Emmanuelle Augeraud-Véron (BSE)
Title: Pollution mean field game and risk aversion
Abstract: This paper studies an optimal pollution model in the framework of a mean-field game, where individuals base their decisions solely on the perception of global pollution. Each representative agent chooses their consumption trajectory while accounting for the aggregated pollution generated by the entire population, rather than local externalities. Within this framework, we investigate how the degree of risk aversion influences optimal behavior and pollution dynamics. In particular, we show how individual risk preferences shape the collective outcome and determine whether higher risk aversion leads to more conservative consumption paths and lower global pollution. The analysis highlights the interplay between macro-level externalities and micro-level preferences in shaping sustainable consumption dynamics.
augerard.pdfAlessandro Calvia (Polimi)
Title: A Mean-Field Game of optimal investment.
Abstract: In this talk, I will introduce a continuous-time mean-field game of optimal investment for a population of firms in a given economy. I will consider the case where the representative company can increase its production capacity via costly investments and aims at optimizing its discounted operating profit, which at equilibrium depends on average production capacity of all the firms in the economy. I will discuss the economic foundation of this model and I will formulate both its finite and infinite time horizon versions. Then, I will analyze a specific linear-quadratic case, where it is possible to prove existence and uniqueness of an equilibrium. This is joint work with Giorgio Ferrari (Universität Bielefeld), Salvatore Federico (Università di Bologna), and Fausto Gozzi (LUISS University).
calvia.pdfFabio Camilli (Uniroma1)
Title: A Mean-Field Game network model for urban planning
Abstract: We consider a mathematical framework aimed at capturing the dynamic evolution of urban landscapes, shaped by the interplay between two populations: the workforce and the business establishments. Visualizing the city as a network, where edges symbolize both residential zones and connectivity pathways, we explore how these populations compete for spatial occupancy while engaging in labor market transactions. The mathematical model is composed of a two-population Mean-Field Game system coupled with an Optimal Transport problem. We establish the existence and uniqueness of solutions and supplement our findings with several numerical simulations.
Camilli.pdfLuciano Campi (Unimi)
Title: Correlated equilibria in mean field games
Abstract: We will give an overview of correlated and coarse correlated equilibria in mean field games in discrete and continuous time, focusing on a specific example (where Nash MFG solution do not exist) and on the linear-quadratic case. If time allows, we will also discuss how to construct approximate equilibria in the corresponding N-player games and some future developments. This talk is based on joint works with O. Bonesini, F. Cannerozzi, M. Fischer and I. Tzouanas.
campi.pdfMarco Cirant (Unipd)
Title: Some results on the local turnpike property in MFG
Abstract: I will consider some MFG systems with quadratic Hamiltonian and nonmonotone coupling, and discuss the local stability of stationary solutions, that is, whether or not they can be long time limits of backward-forward problems. The talk will be based on an ongoing joint work with N. De Bernardi (Padova).
cirant.pdfPierre Cardaliaguet (Dauphine PSL)
Title: Mean Field Games in a Stackelberg problem with an informed major player and application to optimal edging in finance.
Abstract: We investigate a stochastic differential game in which a major payer has a private information, which she discloses through her control to a population of small players playing in a Nash Mean Field Game equilibrium. The major player's cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players. We illustrate the problem by an example of optimal hedging in finance, in which an informed broker faces many traders.
cardaliaguet.pdfElisabetta Carlini (Uniroma1)
Title: Numerical Approximations of the Newton Algorithm for Mean Field Games
Abstract: We propose two numerical approximations for second-order Mean Field Games (MFG) systems, based on the discretization of the Newton algorithm for MFG systems recently introduced by F. Camilli and Q. Tang (2023). Specifically, we develop a Semi-Lagrangian scheme and a finite-difference scheme and compare their performance with the Newton finite-dimensional method applied to the discrete finite-difference system, initially introduced by Y. Achdou and I. Capuzzo Dolcetta (2010). This work is in collaboration with F.J. Silva and A. Zorkot.
carlini.pdfGiulia Cavagnari (Polimi)
Title: Eulerian and Lagrangian formulations of optimal control problems for multi-agent systems
Abstract: We explore different approaches to the study of optimal control problems for multi-particle systems with interactions. One approach frames the problem in the Wasserstein space of probability measures where results from measure theory and optimal transport theory can be used (like the so called Superposition Principle). The other approach (à la Lions) sets the problem in a Hilbert space of parametrizations. Requiring that the Eulerian evolution is the time-dependent law of the Lagrangian process, we prove the equivalence of the value functions in the two formulations, highlighting the pros/cons of them and the key role of an intermediate formulation. These results reveal to be interesting also in light of recent studies on the relations between different definitions of viscosity solutions for Hamilton-Jacobi equations in this context. This is a joint work with Stefano Lisini, Carlo Orrieri (University of pavia - Italy) and Giuseppe Savaré (Bocconi University - Italy).
cavagnari.pdfAlekos Cecchin (Unipd)
Title: On the exponential turnpike phenomenon for mean field game systems
Abstract: This article aims at quantifying the long time behavior of solutions of mean field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov control. Our main contribution is to show well-posedness of the ergodic problem and the exponential turnpike property of dynamic optimizers, which implies exponential convergence to equilibrium for both optimal states and controls to their ergodic counterparts. In contrast with previous works that require some version of the Lasry-Lions monotonicity condition, our main assumption is a weak form of asymptotic monotonicity on the drift of the controlled dynamics and some basic regularity and smallness conditions on the interaction terms. Our proof strategy is probabilistic and based on the construction of contractive couplings between controlled processes and forward-backward stochastic differential equations. The flexibility of the coupling approach allows us to work on the whole space, treat cost functions that are merely Lipschitz in the total variation distance and cover the case of non-constant diffusion coefficients. Moreover, under reinforced regularity assumptions, we are able to show turnpike estimates for the Hessians of solutions to the backward equation. Finally, we mention how these tools allow to address uniform in time propagation of chaos for mean field control problems. Based on joint work with G. Conforti (Padova), A. Durmus (Ecole Polytechnique) and K. Eichinger (Paris Saclay).
cecchin.pdfAnnalisa Cesaroni (Unipd)
Title: Phase transitions and local stability of equilibria in a Kuramoto MFG
Abstract: I will present results obtained in collaboration with Marco Cirant concerning equilibria in a Kuramoto mean-field game model. This model, introduced by R. Carmona, Q. Cormier, and M. Soner, is a game-theoretic version of the classical Kuramoto model, and describes synchronization phenomena in a large population of rational interacting oscillators. Specifically, I will discuss the characterization of equilibria and a local turnpike property.
cesaroni.pdfPaolo Dai Pra (Univr)
Title: Spatial pattern in long range games
Abstract: Long range games extend traditional mean field games by including a macroscopic spatial structure; they include, in particular, graphon and multi-graphon games. After having recalled the notion of long range game and the corresponding essential properties, we consider the long time behavior of a class of these games, and their connections with spin systems with long range interaction in Statistical Mechanics.
daipra.pdfGökçe Dayanıklı (UIUC)
Title: Utilizing deep learning and game theory to find optimal policies for a large number of agents
Abstract: In this talk, we will discuss how we can utilize deep learning to solve complex (dynamic and stochastic) game theoretical problems where there many agents (such as banks, companies, or people) interacting. We will first look at a stochastic optimal control problem for one agent and explain how we can use deep learning to solve this problem. Later, we will move on to the multi-agent setup, and we will discuss and compare two equilibrium notions in game theory: Nash equilibrium and Stackelberg equilibrium. After explaining how a Nash equilibrium can be approximated for dynamic and stochastic games with a large number of agents through mean field games, we will introduce the Stackelberg mean field games between a principal (i.e., regulator) and many agents. Stackelberg mean field game models can be used to find incentives or optimal policies for a large group of noncooperative agents with a motivation to model different real-life problems such as regulating the systemic risk in banking sector, regulating energy markets, or mitigating an epidemic. We will discuss how (intrinsically bi-level) Stackelberg mean field game model can be rewritten to propose a single-level deep learning method to solve this complex problem and conclude with some examples. (joint work w/ Mathieu Lauriere)
Dayanikli.pdfPaola Mannucci (Unipd)
Title: Deterministic Mean Field Games with non coercive Hamiltonian and state constraints.
Abstract: I will talk on a research project about some models of evolutive deterministic MFGs where the Hamiltonian is not coercive in the gradient term because the dynamics of the generic player must fulfill some state constraints (holonomic or non-holonomic) or fail to be strongly controllable. First of all, I will treat the case where the generic agent can control its state only through the acceleration and it is constrained to remain in a given bounded region. In particular I will mainly focus on the case where agents are constrained to remain in a interval because the description of the problem is more clear; however, many of our results still hold in a more general setting. Afterwards, I will outline the model where the generic player is still constrained to remain in a given region and it has some non-holonomic constraints for example it may have forbidden directions. I will study the existence of weak solutions and their representation by means of relaxed equilibria in the Lagrangian interpretation which describes the game in terms of a probability measure on optimal trajectories. In both cases, to get the existence of such equilibria, we consider the associated constrained optimal control problem and we obtain a closed graph result for the multivalued map which associates to an initial condition the set of the optimal trajectories starting from that point. Project in collaboration with Y. Achdou, A. Cutri', C. Marchi, C. Mendico, N.Tchou.
Mannucci.pdfAthena Picarelli (Univr)
Title: Extended mean field control: a finite-dimensional numerical approximation
Abstract: We present a finite-dimensional numerical approximation for a class of extended mean field control problems. Our algorithm learns the value function on the whole Wasserstein domain, as opposed to a fixed initial condition. We leverage the approximation of the mean field problem by a finite-player cooperative optimisation problem, due to the propagation of chaos, together with the usage of finite-dimensional solvers. This avoids the need to directly approximate functions on an infinite-dimensional domain, and allows for more efficient memory usage and faster computation.
picarelli.pdfAlessio Porretta (Uniroma2)
Title: Mean field game models of knowledge diffusion and economic growth
Abstract: There is a huge literature in macroeconomics discussing about endogenous growth models, where production is based on technological knowledge and the economic growth is driven by learning/diffusion of technology. In this talk I will discuss a similar model proposed by R.E. Lucas and B. Moll which leads to a mean field game system coupling individual strategies of time allocation with the evolution of the cumulative distribution function. The main interest for economists lies in the analysis of balanced growth paths to describe sustained growth in long time. In mathematical terms, this involves the analysis of traveling waves solutions in PDE systems where Hamilton-Jacobi equations are coupled with reaction diffusion equations. I will discuss two variants of this model leading to possibly different phenomena. Based on joint works with Luca Rossi (University of Rome 1) and Lenya Ryzhik (Stanford University).
porretta.pdfMichele Ricciardi (LUISS)
Title: Mean Field Games and Master Equation in infinite dimensional state spaces: new results and applications
Abstract: In a framework of a differential game with a large number N of indistinguishable players, the intricate structure of the Nash system can be approximated, for N large with an easier system of coupled PDEs, called Mean Field Games system. It consists of a backward HJB equation for the single agent's value function, coupled with a forward FP equation for the distribution of the population. However, a rigorous proof of this convergence passes through the study of a PDE equation in the space of measures, called Master Equation. The novelty introduced in this talk relies on the state space, which is assumed to be a possibly infinite dimensional Hilbert space. We show two examples where the infinite-dimensional framework is mandatory. The first one is an advertising model, where the stock of advertising goodwill solve a non-Markovian SDE. The second one is a vintage capital model, where the amount of capital goods solves a PDE instead of an ODE. We study the case when the cost functions are linear or quadratic polynomials in the cost functions and the average distribution of the population. Here, the particular structure of the coefficients and the data of the problem allows us to give a precise representation of the solutions for the Nash system, the Mean Field Games and the Master Equation. This representation allows to pass to the limit for N→+∞ and to prove the desired convergence result.
ricciardi.pdf