studying the algorithmic aspects of such equilibria. Can we compute them efficiently, that is, in polynomial time? In the first part of this course we will cover results addressing this question, and more, in general n-player games and the special class of potential (or congestion) games. We will mostly focus on computational questions concerning pure Nash equilibria (PNE), mixed Nash equilibria (MNE) and (coarse) correlated equilibria (CCE).
In the second part of this course we study problems in the area of Computational Economics, with a focus on online selection problems. One example here is the selling of an item on an online platform in which buyers arrive in an unknown order, and where we have to irrevocably decide upon a buyer's arrival if we want to sell to her or not. This problem is closely related to, e.g., the classical secretary problem. We will cover various results and online models in this area, such as combinatorial secretary problems and prophet inequalities, and also discuss connections to online mechanism design.
Tentative outline (subject to changes). Files are sometimes updated.
|11.11||Introduction and overview||Lecture 1, Hand-out||Chapter 13 [R2016]|
|16.11||Tutorial 0 (Background material)|
|18.11||Congestion games I: Computation of PNE||Lecture 2, Hand-out||Chapter 19 [R2016]|
|25.11||Congestion games II: Inefficiency of PNE||Lecture 3, Hand-out|
|02.12||Finite games I: Existence and computation of MNE||Lecture 4, Hand-out||Chapter 20 [R2016]|
|09.12||Finite games II: Computation of approximate MNE||Lecture 5, Hand-out|
|16.12||Computation of (C)CE||Lecture 6, Hand-out||Chapter 17 [R2016]|
|06.01||Online Selection Problems||Lecture 7, Hand-out|
|13.01||Some Mechanism Design||Lecture 8, Hand-out||Chapters 2-3 [R2016]|
|20.01||Online Bipartite Matching||Lecture 9, Hand-out|
|27.01||Matroid Secretary Problems||Lecture 10, Hand-out|
|03.02||Prophet Inequalities||Lecture 11, Hand-out|
|19.02 (14:15)||Q&A session for oral exam|
Lecture 1 (11.11):
- See Chapter 13 of [R2016] for an overview of the hierarchy of equilibrium concepts that we discussed during the lecture.
Lecture 2/3 (18.11/25.11):
- See Chapter 19 of [R2016] for the complexity of computing a PNE when interpreted as a local search problem.
- See this paper of Roughgarden for an overview of the smoothness technique and more of its applications.
Lecture 4 (02.12):
- See Chapter 20 [R2016] for the PPAD complexity of computing an MNE.
- Starting point for reading about the convergence time of fictitious play is this paper by Daskalakis and Pan (and references therein).
Lecture 5 (09.12):
- Paper by Lipton, Markakis and Mehta regarding approximate Nash equilibria with logarithmic support size (discussed during lecture).
Lecture 6 (16.12):
- See Chapter 17 [R2016] for the no-regret dynamics discussed during the lecture, and a proof that the MW algorithm satisfies the no-regret property.
- See Chapter 18 [R2016] for similar (but different) "no-regret dynamics" converging to a correlated equilibrium.
Lecture 7 (06.01):
- For the full proof of the performance guarantee of the secretary algorithm, see, e.g., this note.
Lecture 8 (13.01):
- See Chapters 2 and 3 [R2016] for a more formal treatment of Mechanism Design, and, in particular, a proof of Myerson's lemma.
- See also Part II of [NRTV2008] for a more extensive treaty.
Lecture 9 (20.01):
- Paper by Kesselheim, Radke, Tönnis and Vöcking on online bipartite matching algorithm.
- Paper by Reiffenhäuser with online strategyproof mechanism for unit-demand setting.
Lecture 10 (27.01):
- Paper of Babaioff, Immorlica, Kempe and Kleinberg on the matroid secretary problem.
Lecture 11 (03.02):
- Paper of Kleinberg and Weinberg on matroid prophet inequalities.
- Algorithmic Game Theory by Noam Nisan, Tim Roughgarden, Éva Tardos and Vijay V. Vazirani [NRTV2008]
- Twenty Lectures on Algorithmic Game Theory by Tim Roughgarden [R2016]
- This book is based on Prof. Roughgarden's lecture notes.
Digital editions of these books are accessible from within the IP-range of Saarland University. Hardcopies available in the university library.
Background (prerequisite) material
Throughout the course, we will use some elementary tools from combinatorics, (linear) optimization and probability theory that will not be fully explained in the lectures. A small document containing more details on this material can be found below.
We aim to create a pleasant lecture environment despite the current situation. Personal interactions on Zoom, with your microphone and camera switched on, may contribute to this environment. We also encourage you to ask questions verbally. Note that this is voluntary. You may switch off both your camera and your microphone, and register under a pseudonym. Questions are still possible, in particular using the chat function.
The lectures on Zoom will be recorded, the tutorials not. Access to the recordings will be provided to participants registered to the mailing list. It is strictly forbidden to share access information, as well as any offline copies of the recorded lectures, with others. In general, all material (and access to material) provided via the mailing list is for personal use only, and may not be distributed.