# Seminar on Mean-Field Games Spring 2020

Organizer
Christoph Czichowsky
Coordinator
Robert A. Crowell

Fridays, 15:00-17:00, HG G 26.3

Tentative seminar outline and seminar announcement in the course catalog.

Mean-field games provide a tractable model of large population strategic games. In this reading group and seminar we want to acquaint ourselves with the basic problem formulation, some approaches to study such games, and gain an overview of current research on this topic. The typical participant will be an advanced graduate student, doctoral student or a postdoc, not necessarily an expert on the topic but interested in jointly learning the subject.

## Content

Stochastic differential games with a large number of players are typically intractable. Mean-field games introduced independently by Lasry and Lions, and Huang, Malhame and Caines study Nash equilibria as the number of players $$n$$ tends to infinity and players interact through the distribution of states rather than directly. This modeling philosophy is what inspires the mean-field terminology from statistical physics. A canonical mean-field formulation is as follows. Given the population's distribution of states $$\mu$$ on $$\mathbf{C}([0,T], \mathbb R^d)$$, the representative player controls a stochastic system of the form $$\sup \mathbf E_{\alpha}\Big [\int_0^T f(X, \mu, \alpha_t)dt +g(X)\Big]$$ where $$\mathbf E_\alpha$$ is the expectation under $$\mathbf P_{\alpha}$$, the law of the solution solution to a controlled SDE of McKean-Vlasov type $$dX_t= b_t(X, \mu, \alpha_t)dt +\sigma_t(X, \mu, \alpha_t)dB_t.$$ A mean-field Nash equilibrium is a measure $$\mathbf P_\alpha$$ which is optimal given $$\mu$$ and satisfies in addition the fixed-point property $$\mathbf P_\alpha = \mu$$.

A special case of this model leads us to consider the mean-field PDE system of the form $$\begin{cases} -\partial_t u(t, x) -\Delta u(t, x) + H(t, x, Du(t, x), m(t)) = 0 \\\partial_t m - \Delta m(t, x) -\text{div}(m(t, x)D_pH(t, x, Du(t, x), m(t,x)))=0\end{cases}$$ where $$m$$ is the density of a probability measure. The first PDE is the Hamilton-Jacobi-Bellman equation, it governs optimality. This equation is to understood backward in time, and its study requires a rather intricate theoretical apparatus. The latter equation is the Fokker-Planck PDE, which describes a flow of measures in the space of probabilities. It is an equation forward in time and to be understood in the distributional sense.

The seminar will begin by recalling the basics of stochastic control, both in its probabilistic backward SDE, and in its PDE formulation via verification. We will then study McKean-Vlasov type equations and the connection to the Fokker-Planck equation. Equipped with the necessary background we can introduce the probabilistic formulation of mean-field games and discuss existence and uniqueness results in various degrees of generality. We shall then briefly touch on the connections to the mean-field PDE system, discuss the common noise case and some applications.

The mean-field games literature employs an interesting mix of techniques form PDEs, stochastic analysis and optimal transport theory. The seminar provides a good staring point for students to write a M.Sc. thesis in this or related areas.

A detailed but still tentative outline can be found here. The main references are available in this Polybox folder.

## Prerequisites

Detailed prerequisites for each talk are given in the seminar outline. Student should have a firm understanding of stochastic calculus (e.g. from the lecture Brownian motion and stochastic calculus), as well as probability theory. A basic understanding of PDE theory may be helpful. Students with a background in PDE but only basic stochastic analysis knowledge are warmly welcome to give a talk from a more analytic perspective!

Student should be at a sufficiently advanced level to present selected topics from research papers and books. The seminar will take an intermediate form, somewhere between a student seminar and a graduate reading course. As such, participants should be motivated to prepare a presentation based on research articles, but also willing to consult more senior participants for feedback and help if needed. Roughly the first half of the planned talks is suitable for masters students with a firm understanding of the probabilistic and analytic techniques, the remaining talks will be given by more advanced participants.

Please also consult the details in the course catalog.

#### Information for PhD students

The seminar is open to PhD students. Credit points can be obtained with advisor's approval.

## List of Topics and Lecture Summaries

A tentative list of topics is listed in the seminar outline. Once the outline has been fixed and the talks distributed they will be posted below. Feel free to contact the organizers if you want to give a talk.

Lecture summaries will be perpetually updated over the course of the seminar. They may still be preliminary and contain typos.

Summaries of all the lectures in a single PDF file.

#TopicDateSpeakerLecture Summary
0 Seminar overview Fri. 21.02. Robert Crowell Mean-field games - overview and motivation
1 Stochastic control primer (1) Fri. 28.02 Skander Stephan Stochastic control problem
2 Stochastic control primer (2) Fri. 06.03 Philippe von Wurstemberger Weak formulation and BSDE
3a Games and Nash-equilibria Fri. 13.03 (1) Dmitri Ganjour Games and Nash equilibrium
3b Stochastic differential games Fri. 13.03 (2) Pontus Westermark Nash equilibrium in stochastic differential games
4 McKean-Vlasov SDEs Fri. 20.03 Milo Grillo SDEs of McKean-Vlasov Type
5 Mean-field formulation and examples Fri. 27.03 canceled
6 MFG probabilistic view (1) Fri. 03.04 canceled
7 MFG probabilistic view (2) Fri. 24.04 canceled
8 MFG probabilistic view (3) Fri. 08.05 canceled
9 Large population games without symmetry Fri. 15.05 canceled
10 Applications Fri. 22.05 canceled
11 Applications Fri. 29.05 canceled