The purpose of this conference is to bring together mathematicians and economists interested in studying dynamic stochastic systems in which interactions among large numbers of people influence aggregates that in turn affect the decisions of each individual. Mean field games and optimal control theories cast in continuous time are promising tools for studying such settings. The conference aims to be a forum to meet new ideas and new people and to exchange ideas. Over the years, economics has profitably imported ideas and methods from applied mathematics and control theory. We believe that this conference will foster net imports to economics in this way once again.

**Conference Venue: Room 421, Peking University PHBS Business School Building**

### Agenda:

**May 14 **

08:30 Depart from Hotel

08:45-09:00 Registration

09:00-09:10 Opening speech by Thomas J. Sargent

09:10-10:10 Thomas J. Sargent: Income and wealth inequality

10:10-11:10 Diogo Gomes: An Introduction to Mean Field Games

11:10-11:25 Break

11:25-12:25 Philip Yam: Calculus on Space of Random Variables and Mean Field Theory

12:25-13:30 Lunch and break

13:30-14:30 Fabio Camilli: Partial Differential Equations and Mean Field Games on Networks

14:30-14:45 Break

14:45-15:45 Marco Cirant: Periodic Patterns Arising in Non-monotone Mean-Field Games

15:45- 16:45 Qian Lin: Nash equilibrium payoffs for stochastic differential games

16:45-17:00 Group Photo

17:00-18:30 Dinner

18:30 Depart to Hotel

**May 15**

08:45 Depart from Hotel

09:00-10:00 Diogo Gomes: A Mean-field Game Price Model

10:00-11:00 Philip Yam: Optimal Savings and the Value of Population under Stochastic Environment

11:00-11:15 Break

11:15-12:15 Xue Dong He: On the Equilibrium Strategies for Time Inconsistent Problems in Continuous Time

12:15-13:30 Lunch and break

13:30-14:30 Hanbing Liu: Optimal Control Problem of Parabolic Systems with State Constraints

14:30-14:45 Break

14:45-15:45 Jonathan Payne: Application of Mean Field Game Theory to economics

15:45-17:00 Free discussion and addtional talks

17:00-18:30 Dinner

18:30 Depart to Hotel

**May 16**

08:45 Depart from Hotel

09:00-10:00 Fabio Camilli: A Mean Field Games Approach to Clustering

10:00-11:00 Qing Tang: On Some Nonlocal Mean Field Games

11:00-11:45 Free discussion and addtional talks

11:45-12:00 Closing speech by Thomas J. Sargent

12:00-13:00 Lunch

## Invited Speakers and talks

**Fabio Camilli，Department of Basic and Applied Sciences for Engineering (SBAI) ，Sapienza Università di Roma**

*Talk 1: Partial differential equations and Mean Field Games on networks*

Abstract: In this talk, we will discuss basic aspect of the theory of PDE defined on networks and then, the specific example of the MFG system on networks.

Presented paper: [1] F. Camilli, R. D. Maio, and A. Tosin. Measure-valued solutions to nonlocal transport equations on networks. Journal of Differential Equations, 264(12):7213 – 7241, 2018.

[2] S. Cacace, F. Camilli, Fabio C. Marchi. A numerical method for mean field games on networks. ESAIM: M2AN, 51(1):63–88, 2017.

[3] F. Camilli, C. Marchi, Stationary Mean Field Games systems defined on networks, SIAM J. of Control & Optimization, 54 (2016), no. 2, 1085–1103.

*Talk 2: A Mean Field Games approach to clustering*

Abstract: In this talk, we will discuss the study of some multi-population MFG systems which mimic the classical EM algorithm in cluster theory.

Presented paper: A Mean Field Games approach to clustering, work in progress.

**Marco Cirant，Dipartimento di Scienze Matematiche, Fisiche e Informatiche Università di Parma**

*Talk: Periodic patterns arising in non-monotone Mean-Field Games*

Abstract: This talk will be devoted to the discussion of some Mean-Field Games models where agents have a tendency to aggregate. In these models the typical uniqueness and stability of equilibria fails in the long time regime. It will be shown the existence of solutions that exhibit a periodic behaviour in time. A description of the analytic techniques involved will be presented, together with some numerical experiments.

Reference: M. Cirant, On the existence of oscillating solutions in non-monotone Mean-Field Games. To appear in J. Differential Equations, 2019.

**Diogo Gomes，Professor, Applied Mathematics and Computational Science， Computer, Electrical and Mathematical Science and Engineering Division， King Abdullah University (KAUST)**

*Talk 1: An introduction to mean-field games*

Abstract: Mean-field games comprise a wide range of models that describe large populations of agents in competition. Often these games are described by systems of partial differential equations for which explicit solutions cannot be found, in general. Therefore, fixed-point methods, regularization techniques, and continuation arguments provide systematic tools to study the existence of solutions. In this talk, I will discuss several recent developments on the theory for mean-field game systems and present a concise overview of the main techniques. I will also give some simple and computation-friendly examples.

Reference: D. A. Gomes and J. Saude. Mean field games models|a brief survey. Dynamic Games and Applications, 4(2):110{154, Jun 2014. D. A. Gomes, E. A. Pimentel, and V. Voskanyan. Regularity theory for mean-field game systems. SpringerBriefs in Mathematics, 2016.

*Talk 2: A mean-field game price model*

Abstract: Here, we introduce a price-formation model where a large number of small players can store and trade electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly and compare our model with real data.

Reference: D. Gomes and J. Saude. A mean-field game approach to price formation in electricity markets. Arxiv preprint 07 2018.

**Philip Yam，Department of Statistics，The Chinese University of HongKong**

*Talk 1: Calculus on Space of Random Variables and Mean Field Theory*

Abstract: Mean field games and mean field type control problems (Bensoussan et al. 2013), which aims at searching for the Nash equilibrium of differential games among a large number of players, have become so vital to disciplines ranging from engineering to economics and finance. In this talk, we shall introduce a new calculus technique to study the (infinite- dimensional) forward-backward stochastic differential equations of McKean-Vlasov type arisen from mean field games and mean field type problems. More specifically, after explaining this Hilbert space framework of random variables, we shall provide a sketch of proof of the global unique existence of the solution to mean field systems when the mean field effect is not too large.

Reference: Bensoussan, A., Frehse, J., and Yam, P. (2013). Mean Field Games and Mean Field Type Control Theory (2013). New York: Springer Verlag.

*Talk 2: Optimal Savings and the Value of Population under Stochastic Environment*

Abstract: In the work of Arrow et al. (2007, PNAS), they studied a macroeconomic growth model so that the population dynamic was involved in both the total utility (objective function) of the whole population and in the capital investment process. In essence, they assumed the deterministic evolution for both dynamics, such that the labor force of the population is also incurred through the Cobb–Douglas production function. In this talk, we shall discuss the well-posedness of the extension of their problem, particularly over a finite time horizon, in which we also allow more realistic and generic population growth and incorporate a stochastic environment for both the demography and capital investment. Technically, some novel PDE methodologies have been developed in our work, and we believe that this can be of parallel use in various sophisticated modeling in economic growth theory.

Reference: Arrow, K., Bensoussan, A., Feng, Q., and Sethi, S.P. (2007). Optimal savings and the value of population, Proceedings of the National Academy of Sciences, 47: 18421-6.

**Qing Tang，Department of Mathematics，China University of Geosciences (Wuhan)**

* Talk: On some nonlocal mean field games*

Abstract: In this talk, I will discuss some time-fractional mean field games. We consider the variational structure and optimal transport formulation of a time-fractional second order Mean Field Games (MFG) system with local coupling. The MFG system consists of time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations. In such a situation the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Hence, the results of this paper extend the theory of variational MFG to the subdiffusive situation, providing an Eulerian interpretation of timefractional MFG systems.This is joint work with Fabio Camilli. I will also discuss some of my own recent work in progress regarding mean field games with jump diffusion and applications to finance.

Reference: Q. Tang, F. Camilli, Variational time-fractional Mean Field Games, arXiv:1812.05431math.AP.

**Xue Dong He，Department of Systems Engineering and Engineering Management，The Chinese University of Hong Kong**

*Talk: On the Equilibrium Strategies for Time Inconsistent Problems in Continuous Time*

Reference: He, Xue Dong and Jiang, Zhaoli, On the Equilibrium Strategies for Time Inconsistent Problems in Continuous Time (January 30, 2019).

Available at SSRN: https://ssrn.com/abstract=3308274 or http://dx.doi.org/10.2139/ssrn.3308274

**Hanbing Liu，China university of Geosciences (Wuhan)**

*Talk: Optimal control problem of parabolic systems with state constraints*

Abstracts: In this talk, I will give a survey about optimal control problems of partial differential equations. Then I will introduce a Moreau-Yosida regularization method to obtain the first order optimality condition for optimal control problems governed by parabolic systems with pointwise state constraint in time.

Reference: Liu, Hanbing; Yang, Juan Optimal control of semilinear parabolic systems with state constraint. J. Math. Anal. Appl. 417 (2014), no. 2, 787–803.

Hanbing Liu, Pin Lin, Gengsheng Wang, Output feedback stabilization for heat equations with sampled-data controls, preprint arXiv:1807.01220 [math.OC], 2019.

**Qian Lin，Distinguished Associate Research Fellow of Finance，School of Economics and Management，Wuhan University**

*Talk: Nash equilibrium payoffs for stochastic differential games*

Abstract: In this talk, we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs.

**Jonathan Payne, PhD Candidate of Economics, New York University**

*Talk: Application of Mean Field Game Theory to economics*

Abstract: This talk will discuss the use of Mean Field Game Theory in economics. I will start by outlining which economics models are mean field games and what general forms their HJB and KF equations take. I will then discuss which existence, unique and stability results are known in the economics literature. Finally, I will provide an overview of the computation techniques that economists use to solve Stochastic Mean Field Games.