An Introduction to Mean Field Game Theory

Cardaliaguet, Pierre; Porretta, Alessio

doi:10.1007/978-3-030-59837-2_1

Cited by 54 publications

(84 citation statements)

References 142 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From the theoretical point of view, a general question on the proposed theoretical framework is if the Markov equilibrium can be found in a generalized feedback form, where the closed-loop strategies Θ depends not only on time and state of each agent, but also on the current distribution of the other agents. A possible answer is to look at the Master Equation associated to our model, which, in turn, could be a first step to obtain stronger properties of the equilibria, like subgame perfection, and, possibly, uniqueness (see, e.g., Section 1.4 in Cardaliaguet & Porretta (2020)).…”

Section: Discussionmentioning

confidence: 99%

Mobility decisions, economic dynamics and epidemic

Fabbri¹,

Federico²,

Fiaschi³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper we propose a theoretical model including a susceptible-infectedrecovered-dead (SIRD) model of epidemic in a dynamic macroeconomic general equilibrium framework with agents' mobility. The latter affect both their income (and consumption) and their probability of infecting and of being infected. Strategic complementarities among individual mobility choices drive the evolution of aggregate economic activity, while infection externalities caused by individual mobility affect disease diffusion. Rational expectations of forward looking agents on the dynamics of aggregate mobility and epidemic determine individual mobility decisions.The model allows to evaluate alternative scenarios of mobility restrictions, especially policies dependent on the state of epidemic.We prove the existence of an equilibrium and provide a recursive construction method for finding equilibrium(a), which also guides our numerical investigations.We calibrate the model by using Italian experience on COVID-19 epidemic in the period February 2020 -May 2021. We discuss how our economic SIRD (ESIRD) model produces a substantially different dynamics of economy and epidemic with respect to a SIRD model with constant agents' mobility. Finally, by numerical explorations we illustrate how the model can be used to design an efficient policy of state-of-epidemic-dependent mobility restrictions, which mitigates the epidemic peaks stressing health system, and allows for trading-off the economic losses due to reduced mobility with the lower death rate due to the lower spread of epidemic.

show abstract

Section: Discussionmentioning

confidence: 99%

Mobility decisions, economic dynamics and epidemic

Fabbri¹,

Federico²,

Fiaschi³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In this section we prove that, if the anti-monotonicity of the coupling F (x, m) is sufficiently small, then we can prove the existence of solutions of (1.1) satisfying the turnpike property. The strategy we adopt follows Section 1.3.6 of [6], through the construction of a solution via a fixed point in a suitable weighted space. Then there exists γ > 0 only depending on L, κ (and on the functions F, H), such that if F (x, s) + γs is nondecreasing then any solution (u T , m T ) of problem (1.1) satisfies…”

Section: The Exponential Turnpike Estimatementioning

confidence: 99%

“…Thus m satisfies the second equation as the law of the optimal process, where α corresponds to the optimal feedback strategy. We refer to [6] for an extended introduction to mean field games systems. In all the above presentation, the state space could be differently chosen, together with possibly boundary effects (reflection or absorption effects at the boundary, for instance) but we will assume here the simplest, yet instructive case of periodic setting.…”

Section: Introductionmentioning

confidence: 99%

Long time behavior and turnpike solutions in mildly non-monotone mean field games

Cirant

Porretta

2021

ESAIM: COCV

Self Cite

View full text Add to dashboard Cite

We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

show abstract

“…Of course, we have γ ≤ γ 0 given by Proposition 3.7 (indeed, a unique stationary state is used here). This value γ depends on F, H through the constant L in (11) and through assumptions ( 6), (5), in the sense that it depends on the constants c K , ℓ K , α K for a value of K only depending on L, κ. No special effort is devoted, in the proof below, to catch a refined estimate of γ; as it will be clear, several arguments will need γ to be smaller than generic constants appearing in the global (in time) estimates of the solutions.…”

Section: The Exponential Turnpike Estimatementioning

confidence: 99%

“…Thus m satisfies the second equation as the law of the optimal process, where α corresponds to the optimal feedback strategy. We refer to [5] for an extended introduction to mean field games systems. In all the above presentation, the state space could be differently chosen, together with possibly boundary effects (reflection or absorption effects at the boundary, for instance) but we will assume here the simplest, yet instructive case of periodic setting.…”

Section: Introductionmentioning

confidence: 99%

Long time behaviour and turnpike solutions in mildly non-monotone mean field games

Cirant¹,

Porretta²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider mean field game systems in time-horizon (0, T ), where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth.Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0, T ), (ii) the convergence of the system in (0, T ) towards the system in (0, ∞), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution.This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

show abstract

An Introduction to Mean Field Game Theory

Cited by 54 publications

References 142 publications

Mobility decisions, economic dynamics and epidemic

Mobility decisions, economic dynamics and epidemic

Long time behavior and turnpike solutions in mildly non-monotone mean field games

Long time behaviour and turnpike solutions in mildly non-monotone mean field games

Contact Info

Product

Resources

About