scholarly journals Continuous-Time Mean Field Games with Finite State Space and Common Noise

2021 ◽  
Author(s):  
Christoph Belak ◽  
Daniel Hoffmann ◽  
Frank T. Seifried
Keyword(s):  
Continuous Time ◽  
Mean Field ◽  
Mean Field Games ◽  
Mathematical Framework ◽  
Equilibrium System ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
State Process ◽  
Common Noise ◽  
Appl Math

AbstractWe formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise, including a rigorous probabilistic construction of the state process and existence and uniqueness results for the resulting equilibrium system. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (Appl Math Optim 68(1): 99–143, 2013) and Cecchin and Fischer (Appl Math Optim 81(2):253–300, 2020).

scholarly journals Comparing the best-reply strategy and mean-field games: The stationary case

2020 ◽  
pp. 1-32
Author(s):  
MATT BARKER ◽  
PIERRE DEGOND ◽  
MARIE-THERESE WOLFRAM
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Mean Field ◽  
Mean Field Games ◽  
Numerical Comparison ◽  
Stationary Case ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Interacting Agents

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

scholarly journals Stochastic Graphon Games: I. The Static Case

2021 ◽  
Author(s):  
René Carmona ◽  
Daniel B. Cooney ◽  
Christy V. Graves ◽  
Mathieu Laurière
Keyword(s):  
Nash Equilibria ◽  
Mean Field ◽  
Equilibrium Strategy ◽  
Mean Field Games ◽  
Static Case ◽  
Network Games ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Network Game ◽  
Approximate Nash Equilibria

We consider static finite-player network games and their continuum analogs graphon games. Existence and uniqueness results are provided as well as convergence of the finite-player network game optimal strategy profiles to their analogs for the graphon games. We also show that equilibrium strategy profiles of a graphon game provide approximate Nash equilibria for the finite-player games. Connections with mean field games are discussed. A motivating application of Cournot competition is presented, and explicit computation of its Nash equilibrium is provided.

scholarly journals On mean field games with common noise and McKean-Vlasov SPDEs

2019 ◽  
Vol 37 (4) ◽  
pp. 522-549 ◽  
Author(s):  
Vassili N. Kolokoltsov ◽  
Marianna Troeva
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Common Noise

Mean Field Game System with a Common Noise

2019 ◽  
pp. 85-127
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Existence And Uniqueness ◽  
Mean Field ◽  
Mean Field Games ◽  
Random Functions ◽  
Common Noise ◽  
The Common ◽  
The Mean ◽  
Random Flow ◽  
Uniqueness Of A Solution ◽  
Classical Strategy

This chapter talks about the unique solvability of the mean field games (MFGs) system with common noise. In terms of a game with a finite number of players, the common noise describes some noise that affects all the players in the same way, so that the dynamics of one given particle reads a certain master equation. It explains the use of the standard convention from the theory of stochastic processes that consists in indicating the time parameter as an index in random functions. Using a continuation like argument instead of the classical strategy based on the Schauder fixed-point theorem, this chapter investigates the existence and uniqueness of a solution. It discusses the effect of the common noise in randomizing the MFG equilibria so that it becomes a random flow of measures.

scholarly journals Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Mean Field ◽  
Partial Differential ◽  
Existence And Uniqueness Results ◽  
Doubly Stochastic ◽  
Uniqueness Results

Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

scholarly journals Asymptotic analysis of mean field games with small common noise

2018 ◽  
Vol 106 (3-4) ◽  
pp. 205-232
Author(s):  
Saran Ahuja ◽  
Weiluo Ren ◽  
Tzu-Wei Yang
Keyword(s):  
Asymptotic Analysis ◽  
Mean Field ◽  
Mean Field Games ◽  
Common Noise
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