scholarly journals A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

2021 ◽  
Author(s):  
Daniel Lacker ◽  
Agathe Soret
Keyword(s):  
Large Population ◽  
Mean Field ◽  
Laplacian Matrix ◽  
Linear Quadratic ◽  
Large Graphs ◽  
Mean Field Game ◽  
Dense Graph ◽  
Correlation Decay ◽  
The Mean

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

scholarly journals Convergence to the mean field game limit: A case study

2020 ◽  
Vol 30 (1) ◽  
pp. 259-286 ◽  
Author(s):  
Marcel Nutz ◽  
Jaime San Martin ◽  
Xiaowei Tan
Keyword(s):  
Mean Field ◽  
Mean Field Game ◽  
The Mean

scholarly journals Backward-forward linear-quadratic mean-field Stackelberg games

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kehan Si ◽  
Zhen Wu
Keyword(s):  
Large Population ◽  
Consistency Condition ◽  
Market Price ◽  
Mean Field ◽  
Mapping Method ◽  
Open Loop ◽  
Stackelberg Games ◽  
Linear Quadratic ◽  
Population System ◽  
The Cost

AbstractThis paper studies a controlled backward-forward linear-quadratic-Gaussian (LQG) large population system in Stackelberg games. The leader agent is of backward state and follower agents are of forward state. The leader agent is dominating as its state enters those of follower agents. On the other hand, the state-average of all follower agents affects the cost functional of the leader agent. In reality, the leader and the followers may represent two typical types of participants involved in market price formation: the supplier and producers. This differs from standard MFG literature and is mainly due to the Stackelberg structure here. By variational analysis, the consistency condition system can be represented by some fully-coupled backward-forward stochastic differential equations (BFSDEs) with high dimensional block structure in an open-loop sense. Next, we discuss the well-posedness of such a BFSDE system by virtue of the contraction mapping method. Consequently, we obtain the decentralized strategies for the leader and follower agents which are proved to satisfy the ε-Nash equilibrium property.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

New data on the biology and habitat preferences of the oligochaete species Ripistes parasita (Annelida: Clitellata: Naididae): a case study in a temporary woodland pond

Biologia ◽  
2015 ◽  
Vol 70 (5) ◽  
Author(s):  
Mariola Krodkiewska ◽  
Aneta Spyra
Keyword(s):  
Stepwise Regression ◽  
Habitat Preferences ◽  
Large Population ◽  
Industrial Area ◽  
Early Spring ◽  
Nitrogen And Phosphorus ◽  
Life History Pattern ◽  
The Mean ◽  
Woodland Ponds

AbstractStudies carried out in woodland ponds located in an industrial area of southern Poland revealed the occurrence of a large population of Ripistes parasita (Schmidt, 1847) in one of them. This is a naidid species that is not usually abundant in oligochaete communities. Its ecology and biology is poorly known and thus the aim of this study was to characterise the environmental conditions influencing the occurrence of R. parasita and to assess its population dynamics and life history pattern. R. parasita occurred in a pond with soft water and a low level of mineralisation, a pH ranging from 6.1 to 7.0, and a high content of nitrogen and phosphorus. During our investigation, considerable seasonal changes in the occurrence and population density were detected. Specimens of this species inhabited alder leaf deposits in winter and early spring while in summer and autumn they occupied in large number floating Nuphar lutea leaves. A stepwise regression analysis showed a relationship between the temperature and dissolved oxygen content in the water and the density of R. parasita. The R. parasita reproduced asexually by paratomy (between May and November). The mean doubling time (days) for the population was 22.4. Only a few individuals (less than 1% of the population) matured in September and October.

scholarly journals Schrödinger approach to Mean Field Games with negative coordination

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Time Limit ◽  
External Potential ◽  
Mean Field Games ◽  
Relative Importance ◽  
Mean Field Game ◽  
Non Linear ◽  
The Mean ◽  
The Way

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schrödinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential vary, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.

scholarly journals Selection of equilibria in a linear quadratic mean-field game

2020 ◽  
Vol 130 (2) ◽  
pp. 1000-1040 ◽  
Author(s):  
François Delarue ◽  
Rinel Foguen Tchuendom
Keyword(s):  
Mean Field ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
Mean Field Game ◽  
Selection Of

scholarly journals Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1669
Author(s):  
Jun Moon ◽  
Wonhee Kim
Keyword(s):  
Nash Equilibrium ◽  
Mean Field ◽  
Jump Diffusion ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
Field Type ◽  
The Mean ◽  
Zero Sum ◽  
Objective Functional

We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

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