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 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 Markov--Nash Equilibria in Mean-Field Games with Discounted Cost

2018 ◽  
Vol 56 (6) ◽  
pp. 4256-4287 ◽  
Author(s):  
Naci Saldi ◽  
Tamer Başar ◽  
Maxim Raginsky
Keyword(s):  
Nash Equilibria ◽  
Mean Field ◽  
Mean Field Games ◽  
Discounted Cost

scholarly journals Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

2013 ◽  
Vol 24 (1) ◽  
pp. 93-115 ◽  
Author(s):  
Pierre Degond ◽  
Jian-Guo Liu ◽  
Christian Ringhofer
Keyword(s):  
Large Scale ◽  
Nash Equilibria ◽  
Mean Field ◽  
Mean Field Games

scholarly journals Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

2021 ◽  
Author(s):  
Matteo Basei ◽  
Haoyang Cao ◽  
Xin Guo
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Mean Field Games ◽  
Model Parameters ◽  
Management Problem ◽  
Player Game ◽  
The Impact ◽  
Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to infinity, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

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