# Remarks on Nash equilibria in mean field game models with a major player

@article{Cardaliaguet2018RemarksON, title={Remarks on Nash equilibria in mean field game models with a major player}, author={Pierre Cardaliaguet and Marco Cirant and Alessio Porretta}, journal={arXiv: Optimization and Control}, year={2018} }

For a mean field game model with a major and infinite minor players, we characterize a notion of Nash equilibrium via a system of so-called master equations, namely a system of nonlinear transport equations in the space of measures. Then, for games with a finite number N of minor players and a major player, we prove that the solution of the corresponding Nash system converges to the solution of the system of master equations as N tends to infinity.

#### 18 Citations

Linear-quadratic mean field games with a major player: Nash certainty equivalence versus master equations

- Computer Science, Mathematics
- Commun. Inf. Syst.
- 2021

Mean field games with a major player were introduced in (Huang, 2010) within a linear-quadratic (LQ) modeling framework. Due to the rich structure of major-minor player models, the past ten years… Expand

Linear quadratic mean field games with a major player: The multi-scale approach

- Mathematics, Computer Science
- Autom.
- 2020

This paper considers linear quadratic mean field games with a major player and analyzes an asymptotic solvability problem and shows that the two decentralized strategies can be interpreted as the best responses of a major players and a representative minor player embedded in an infinite population. Expand

Strategic advantages in mean field games with a major player

- Computer Science, Mathematics
- 2020

It is shown how to model mean field games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. Expand

Closed-loop convergence for mean field games with common noise

- Mathematics
- 2021

This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points,… Expand

On the convergence of closed-loop Nash equilibria to the mean field game limit

- Mathematics
- 2018

This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of $n$-player stochastic differential games converge to the mean field game as… Expand

Splitting methods and short time existence for the master equations in mean field games

- Mathematics
- 2020

We develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with… Expand

Convex Analysis for LQG Systems with Applications to Major Minor LQG Mean Field Game Systems

- Mathematics, Computer Science
- Syst. Control. Lett.
- 2020

A convex analysis approach for solving LQG optimal control problems and applying it to major-minor (MM) and mean-field game (MFG) systems provides a tool for dealing with complex and non-standard systems. Expand

Finite population games of optimal execution

- Economics, Mathematics
- 2020

We investigate finite population games of optimal execution, taking place at a market with friction. The models over which we develop our results are akin to the standard Almgren-Chriss model with… Expand

Linear quadraticmean field gameswith amajor player : Themulti-scale approach

- 2019

This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming… Expand

$\epsilon$-Nash Equilibria for Major–Minor LQG Mean Field Games With Partial Observations of All Agents

- Mathematics, Computer Science
- IEEE Transactions on Automatic Control
- 2021

For the general case of PO MM LQG MFG systems, the existence of $\epsilon$-Nash equilibria, together with the individual agents’ control laws yielding the equilibrio, are established via the separation principle. Expand

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